the search for absolute infinity

[phoenixthoth]

TUZFC
the general idea is to find a way to axiomatize a universal set into existence in a way that doesn't contradict other axioms.

there are potential ways this might be done, including
1. changing the subsets axiom
2. using ternary logic and changing all axioms


1 would go something like this: there is a set with the usual subset properties UNLESS the existence of that subset leads to a contradiction.

2 would be to use 3 valued or ternary logic. see the two articles here:
http://plato.stanford.edu/entries/logic-fuzzy/
http://plato.stanford.edu/entries/logic-manyvalued/

there isn't a unique way to do fuzzy logic, but let's at least assume that ternary logic generalizes binary logic in the following way:
\begin{array}{cccccccc}
A & B & \symbol{126}A & A\vee B & A\wedge B & A\rightarrow B &
A\leftrightarrow B & \left( A\wedge \left( A\rightarrow B\right) \right)
\rightarrow B \\
T & T & F & T & T & T & T & T \\
T & M & F & T & M & M & M & M \\
T & F & F & T & F & F & F & T \\
M & T & M & T & M & T & M & T \\
M & M & M & M & M & M & M & M \\
M & F & M & M & F & M & M & M \\
F & T & T & T & F & T & F & T \\
F & M & T & M & F & T & M & T \\
F & F & T & F & F & T & T & T
\end{array}

the main observation is that russell's paradox is based on a tautology that isn't a tautology in ternary logic. also note that the standard modus ponens above also fails to be a tautology. however, one may be able to resuce this fact by eliminating ternary logic from all axioms except the subsets axiom in the following way:
in non SS (subsets) axioms, if there is a well formed formula W, and V() is an operator that sends a wff to its truth value, then by replacing appearances of W in the axiom by V(W)=T, we get similar results as the axiom "intends" while still allowing V(W) to be occasionally M. for example, while A<->B if A and B are either both true or both false, V(A<->B)=M if either V(A)=M or V(B)=M. by replacing A<->B with V(A<->B)=T, we get the usual results.

in the case of SS, we can replace it by this:
SS2: \exists x\forall yV\left( \left( y\in x\leftrightarrow y\in a\wedge A\left( y\right) \right) \right) \neq F.

this would not contradict the following axiom:
US: \exists x\forall yV\left( y\in x\right) =T
at least by russell's paradox. there may be other ways US contradicts TZFC, ternary-ZFC.

a list of axioms. in TUZFC, versions 2 would be more appropriate:
1. axiom of extensionality:

\forall x\left( x\in a\leftrightarrow x\in b\right) \rightarrow a=b

axiom of extensionality version 2:

V\left( \forall x\left( V\left( x\in a\leftrightarrow x\in b\right)
=T\right) \rightarrow a=b\right) =T



2. axiom of the unordered pair:

\exists x\forall y\left( y\in x\leftrightarrow y=a\vee y=b\right)

axiom of the unordered pair version 2:

V\left( \exists x\forall y\left( V\left( y\in x\leftrightarrow y=a\vee
y=b\right) =T\right) \right) =T

3. axiom of subsets:

\exists x\forall y\left( y\in x\leftrightarrow y\in a\wedge A\left(
y\right) \right)

axiom of subsets version 2:

V\left( \exists x\forall yV\left( y\in x\leftrightarrow y\in a\wedge
A\left( y\right) \right) =T\right) =T

axiom of subsets version 3:

V\left( \exists x\forall yV\left( \left( y\in x\leftrightarrow y\in a\wedge
A\left( y\right) \right) \right) \neq F\right) =T



4. axiom of the sum set:

\exists x\forall y\left( y\in x\leftrightarrow \exists z\in a\left( y\in
z\right) \right)

axiom of the sum set version 2:

V\left( \exists x\forall y\left( V\left( y\in x\leftrightarrow
\exists z\in a\left( y\in z\right) \right) =T\right) \right) =T



5. axiom of the power set:

\forall x\exists y\left( \forall z\left( z\in y\leftrightarrow z\subset
x\right) \right)

axiom of the power set version 2:

V\left( \forall x\exists y\left( \forall zV\left( z\in
y\leftrightarrow z\subset x\right) =T\right) \right) =T



6. axiom of the empty set:

\exists x\forall y\left( y\notin x\right)

axiom of the empty set version 2:

V\left( \exists x\forall yV\left( y\in x\right) =F\right) =T



7. axiom of infinity:

\exists x\left( \O \in x\wedge \forall y\in x\left( y^{\prime }\in x\right)
\right)

axiom of infinity version 2:

V\left( \exists x\left( V\left( \O \in x\wedge \forall y\in
x\left( y^{\prime }\in x\right) \right) =T\right) \right) =T



8. axiom of the universal set

V\left( \exists x\forall yV\left( y\in x\right) =T\right) =T



9. axiom of replacement:

\exists x\forall y\in a\left( \exists zA\left( y,z\right) \rightarrow
\exists z\in xA\left( y,z\right) \right)

axiom of replacement version 2:

V\left( \exists x\forall y\in a\left( V\left( \exists zA\left(
y,z\right) \rightarrow \exists z\in xA\left( y,z\right) \right) =T\right)
\right) =T



10. axiom of foundation/regularity:

\exists xA\left( x\right) \rightarrow \exists x\left( A\left( x\right)
\wedge \forall y\in x\left( !A\left( y\right) \right) \right)

axiom of foundation/regularity version 2:

V\left( \exists xA\left( x\right) \rightarrow \exists x\left(
V\left( A\left( x\right) \wedge \forall y\in x\left( !A\left( y\right)
\right) \right) =T\right) \right) =T



11. axiom of choice (typo?):

\forall x\in a\exists A\left( x,y\right) \rightarrow \exists y\forall x\in
aA\left( x,y\left( x\right) \right) .

axiom of choice version 2:

V\left( \forall x\in a\exists A\left( x,y\right) \rightarrow
\exists y\forall x\in aA\left( x,y\left( x\right) \right) \right) =T

[Organic]

Here is another possability:

General Information Framework (GIF) set theory


Set (which notated by ‘{‘ and ‘}’) is an object that used as General Information Framework.

Set's property depends on its information’s type.

There are 2 basic types of information that can be examined through GIF.

1.a) Empty set ={}
2.a) Non-empty set

(2.a) has 3 non-empty set’s types:

1.b) Finitely many objects ( {a,b} ).
2.b) Infinitely many objects ( {a,b,…} ).
3.b) Infinite object ( {__} ).

(3.b) Is the opposite of the Empty set, therefore {__}=Full set.

GIF has two limits:

The lowest limit is {}(=Empty set).

The highest limit is {__}(=Full set).

Both limits are unreachable by (1.b) or (2.b) non-empty set’s types.

Or in another words:

Any information system exists in the open interval of ({},{__}).


Infinitely many objects ( {a,b,…} ) cannot be completed, therefore words like ‘all’ or ‘complete’ cannot be used with sets that have infinitely many objects.

{} or {__} are actual infinity.

{a,b,...} is potential infinity.

An example:

http://www.geocities.com/complementarytheory/LIM.pdf


Question:

So what can we do with this theory that we can't do with standard set theory?

A non-formal answer (yet):

Please look at this two articles:

http://www.geocities.com/complementarytheory/ET.pdf

http://www.geocities.com/complementarytheory/CATheory.pdf


At this stage you have to look at them as non-formal overviews, but with a little help from my friends, they are going to be addressed in a rigorous formal way.

All the energy that was used to research the transfinite universes, is going to be used to research the information concept itself, including researches that explore our own cognition's abilities to create and develop the Math language itself.

By GIF set theory our models does not have to be quantified before we can deal with them, because GIF set theory has the ability to deal with any information structure in a direct way, which keeps its dynamic natural complexity during the research.

Concepts like symmetry-degree, Information's clarity-degree, uncertainty, redundancy and complementarity, are some of the fundamentals of this theory.


I am going to open a new thread for this.


Organic

[phoenixthoth]

the hope is that since the subsets axiom is the only one structurally modified that will mean that we can still apply modus ponens as if it were a tautology; care will have to be taken with the subsets axiom. i will investigate this further.

the hope is that U, the set mentioned in the second infinity axiom, has self-awareness structure. for definition, including a link to a relevant article on a proposed TOE, see this web site:

SAS conjectures: http://207.70.190.98/scgi-bin/ikonboard.cgi?;act=ST;f=2;t=196

old search for absolute infinity (scattered remains): http://207.70.190.98/scgi-bin/ikonboard.cgi?;act=ST;f=2;t=129

the idea behind U having self-awareness structure is that if any sets have it, then U must also. while you may object to living in a mathematical structure, let me point out that they can have a dual static and dynamic aspect, not unlike certain dualities in quantum mechanics. for example, f(x)=x^2 is nonconstant while if you view it as a set, {(0,0),(1,1),(2,4),...}, it is static.

[phoenixthoth]

it doesn't make much sense to write V( )=T around each axiom...

seems that the axiom of extensionality would have to be modified for fuzzy subsets like the S in russell's paradox to something like this:
extensionality version 3:
\forall x\left( V\left( x\in a\right) =V\left( x\in b\right) \right) \rightarrow a=b.

without this, it seems difficult to prove that S=S since
V(S&isin;S)=M.

axiom of subsets version 4:
\exists x\forall yV\left( y\in x\right) =V\left( y\in a\wedge A\left( y\right) \right)

[Organic]

Hi phoenixthoth,

Does U defined by 'infinitely many ...'?

[phoenixthoth]

that is one way to describe U.

U is the set mentioned in axiom 8 above.

[Organic]

Another subject: In my opinion, self-awareness is based on the ability to associate between opposite concepts.

When this ability points to itself, but remains open to anything which is not itself, then and only then we can talk about its awareness.

What do you think?

[phoenixthoth]

U could have awareness of all sets but nothing besides that, perhaps.

i think what you're talking about would be regarding proof of awareness. one may be aware but have no frame of reference to be able to prove, even to itself, that it is aware.

i believe sets like U can have a dual dynamic and static nature, depending on your perspective.

[Organic]

Then what is the meaning of the word 'aware' if 'awareness' does not 'aware' to its 'awareness'?

[phoenixthoth]

it is aware but it can't prove it. i think if a math structure has awareness, it can't prove it. however, if you feed it into a computer, maybe it can prove it is aware.

[Russell E. Rierson]

[ abstract representation]--->[semantic mapping]--->[represented system]


An abstract representation is exactly that, "abstract". It is not a space, or time, but is instead a product of consciousness, or a mental construct. Topologically it is equivalent to a "point". The abstract description contains the concrete topology. Likewise, the concrete contains the abstract.

A duality.

A point contains an infinite expanse of space and time?

Could it be, that the "absolute" infinity, is actually a dimensionless point?


Since it is possible for a "computation" to be self aware, there must be platonic forms that are types of self aware algorithms:


The description of any entity inside the real universe can only be with reference to other things in the universe. Space is then relational, and the universe, self referential. For example, if an object has a momentum, that momentum can only be explained with respect to another object within the universe. Space then becomes an aspect of the relationships between things in reality. It becomes analogous to a sentence, and it is absurd to say that a sentence has no words in it. So the grammatical structure of each sentence[space] is defined by the relationships that hold between the words in it. For example, relationships like object-subject or adjective-noun. So there are many different grammatical structures composed of different arrangements of words, and the varied relationships between them.

Language describes the universe, because the universe is isomorphic to a description on some level, and reality can only refer to itself, because, there is nothing outside of ..."total existence" which becomes equivalent to a self referential system, which must be a self aware system. Since descriptions make distinctions, or references to other entities, and distinctions are tautologically logical, [A or ~A], reality is logical, in that its contents can be described by a language. The contents within reality are distinctive entities, individually different from the others, yet consisting of the same foundational substance.

According to Berkeley, perception is consistent due to the fact that a type of mental universal self consistency must apply to the collective whole of individual perceptions. A type of universal being of superior intelligence who creates a world by the power of thought, in which every object becomes, for the percipient, the collected results of many perceptions, or bits of information. That is to say, sensory objects are compositions derived from many perceptual experiences over a period of time, originating from a universal compositional entity, or "BEING". These perceptions are impressed upon each individual mind with order and consistency. Since this universal "Self Awareness" must sustain Creation at all times, everything is always perceived by this self referring, self referential entity, ergo, total reality continues "to exist", even though it may cease to be experienced by any individual self aware perceiver.

[phoenixthoth]

perhaps the difficulty in proving that U exists in a noncontradictory way has to do with proving that awareness exists anywhere.

[Russell E. Rierson]

There must be Goedelian limitations on a consistent and "complete" definition of self awareness? [*(]

If "To Exist" means "TO BE", it is a first principle which is an absolute, or synonymously, an abstract beginning; that does not presuppose anything, it is not be mediated by anything which means that it is the ontological substrate for the whole "she-bang". It does not have a basis; rather it is to be itself the basis of the entirety of existence itself. The Total Existensial Entity is called "TCE" [Total Compositional Existence].

It cannot possess any determination relatively to anything outside itself, so too it cannot contain within itself any determination, any content, for any such would be a distinguishing and an interrelationship of distinct moments, and consequently a mediation. Ergo the beginning is "Pure Being."


While the concept of Being is an empty concept, whose content is nothing, it becomes clear that the concept of Nothing, has equivalently, the same content as the concept of Being, but which seems to stand in diametric opposition to it. Nothingness is . . . of the same determination, or rather, an abscence of determination, and thus altogether the same as, the pure essence of Being. On the other hand, paradoxically, Being and Nothing are not the same. So there is the difference of Being and Nothing passing into identity, and the identity passing into difference. Ergo, their truth is, this movement of the immediate vanishing of one in the other: becoming, a movement in which both are distinguished, but by a difference which has equally immediately resolved itself. A self resolving paradox. The new concept is generated by the "sublation" of the first two. This process of generating a third concept as expressing the identity and difference of the first two is fundamental to the discipline of logic?

[Organic]

Let us examine this idea:


Question: Whet is silence?


Some possible answers:

1) “I don’t know”.
2) “No sound waves at all”.
3) “3 o’clock in Saturday’s morning”.
4) No answer has been given.

(1) Is an honest answer.
(2) Is a scientific answer.
(3) Is a subjective answer.
(4) Is a direct demonstration of silence.

Now, (1) to (3) answers are on silence.

(4) Is a direct response by demonstration.

There is a preventive ratio between (1) to (3) answers and silence itself,
and it leads us to this paradox:

If you answer then there is no silence.

If you keep silence then you don’t give an answer.


I think that “silence’s paradox” can be translated to “awareness’s paradox”,
which means, no model of awareness is awareness itself.

[phoenixthoth]

the answer is this:



in words, no answer.

this has to do with proof of awareness vs having awareness without proof of it. we can cavilierly define what it means to be infinite and we do it in such a way that an infinite set was axiomatized into existence by one of the infinity axioms (7 or 8 above, i think); so why can't we make up axioms of a so-called SAS? define under 20 properties and then see if axiomatizing such a set into existence like the infinity axiom contradicts anything.

[Organic]

phoenixthoth,

Please let me ask you once more:

You say that U is (something) of all sets.

By connecting the word 'all' the the letter 's' (sets), as much as i can see, the result cannot be but 'infinitely many ...'.

Am i right?

[phoenixthoth]

infinity is a defined concept. what do you mean by infinitely many? well, some people think it means that for no n in N is x in bijection with n. (n={0,...,n-1}). by the other infinity axiom, that x has infinitely many objects. the defining property of U is that it is true that for all sets x, x &isin; U. now, one has to prove that there are infinitely many sets by the axioms. the statement reduces to this:
1. if there are infinitely many sets, whatever infinitely many means, then U has infinitely many objects.
1. if there are not infinitely many sets, whatever infinitely many means, then U does not have infinitely many objects.

[Hurkyl]

I said I'd look at this, and I will, once I'm not quite as busy.

I just didn't want you to think I'd forgotten about this.

[phoenixthoth]

i'm working on an update so it might be best if you wait for at least 12 hours, if not more.

hint: i found one possible resuce of modus ponens along with a way to crispify (fry) some of the fuzzy logic out of the axioms i don't want it in. this involves adding some connectives in a way similar to "On a set theory with uncertain membership relations", located here:
http://kurt.scitec.kobe-u.ac.jp/~kikuchi/papers/his2003.pdf

this apparently has been done without ternary logic in another way by quine called "new foundations." i don't believe he uses ternary logic at the level of axioms.

http://math.boisestate.edu/~holmes/holmes/nf.html

for those of you looking for an online intro to set theory, along with forcing techniques which can evidently do things like prove consistency of axioms, check out this article:
http://staff.science.uva.nl/~vervoort/AST/ast.pdf

(if i want to get a phd, are there any schools around who do dynamics and set theory/logic? i'm not sure which of the two areas i'd want to go into and i've always liked both...)

[phoenixthoth]

notation elements and subsets:

\in _{V}is a relation on sets such that a\in _{V\left( a\in b\right) }b:=V\left( a\in b\right) . so, for example, if V\left( a\in b\right) =M, then we can just write a\in _{M}b. if V\left( a\in b\right) =T or
V\left( a\in b\right) =F, we'll just use the usual notation a\in b or a\notin b instead of a\in _{T}b and a\in _{F}b.

connectives:

A\rightarrow _{T}B will mean that V\left( A\rightarrow B\right) =T and, similarly, A\leftrightarrow _{T}B means V\left( A\leftrightarrow B\right) =T. note that if A\leftrightarrow _{T}B, then either V\left( A\right) =V\left( B\right) =T or V\left( A\right) =V\left( B\right) =F.

another logical connective is \leftrightarrow _{=} which is defined this way:\ A\leftrightarrow _{=}B iff V\left( A\right) =V\left( B\right) . A\leftrightarrow _{=}B if and only if A and B have the same truth value.

a third new logical connective is \leftrightarrow _{\neq F}which means is V\left( A\leftrightarrow _{\neq F}B\right) =T iff V\left( A\leftrightarrow B\right) \neq F, else V\left( A\leftrightarrow _{\neq F}B\right) =F.


second truth table:


\begin{array}{cccccccc}
A & B & A\rightarrow B & A\rightarrow _{T}B & A\leftrightarrow B &
A\leftrightarrow _{T}B & A\leftrightarrow _{=}B & A\leftrightarrow _{\neq F}B
\\
T & T & T & T & T & T & T & T \\
T & M & M & F & M & F & F & T \\
T & F & F & F & F & F & F & F \\
M & T & T & T & M & F & F & T \\
M & M & M & F & M & F & T & T \\
M & F & M & F & M & F & F & T \\
F & T & T & T & F & F & F & F \\
F & M & T & T & M & F & F & T \\
F & F & T & T & T & T & T & T
\end{array}


observations: (here, A and B can have any ternary truth value)

1. \left( A\leftrightarrow _{T}B\right) \rightarrow \left( A\leftrightarrow _{=}B\right) is a tautology.

2. (modified modus ponens) \left( A\wedge \left( A\rightarrow _{T}B\right) \right) \rightarrow B is a tautology.

3. B\leftrightarrow _{T}\symbol{126}B is always false.

4. B\leftrightarrow _{=}\symbol{126}B is true iff V\left( B\right) =M, otherwise it is false.

5. (contradiction 1) \left( A\rightarrow \left( B\leftrightarrow \symbol{126}B \right) \right) \rightarrow \symbol{126}A is not a tautology.

6. (contradiction 2) \left( A\rightarrow \left( B\leftrightarrow _{T} \symbol{126}B\right) \right) \rightarrow \symbol{126}A is not a tautology.

7. (contradiction 3) \left( A\rightarrow \left( B\leftrightarrow _{=} \symbol{126}B\right) \right) \rightarrow \symbol{126}A is not a tautology.

8. (contradiction 4) \left( A\rightarrow \left( B\leftrightarrow _{\neq F} \symbol{126}B\right) \right) \rightarrow \symbol{126}A is not a tautology.

restatement of all nonchoice axioms with this notation:

axiom of extensionality version 2 (in version 2, it is not clear that a fuzzy set equals itself):
\forall x\left( x\in a\leftrightarrow _{T}x\in b\right) \rightarrow a=b

axiom of extensionality version 3:
\forall x\left( x\in a\leftrightarrow _{=}x\in b\right) \rightarrow a=b

2. axiom of the unordered pair version 2:
\exists x\forall y\left( y\in x\leftrightarrow _{T}y=a\vee y=b\right)


3. axiom of subsets version 2:
\exists x\forall y\left( y\in x\leftrightarrow _{T}y\in a\wedge A\left( y\right) \right)

axiom of subsets version 3:
\exists x\forall y\left( y\in x\leftrightarrow _{\neq F}y\in a\wedge A\left( y\right) \right)

axiom of subsets version 4:
\exists x\forall y\left( y\in x\leftrightarrow _{=}y\in a\wedge
A\left( y\right) \right)

4. axiom of the sum set version 2:
\exists x\forall y\left( y\in x\leftrightarrow _{T}\exists z\in
a\left( y\in z\right) \right)

5. axiom of the power set version 2:
\forall x\exists y\forall z\left( z\in y\leftrightarrow
_{T}z\subset x\right)

axiom of the power set version 3:
\forall x\exists y\forall z\left( z\in y\leftrightarrow
_{=}z\subset x\right)

6. axiom of the empty set:
\exists x\forall y\left( y\notin x\right)

7. axiom of infinity:
\exists x\left( \O \in x\wedge \forall y\in x\left( y^{\prime }\in x\right) \right)

axiom of infinity version 2:
\exists x\left( \O \in x\wedge \forall y\in x\left( y^{\prime }\in x\right) \right)

8. axiom of the universal set
\exists x\forall y\left( y\in x\right)

9. axiom of replacement version 2:
\exists x\forall y\in a\left( \exists zA\left( y,z\right)
\rightarrow _{T}\exists z\in xA\left( y,z\right) \right)

10. axiom of foundation/regularity version 2:
\exists xA\left( x\right) \rightarrow _{T}\exists x\left( A\left( x\right) \wedge \forall y\in x\left( !A\left( y\right) \right) \right)

[phoenixthoth]

10 should say \exists xA\left( x\right) \rightarrow _{T}\exists x\left( A\left( x\right) \wedge \forall y\in x\left( !A\left( y\right) \right) \right) but it wouldn't let me edit it.

subsets: (rough draft)
x\subset y means that \forall z\left( z\in x\rightarrow _{T}x\in y\right) while x\subset _{M}y means \forall z\left[ \left[ z\in _{M}x\rightarrow _{T}\left( z\in _{M}y\vee z\in y\right) \right] \wedge \left[ z\in x\rightarrow _{T}x\in y\right] \right] . \ note that \left( \left(
A\rightarrow _{T}B\right) \wedge \left( A\rightarrow _{T}B\right) \right) \leftrightarrow _{T}\left( A\leftrightarrow _{T}B\right) .

[Organic]

And what if U is reality itself, and in this case no model of U is U itself?

[phoenixthoth]

no one is saying the model of U is U. U will be studied with theorems and such but those theorems are not what U is. U's nature will probably never be completely known and only general statements can be made about U.

another note is that V={x|x=x} is, in ZFC, a proper class; ie, it is not a set. V is the class of all sets.

another observation (A and B can have any truth value):

\left( A\rightarrow _{T}B\right) \wedge \left( B\rightarrow _{T}A\right) is tautologically equivalent to A\leftrightarrow _{T}B and they are true only if V\left( A\right) =V\left( B\right) =T, else they are false.

SUBSETS

theorem: x=y iff (binarily) \left( x\subset y\wedge y\subset x\right) .

proof: suppose x=y and let z\in x be given. since x=y, z\in y. hence, z\in x\rightarrow _{T}z\in y for all z and x\subset y. similarly, y\subset x. now suppose \left( x\subset y\wedge y\subset x\right) and let z\in x be given. in order to show that z\in x \leftrightarrow_{T} z\in y, it suffices to prove \left( z\in x\rightarrow _{T}z\in y\right) \wedge \left( z\in y\rightarrow _{T}z\in x\right) .
since x\subset y, we have z\in x\wedge \left( z\in x\rightarrow _{T}z\in y\right) . since A\wedge \left( A \rightarrow _{T} B \right) \rightarrow B is a tautology, we can conclude that z\in y. hence, z\in x\rightarrow
_{T}z\in y. similarly, z\in y\rightarrow _{T}z\in x.

note: when we assume z\in x, in our notation, that is equivalent to assuming that V\left( z\in x\right) =T. if V\left( z\in x\right) =M or V\left( z\in x\right) =F, we write z\in _{M}x and z\notin x.

sacrifices: proofs by (standard) contradiction as well as sets that are only crisp. russell's set S is the prototypical example of a fuzzy set because S\in _{M}S.

gains: so far, the universal set U.

[Organic]

What do you think about us (self awared complex systems) as associators between potential existence (some model) and actual existence (some reality).

[phoenixthoth]

Originally posted by Organic
What do you think about us (self awared complex systems) as associators between potential existence (some model) and actual existence (some reality).

i think the same paradigm exists within us. there is an underlying reality of our Selves (designated with a capital S) and then the "associator of potential existence", our selves (lower case s), which is also named our ego. Reality vs subjectivity, context vs content. for sets, the largest possible set that is the context for all content is U. there are "bigger" contexts, such as the category of all sets and the category of all categories. binary logic is in some sense, an even bigger context in its generality. but then, ternary logic would be a bigger context still. there's a hierchy to existence where we have the model of something vs the reality of that something not unlike the difference between self-awareness and awareness itself or ego vs the higher self/true self.

[Organic]

In my opinion our most important ability is to associate between opposite potential and/or actual things in a way that they do not destroy each other when associated.

What do you think?

[phoenixthoth]

the two should be complementary.

[Organic]

That's why i use Complementary Logic:

http://www.geocities.com/complementarytheory/CompLogic.pdf

http://www.geocities.com/complementarytheory/4BPM.pdf

http://www.geocities.com/complementarytheory/RealModel.pdf

[phoenixthoth]

i haven't read all those articles yet but i will.

new notation: when we don't want to specify the truth value of a\in b, we will write a\in _{?}b (for want of better notation). hence V\left( a\in _{?}b\right) \in \left\{ F,M,T\right\} whereas V\left( a\in b\right) \in \left\{ F,T\right\} , V\left( a\in _{M}b\right) \in \left\{ F,T\right\} and V\left( a\notin b\right) \in \left\{
F,T\right\} .

thus we are dealing with crisp logical formulas when we write axioms such as the extensionality: \forall x\left( x\in a\leftrightarrow x\in b\right) \rightarrow a=b but not with subsets:

axiom of subsets version 5:
\exists x\forall y\left( y\in _{?}x\leftrightarrow _{=}y\in
_{?}a\wedge A\left( y\right) \right) .

i'm wondering now about extensionality as it applies to fuzzy sets. i want two sets to be equal not just if they have the same elements in a crisp sense:
axiom of extensionality version 3:
\forall x\left( x\in _{?}a\leftrightarrow _{=}x\in _{?}b\right) \rightarrow a=b.

[Organic]

By ZF set theory we know that {a,a,a,b,b,b,c,c,c} = {a,b,c}

It means that concepts like redundancy and uncertainy are out of the scpoe of set's concet in its basic form.

When we allow these concepts to be inherent properties of set's concept, then we enrich our abilities to use set's concept, for example:

<-Redundancy->
c c c ^<----Uncertainty
b b b | b b
a a a | a a c a b c
. . . v . . . . . .
| | | | | | | | |
| | | |___|_ | |___| |
| | | | | | |
|___|___|_ |_______| |_______|
| | |


Where:

c c c
b b b
a a a
. . .
| | |
| | | = {a XOR b XOR c, a XOR b XOR c, a XOR b XOR c}
| | |
|___|___|_
|

b b
a a c
. . .
| | |
|___|_ | = {a XOR b, a XOR b, c}
| |
|_______|
|

a b c
. . .
| | |
|___| | = {a, b, c}
| |
|_______|
|

I think that any iprovment in set's concept has to include redundancy and uncertainty as inherent proprties of set's concept.

What do you think?


Orgainc

[Hurkyl]

I think that expanding a set's concept to include redundancy yields some concept different from the set concept. In fact, we have a name for it; a multiset (http://mathworld.wolfram.com/Multiset.html).

[Organic]

Hi Hurkyl,

My point of view leading me to what I call Complementary logic, which is a fading transition between Boolean logic (0 Xor 1) and non-boolean logic (0 And 1), for example:

Number 4 is fading transition between multiplication 1*4 and addition ((((+1)+1)+1)+1) ,and vice versa.

These fading can be represented as:

(1*4) ={1,1,1,1} <------------- Maximum symmetry-degree,
((1*2)+1*2) ={{1,1},1,1} Minimum information's clarity-degree
(((+1)+1)+1*2) ={{{1},1},1,1} (no uniqueness)
((1*2)+(1*2)) ={{1,1},{1,1}}
(((+1)+1)+(1*2)) ={{{1},1},{1,1}}
(((+1)+1)+((+1)+1))={{{1},1},{{1},1}}
((1*3)+1) ={{1,1,1},1}
(((1*2)+1)+1) ={{{1,1},1},1}
((((+1)+1)+1)+1) ={{{{1},1},1},1} <------ Minimum symmetry-degree,
Maximum information's clarity-degree
(uniqueness)
============>>>

Uncertainty
<-Redundancy->^
3 3 3 3 | 3 3 3 3
2 2 2 2 | 2 2 2 2
1 1 1 1 | 1 1 1 1 1 1 1 1 1 1
{0, 0, 0, 0} V {0, 0, 0, 0} {0, 1, 0, 0} {0, 0, 0, 0}
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
| | | | |__|_ | | |__| | | |__|_ |__|_
| | | | | | | | | | | |
| | | | | | | | | | | |
| | | | | | | | | | | |
|__|__|__|_ |_____|__|_ |_____|__|_ |_____|____
| | | |

4 =
2 2 2
1 1 1 1 1 1 1
{0, 1, 0, 0} {0, 1, 0, 1} {0, 0, 0, 3} {0, 0, 2, 3}
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
|__| |__|_ |__| |__| | | | | |__|_ | |
| | | | | | | | | | |
| | | | |__|__|_ | |_____| |
| | | | | | | |
|_____|____ |_____|____ |________| |________|
| | | |


{0, 1, 2, 3}
. . . .
| | | |
|__| | |
| | |
|_____| |
| |
|________|
|


Multiplication can be operated only between objects with structural identity .

Also multiplication is noncommutative, for example:

2*3 = ( (1,1),(1,1),(1,1) ) or ( ((1),1),((1),1),((1),1) )

3*2 = ( (1,1,1),(1,1,1) ) or ( ((1,1),1),((1,1),1) ) or ( (((1),1),1),(((1),1),1) )

More about the above you can find here (the first part of it is your definitions):

http://www.geocities.com/complementarytheory/ET.pdf

More about Complementary logic, you can find here:

http://www.geocities.com/complementarytheory/CompLogic.pdf

http://www.geocities.com/complementarytheory/4BPM.pdf



Organic

[phoenixthoth]

seems like !z&isin;?z doesn't lead to a paradox but the usual russell's paradox is still a paradox.

[master_coda]

Organic,

I think the point that Hurkyl was trying to make was that if you take the concept of a set and add more properties to it, such as redundancy, you aren't really just using a set anymore.

That's perfectly fine, of course. That's why we have objects like ordered sets and multisets.

[Hurkyl]

I hate to encourage a hijacking (iow, promote your theory in your thread)... but it seems like this should also be valid (among other things):

2 * 3 = <<1>, 1> * <1, <1, 1>> = <<<1>, 1>, <<<1>, 1>, <<1>, 1>>>

[Hurkyl]

Phoenix: I was doing a bit of thinking, and I wonder if you shouldn't just study fuzzy sets:

For instance, consider this model:

We assign real numbers to truth values: T = 1, M = 0.5, F = 0. Your sets are fuzzy sets whose range must be {0, 0.5, 1}. In other words, a set is merely a function from a domain into {0, 0.5, 1}.

We can then equip a fuzzy set with an extra gadget we call it's "default value"; any element that doesn't appear in the domain of the fuzzy set gets assigned the default value.

for instance, consider the fuzzy set over {0, 1, 2, 3}:

S = { (0, T), (1, T), (2, F), (3, M) }

That is, in your set land, 0 and 1 are elements of S, 2 is not an element of S, and 3 is a maybe-element of S. Or, S(0) = 1, S(1) = 1, S(2) = 0, S(3) = 0.5

Now, we add the default value "M" to T to get (I'm inventing notation now):

S = {M | (0, T), (1, T), (2, F), (3, M) }

So now all of the above values are the same, but we also have (and this is technically new notation): 4 is a maybe-element of S; that is S(4) = 0.5, and similarly for everything that's not 0, 1, or 2.


And then, via magic, we have a "universal set" produced by taking the empty set (which is a fuzzy set with the empty domain!!!) and equipping it with default value T.


We could replace the range of truth values {0, 0.5, 1} with any domain we like, really... it seems there's no need to resort to ternary logic with this approach, we can accomplish the same sort of thing with crisp sets.

We get consistency relative to ZFC for free this way... I wonder how the axiomization of this theory would work?


Allow me to reformulate it in a slightly different way to simplify things.

A p-set is an ordered pair (S, d) where we have \mathrm{Set}(S) and d \in \{\mathrm{true}, \mathrm{false} \}, and we define p-membership as a \in_p (S, d) \leftrightarrow a \in S \oplus d. (where \oplus is exclusive-or; I didn't want to figure out how to make the usual symbol)


That is, d is a sort of polarity; if d = \mathrm{true}, then the "default" value is true, and S contains the elements that aren't a p-member of (S, d), and if d = \mathrm{false}, then the "default" value is false, and S contains the elements that are a p-member of (S, d)


The universal p-set is then (\varnothing, \mathrm{true}).

[Organic]

I updated my previous post.

Please look at it.

Thank you.

Hurkyl, I think my two last posts are conncted also to the idea of 3 valued logic, but I will open a new thread for it.



Organic

[phoenixthoth]

Originally posted by Hurkyl
Phoenix: I was doing a bit of thinking, and I wonder if you shouldn't just study fuzzy sets:

For instance, consider this model:

We assign real numbers to truth values: T = 1, M = 0.5, F = 0. Your sets are fuzzy sets whose range must be {0, 0.5, 1}. In other words, a set is merely a function from a domain into {0, 0.5, 1}.

We can then equip a fuzzy set with an extra gadget we call it's "default value"; any element that doesn't appear in the domain of the fuzzy set gets assigned the default value.

for instance, consider the fuzzy set over {0, 1, 2, 3}:

S = { (0, T), (1, T), (2, F), (3, M) }

That is, in your set land, 0 and 1 are elements of S, 2 is not an element of S, and 3 is a maybe-element of S. Or, S(0) = 1, S(1) = 1, S(2) = 0, S(3) = 0.5

Now, we add the default value "M" to T to get (I'm inventing notation now):

S = {M | (0, T), (1, T), (2, F), (3, M) }

So now all of the above values are the same, but we also have (and this is technically new notation): 4 is a maybe-element of S; that is S(4) = 0.5, and similarly for everything that's not 0, 1, or 2.


And then, via magic, we have a "universal set" produced by taking the empty set (which is a fuzzy set with the empty domain!!!) and equipping it with default value T.


We could replace the range of truth values {0, 0.5, 1} with any domain we like, really... it seems there's no need to resort to ternary logic with this approach, we can accomplish the same sort of thing with crisp sets.

We get consistency relative to ZFC for free this way... I wonder how the axiomization of this theory would work?


Allow me to reformulate it in a slightly different way to simplify things.

A p-set is an ordered pair (S, d) where we have \mathrm{Set}(S) and d \in \{\mathrm{true}, \mathrm{false} \}, and we define p-membership as a \in_p (S, d) \leftrightarrow a \in S \oplus d. (where \oplus is exclusive-or; I didn't want to figure out how to make the usual symbol)


That is, d is a sort of polarity; if d = \mathrm{true}, then the "default" value is true, and S contains the elements that aren't a p-member of (S, d), and if d = \mathrm{false}, then the "default" value is false, and S contains the elements that are a p-member of (S, d)


The universal p-set is then (\varnothing, \mathrm{true}).

let me see if i got this right. how would we embed a normal set into its p-set? would a set K be mapped into (K,false)?

[Hurkyl]

Right:

\forall K \in \mathrm{Set} \forall a: a \in K \Leftrightarrow a \in_p (K, \mathrm{false})

[phoenixthoth]

something i'm going to think about, too, is whether (Ø,true) violates russell's paradox and whether (K,true) is a set.

[master_coda]

Originally posted by Hurkyl
A p-set is an ordered pair (S, d) where we have \mathrm{Set}(S) and d \in \{\mathrm{true}, \mathrm{false} \}, and we define p-membership as a \in_p (S, d) \leftrightarrow a \in S \oplus d. (where \oplus is exclusive-or; I didn't want to figure out how to make the usual symbol)


That is, d is a sort of polarity; if d = \mathrm{true}, then the "default" value is true, and S contains the elements that aren't a p-member of (S, d), and if d = \mathrm{false}, then the "default" value is false, and S contains the elements that are a p-member of (S, d)


The universal p-set is then (\varnothing, \mathrm{true}).

But how many properties of "standard" ZFC sets would also apply to p-sets? Any ZFC set A has an "equivalent" p-set (A,\mathrm{false}) but not every p-set has an equivalent ZFC set.

So even though you can create a universal set, we still need to figure out what we can do with it.


PS. I think the LaTeX symbol you were looking for is \veebar. I like it better than \oplus.

[Hurkyl]

Well, you can do union and intersection. I dunno if you can actually do anything interesting with them, though; I'm not entirely sure what Phoenixthoth's goal is in his pursuit.

[phoenixthoth]

my ultimate goal is to look at self-aware structures as per that article by max tegmark. if we track all possible configurations of the universe over time we get some kind of set, possibly a manifold. since we are in this set, and we are self aware, it makes sense that some sets and/or logical structures have SA-structure. if U is the set of all possible configurations of the universe, i was wondering if U would have SA or if it just contains things that do. if it does, would it, in a sense, be omniscient of all its contents? i'm not even sure how would would define SAS's. perhaps they'll be left undefined, not unlike the word set, and axioms will be given that govern them.

for instance... if x is a set with SAS and x can be bijected to y, then y also has SAS.

however, if SAS has to do with more than number of elements but something more complicated, then isomorphic (wrt some operation(s)) would be more appropriate. the thing is that static sets can't prove to us they are self-aware or even express that fact. a changing three dimensional manifold becomes a static 4D manifold. what's changing is our awareness of it which some to be a concept not bound by time. well, this is way off course now...

would there be different kinds of SA-structure? and a way to measure SA-structure?

i have a thread with random conjectures here:
http://207.70.190.98/scgi-bin/ikonboard.cgi?;act=ST;f=2;t=196

back to a universal set... my question was the same as master_coda's, which was whether (K,true) always corresponds to a set.

seems like (Ø,true) is just like {x|x=x} which is a proper class.

[Hurkyl]

Right. The p-membership of (S, \mathrm{true}) is always a proper class for any set S.

The major drawback of p-sets, as I've defined them, seems to be that it cannot be the case that both the p-membership and p-nonmembership are both proper classes.

However, my gut says it's at least subtheory of your goal, and would be worth studying.

[phoenixthoth]

second truth table (REPEAT):


\begin{array}{cccccccc}
A & B & A\rightarrow B & A\rightarrow _{T}B & A\leftrightarrow B &
A\leftrightarrow _{T}B & A\leftrightarrow _{=}B & A\leftrightarrow _{\neq F}B
\\
T & T & T & T & T & T & T & T \\
T & M & M & F & M & F & F & T \\
T & F & F & F & F & F & F & F \\
M & T & T & T & M & F & F & T \\
M & M & M & F & M & F & T & T \\
M & F & M & F & M & F & F & T \\
F & T & T & T & F & F & F & F \\
F & M & T & T & M & F & F & T \\
F & F & T & T & T & T & T & T
\end{array}

define a new conditional:

\begin{array}{ccccc}
A & B & A\rightarrow _{+}B & A\leftrightarrow _{+}B & A\leftrightarrow _{+}
\symbol{126}A \\
T & T & T & T & F \\
T & M & T & T & F \\
T & F & F & F & F \\
M & T & T & T & T \\
M & M & T & T & T \\
M & F & T & T & T \\
F & T & T & F & F \\
F & M & T & T & F \\
F & F & T & T & F
\end{array}


the axioms of subsets would read this:

axiom of subsets version 6:

\exists x\forall y\left( y\in _{?}x\leftrightarrow _{+}y\in
_{?}a\wedge A\left( y\right) \right) .



if there is a universal set U, and we let a=U and A\left( y\right) =y\notin y, note that y\notin y is a binary statement and y\in _{?}x is a ternary statement. here's how russell's paradox, which was a problem for
those other biconditionals, would work:


\begin{array}{ccc}
S\in _{?}S & S\notin S & S\in _{?}S\leftrightarrow _{+}S\notin S \\
T & F & F \\
M & F & T \\
F & T & F
\end{array}


axiom of foundation states this:

\forall a\left[ a\neq \emptyset \rightarrow \exists x\in a\left( x\cap a=\emptyset \right) \right] . if you consider \left\{ a\right\} , you can show that for no ZFC sets is the following true: a\in a. this axiom is inconsistent with the universal axiom because U\in U. (note that
S\in S is false because S\in _{M}S.) perhaps a way to restate the universal set axiom is this:

\exists !x\left( x\in x\right) . one then has to modify the foundation axiom to this:

\forall a\left[ a\neq \emptyset \wedge a\neq \left\{ U\right\} \rightarrow \exists x\in a\left( x\cap a=\emptyset \right) \right] or some such. instead of {U}, it would have to be any set containing U as an element, i think.

[phoenixthoth]

if V\left( y\in _{?}a\wedge A\left( y\right) \right) =M then that gives no information on xRy where R is \in or \in _{M} or \notin .

i'm finding it difficult to find any fuzzy sets besides the set S in russell's paradox which may be a good thing.

[phoenixthoth]

all feedback appreciated. there should be a file attached which is a zipped pdf.

[Hurkyl]

I haven't tried to sit down and digest all of it yet... I've recently decided to take up the task if trying to fully digest the axiom of foundation; i.e. how to prove:

\forall S \forall x \in S : S \notin x
\forall S \forall x \in S \forall y \in x: S \notin y

et cetera (including transfinite membership chains)


As one of the PDFs you linked mentions, you don't necessarily need to have the axiom of foundation in your set theory anyways.


I thought a bit more about ternary logic, and I think (though I may be wrong) that it should be approached in this way:

Use binary logic simulating ternary logic; e.g. one might have the binary function Q(P) which simulates the ternary fact V(P) = M.

(Incidentally, it seems that you are using this approach, whether consciously or not)

If not that, I'm beginning to think that ternary deduction might be richer than ternary truth tables, and that making truth tables isn't the right way to approach making deductions.

[Hurkyl]

e.g. I can prove the first of these:

Assume x \in S \wedge S \in x. Consider T := \{S, x\}

By the axiom of foundation, one of S \cap T and x \cap T must be the null set.

However, S \cap T = \{x\}, and x \cap T = \{S\}, which is a contradiction.

[phoenixthoth]

Originally posted by Hurkyl
e.g. I can prove the first of these:

Assume x \in S \wedge S \in x. Consider T := \{S, x\}

By the axiom of foundation, one of S \cap T and x \cap T must be the null set.

However, S \cap T = \{x\}, and x \cap T = \{S\}, which is a contradiction.

F2, foundation 2, is
\forall a\left( \left( a\neq \emptyset \wedge U\notin a\right) \rightarrow \exists x\in a\left( x\cap a=\emptyset \right) \right)

this would mean that your T (a in the axiom) wouldn't be an applicable use of F2 (edit: if S or x was U) though it would be of the usual axiom of foundation. otherwise if U is not one of them, you can't have a chain of memberships that start and end with the same set i don't think.

but if this proves to be a problem, i'll have to see what it would unravel to loose foundation. i'm only partway through an article on set theory and he said foundation will have bigger implications but for now, it is used to show that no set is an element of itself (which is why i needed to change it to allow for U in a way that if U weren't around it would be the same axiom).

the main boon in this system is that most things remain crisp with the use of the new membership symbols so i can still use deduction and contradiction. there was another article where someone almost did what i did, by computer scientists no less, in which they invented two new membership symbols. i emailed one of them my ideas but never got a reply. they were really close to turning their guns on russell's paradox but i think that's been considered a dead horse for like 150 years now. anyway, with the crispiness of things mostly intact, except with the subsets axiom, i'm hoping that everything will turn out all right in the end.

in fact, i can't find many fuzzy sets in this theory. for example, i'm not sure i wouldn't have to modify the pairing axiom in order to get a set like x={(Ø,M), ({Ø},T)} where that means that the truth value of Ø&isin;x is M and the truth value of {Ø}&isin;x is T, using the old membership symbols here. in the new system, x&isin;y, x&isin;My, and x\notin y are mutually exlusive.

i'm glad i didn't call this thread naked ladies! thank you very much for all your feedback.

edit: if you would, please expound on what you mean by ternary deduction without truth tables. i think that would be a great boon to tuzfc (ternary universal zfc) though i only have an incling for what that would entail (pun intended). i don't know if i'm jumping the gun here, but, to be conservative, let's say that in 3 years there is a publishable paper in this; well, it could be a joint paper with tuzfc and ternary deduction combined somehow.

if this works out, i would like to say the cardinal number of U is capital omega or capital alpha, alternatively. also, what is the standard letter for the third truth value? is it M? one could be kind of ridiculous and call this m-theory but rather than be ambiguous about what this M stands for, i'll say it stands for 'maybe' and the eastern concept called mu which is actually why i called it M. in truth, this eastern approach is what proved to be a major inspiration to the resolution of certain paradoxes not unlike how it 'resolves' certain koans.

[Organic]

What are the results of http://mathworld.wolfram.com/Indeterminate.html
when using your theory.

[phoenixthoth]

i have more for you to digest once the inital part has cleared.

an example:

if x is a proper subset of U, then there is no map from x onto U.

this entails that for all sets a, U\backslash \left\{ a\right\} is not in bijection with U and U\backslash \left\{ a\right\} \neq U which is a divergence from the usual case of infinite sets where if x is an infinite set then x\backslash \left\{ a\right\} is in bijection with x and, more generally, if a is a set of inferior cardinal number than x,then x is in bijection with x\backslash a.

so perhaps this can be a precise formulation of the difference between potential infinity, actual infinity, and finity.

this makes sense because you wouldn't expect that if you remove something from U that it should be like U. one can show that if anything is like U, ie in bijection with it, then it is U.

edit: one can also show that all proper subsets of U have strictly less cardinality.

i envision a bottom and a top and the real inaccessible cardinals are the ones in between.

[phoenixthoth]

organic,
i think that when one performs those operations with Ø and U, the answer will be indeterminate in some cases. the multiplication would refer to cardinality multiplication. the difference to set difference and the division could refer to another thread i started on division. exponentiation refers to cardinal exponentiation.

edit:
another result: if there is a map from x onto P(x) then either x=U or x is fuzzy.

thus, the only sets for which x might be mapped onto P(x), besides U, are necessarily fuzzy sets. since so few fuzzy sets seem to exist without more axioms, this is a good thing and intuition is not being violated too terribly yet.

i can also show exactly where cantor's diagonal argument fails in general.

[Hurkyl]

All right, beginning at the top. [:)]

Since deduction in ternary logic is pretty much essential to the whole thing, it's definitely worth spending a bit of time on just that. I've thought more about it, and I understand how it works now.


Recall in binary logic, we might write a deduction (like modus ponens)

P&rarr;Q
P
-----
Q


Since there are only two truth values, this notation is sufficient... we need a way to indicate truth value in ternary logic in a deduction. I think the V(P)=T notation isn't bad for formulas, but I think a different notation is clearer for deduction. Consider modus ponens in your ternary logic:

T: P&rarr;Q
M,T: P
--------
T: Q

This notation means that we know V(P&rarr;Q) = T, V(P) in {M, T}, and we deduce V(Q)=T.

This is actually a valid deduction; if we check the truth table, if V(P&rarr;Q)=T and V(P) in {M, T}, then we must have V(Q) = T! Modus ponens is still useful in your ternary logic!

Also, we have:

M: P&rarr;Q
T: P
--------
M: Q

or more generally, the deduction schema

X: P&rarr;Q
T: P
--------
X: Q

where X ranges through {F, M, T}.

And other familiar rules, like:

T: P&rarr;Q
T: Q&rarr;R
--------
T: P&rarr;R

M: P&rarr;Q
M: Q&rarr;R
--------
M: P&rarr;R

T: P&rarr;Q
M: Q&rarr;R
--------
M,T: P&rarr;R

(For this one, we can only conclude that P&rarr;R is not false)

X: A&rarr;~A
--------
X: ~A

T: A&rarr;F
--------
F: A

T: P&rarr;Q
F,M: Q
--------
F: P

And so on; all of our favorites (I think) have a version or two valid for ternary deduction.


More as I think about it.

[Hurkyl]

Hrm...

I've found something disturbing about your axiom of subsets:


\forall a \exists x \forall y:
(y \in_? x \leftrightarrow_+ (y \in_? a \wedge A(y)))


Suppose we have said set x, and we know that, for a particular y, V(y \in_? a)=T, and A(y) = T...

We cannot conclude that V(y \in_? x)=T!!! The axiom is still satisfied if V(y \in_? x)=M... we've obliterated crisp ZF. [:(]

[Hurkyl]

I have an idea for additional predicates. The great thing about binary deduction is that any deduction can be entirely encoded into a formula via \wedge and \rightarrow; something similar is needed for ternary logic.


Let's start with the simplest; suppose we were able to prove:

T: A
------
...
------
T: B

We want to write this as a formula F to simplify further proofs, so we can apply the deduction

T: A
T: F
-----
T: B

in one step.


Here's a truth table:


\begin{array}{c|c|c}
A & B & A \circ B \\
\hline
T & T & T \\
T & M & X \\
T & F & X \\
M & T & T \\
M & M & T \\
M & F & T \\
F & T & T \\
F & M & T \\
F & F & T \\
\end{array}


Where X could be either F or M; at the moment they don't matter. As you can tell, I haven't decided on a symbol for this operation either. [:)]

Does this do what we want? If we're given that V(A \circ B) = T, then if V(A)=T we can then conclude V(A)=F. However, if V(A) \in \{F, M\}, we can conclude nothing about V(B).

Example!

I mentioned earlier that:


\begin{array}{ll}
T: & A \rightarrow \neg A \\
\hline
T: & \neg A
\end{array}


was a valid deduction. Let's try out our new operation:


\begin{array}{c|c|c|c}
A & \neg A & A \rightarrow \neg A & (A \rightarrow \neg A) \circ \neg A \\
\hline
T & F & F & T \\
M & M & M & T \\
F & T & T & T \\
\end{array}


Joy! (A \rightarrow \neg A) \circ \neg A is a tautology, as advertised!

And, as we can see from the truthtable, if we were given (A \rightarrow \neg A) \circ \neg A and A \rightarrow \neg A were both true, we can immediately conclude \neg A is true, without having to do the (trivial) work to prove it again.

For fun, consider the truth table for (A \circ B) \wedge (B \circ A):


\begin{array}{c|c|c|c|c}
A & B & A \circ B & B \circ A & (A \circ B) \wedge (B \circ A) \\
\hline
T & T & T & T & T \\
T & M & X & T & X \\
T & F & X & T & X \\
M & T & T & X & X \\
M & M & T & T & T \\
M & F & T & T & T \\
F & T & T & X & X \\
F & M & T & T & T \\
F & F & T & T & T \\
\end{array}


This gives an "if and only if"-like thing; we have the following deductions:


\begin{array}{ll}
T: & (A \circ B) \wedge (B \circ A) \\
T: & A \\
\hline
T: & B
\end{array}



\begin{array}{ll}
T: & (A \circ B) \wedge (B \circ A) \\
F,M: & A \\
\hline
F,M: & B
\end{array}


But while \circ is useful as a replacement for \rightarrow, I'm not so sure that this thing is useful as a replacement for \leftrightarrow.

[phoenixthoth]

Originally posted by Hurkyl
Hrm...

I've found something disturbing about your axiom of subsets:


\forall a \exists x \forall y:
(y \in_? x \leftrightarrow_+ (y \in_? a \wedge A(y)))


Suppose we have said set x, and we know that, for a particular y, V(y \in_? a)=T, and A(y) = T...

We cannot conclude that V(y \in_? x)=T!!! The axiom is still satisfied if V(y \in_? x)=M... we've obliterated crisp ZF. [:(]

i'm proud to have single-handedly obliterated ZF! actually, i do know what you mean and that's a bad thing.

i agree with this and i'm kicking myself for not noticing it. ok. let me run this by you. there are approximately three places i'd like to change the truth table of the plus-biconditional. for one thing, it means it's no longer based on the plus-conditional with the "and" operation, which really isn't serious. my ad hoc justification is this:
all i want to do is find a model for the absolute infinity that doesn't contradict ZF and hopefully not ZFC and any way to do this should be equivalent. i believe this error can be corrected in the following way.

the following letters in a row represent A and B and then the value of the A<->+B. the A and B refer to what's on the left and right hand side of the subsets axiom.
(optional) changing TMT to TMF. in the subsets axiom 2, when i state the plus conditional, it is assumed as per tradition that it has truth value T. hence if the right hand side has value M then the left hand side is not T.
forsure: changing MTT to MTF. if the right hand side in subsets 2 is T then the left hand side must now be T also. however, if the right hand side is F, then the left hand side is X where X is in {M,F}. i don't know if this is a bad thing.
(optional): changing FMT to FMF.

i need to have this: MFT and this: MMT to avoid russell's paradox.

the new truth table would then be this:\begin{array}{ccc}
y\in _{?}x & y\in _{?}a\wedge A\left( y\right) & y\in _{?}x\leftrightarrow_{+}y\in _{?}a\wedge A\left( y\right) \\
T & T & T \\
T & M & F \\
T & F & F \\
M & T & F \\
M & M & T \\
M & F & T \\
F & T & F \\
F & M & F \\
F & F & T
\end{array}

now it is unambiguous except in one case. if B is true and the plus-biconditional is true, then A is true. if B is M and the plus-biconditional is true, then A is M. if B is false and the plus-biconditional is true, then A is false or M. this last one bothers me. it means that every set has an indeterminate non-true membership in every other set. but this ain't that bad. or is it?

another thing i was thinking is of cooking up a deep-fat fryer: a "function" like the powerset operation that sends a set to a set of all elements that are in it fully; thus the potentially fuzzy set gets crispy. so if V(y&isin;?x)=M, and i leave the plus-biconditional, which is now a terrible notation, alone, y will not be in the cooked/fried version of x yet it would M'ly be in x. this is an aside and a diversion for now, i think. the notation in set theory would then really apply to all cooked versions of sets not unlike how in some integration theories (L^p spaces) a function is written but it secretly means the equivalence class of all functions who differ on a set of measure 0.

[Russell E. Rierson]

http://en.wikipedia.org/wiki/Singleton_set

In mathematics , a singleton is a set with exactly one element. For example, the set {0} is a singleton. Note that a set such as {{1,2,3}} is also a singleton: the only element is a set (which itself is however not a singleton).
A set is a singleton if and only if its cardinality is 1. In the set-theoretic construction of the natural numbers the number 1 is defined as the singleton {0}.







Structures built on singletons often serve as terminal objects or zero objects of various categories
The statement above shows that every singleton S is a terminal object in the category of sets and functions. No other sets are terminal in that category.
Any singleton can be turned into a topological space in just one way (all subsets are open). These singleton topological spaces are terminal objects in the category of topological spaces and continuous functions. No other spaces are terminal in that category.
Any singleton can be turned into a group in just one way (the unique element serving as identity element These singleton groups are zero objects in the category of groups and group homomorphisms . No other groups are terminal in that category.



The problem with set theory appears to be the primitive concept of : inside / outside

A or not-A

Set theory is based on 2-valued logic of course. Since the absolute infinite is the power set of everything that exists, ergo, an element cannot be removed from it by definition, hence the talk of the removal of elements becomes wishful thinking. A fruitless exercise in futility, for the imagination. Also, the problem arises where, if, an element such as a singleton, is removed from the absolutely infinite, and it[absolute infinity] somehow becomes less than its previous absolute condition, that seriously implies that the absolute infinity is totally non-replenishable, but, actually, if the absolute is generalized as the Platonic state of affairs known as total nothingness, then it no longer has this unprecidented weakness inherited from ZF set theory which is dooomed to die a slow and agonizing death, as die hard academians continue to flog the dead ZF "horse" for all it's worth.



An "excellent idea":

http://www.mmsysgrp.com/stefanik.htm


Thesis: Scientific objectivity is best characterized by the concept of invariance as explicated in category theory than the concept of truth as explicated in mathematical logic.



If reality is self referential, it observes itself, through particle interaction AND conscious observers?

[Hurkyl]

The place to put forth your own theories is not in someone else's thread.

[Russell E. Rierson]

My apologies Hurkyl, please continue with your interesting discussion.

[phoenixthoth]

on 1-6-04, someone posted a similar idea on sci.math.research here: http://mathforum.org/epigone/sci.math.research/vermsmixbler

my old logic teacher at cal is looking over this crackpot theory with the corrections hurkyl suggested. he said he vaguely remembers me from (?) 7 years ago.

i am looking for things that should be true about the universal set U. i have a small collection of things i thought ought to be true and none of them were that difficult to prove, though that may be because i'm using fallacious arguments. so if you can think of things that ought to be true of U, feel free to post them.

i know russell and he was referring to something i told him elsewhere that i think ought to be true: if you remove even one element from U, you get something of strictly smaller cardinality. but that relies on a proof that i'm not sure about (it's not in tuzfcver2 but is in my latest version). anyways, i think that should be true: no subset of U, even one less by a singleton, should have the same cardinality as U. this is an example of the kind of theorems about U i'm looking for. russell, thanks for the feedback and hurkyl, thanks for the moderation.

[phoenixthoth]

U can be turned into a boolean ring in the following way:
the additive identity is Ø.
the multiplicative identity is U.
x*y=the intersection of x and y.
x+y is the symmetric difference: x&cup;y\(x*y).

then -x=x and x^2=x.

since U is a ring, one can then prove things about all sets by showing that the set of things with a property forms an ideal with the multiplicative identity in it, which then proves that that ideal is a nonproper ideal equal to the whole ring U, which means that that defining property of the ideal holds for all sets.

example: let p(x) be a property of set x.
let J={x in U : p(x)}. to prove prove p holds for all sets x, one can do it this way:
1. prove J is an additive subgroup of U with the same +.
2. prove J is a subring of U with the same *.
3. for any r in U, show that rJ=J.
4. prove that U&isin;J.

from standard ring theory, it follows that J=U and, hence, p holds for all sets.

can anyone give me some simple property p (that is known to be true for all sets) to try this method on? maybe i can compare the length and difficulty in doing 1-4 to the standard proof...

the current version of my paper with hurkyl's corrections is here:
http://www.alephnulldimension.net/matharticles/tuzfcver6.pdf

[Organic]

Dear phoenixthoth,


Let us say that multiplication and addition are complement properties.

For example: http://www.geocities.com/complementarytheory/ASPIRATING.pdf

My question is: What is U from this point of view?


Yours,

Organic

[phoenixthoth]

i'm not sure what complementary means but U is the unit in the ring.

U=1.

this is because a*1=a*u=a intersect U = U intersect a=1*a=a.

from http://en2.wikipedia.org/wiki/Ring_(algebra)
If S is a set, then the power set of S becomes a ring if we define addition to be the symmetric difference of sets and multiplication to be intersection. This is an example of a Boolean ring.

symmetric difference is like the logical XOR operation:
http://en2.wikipedia.org/wiki/Symmetric_difference

and 1+x=the complement of x=U\x.

[Organic]

Please look at the attached jpg:

Let White be Addition.

Let black be Multiplicaction.

Let Complement be Prevent AND create.

By Complementary Logic, Addition AND Multiplication are complement operations.

So, what is U from this point of view?

[Hurkyl]

So, what is U from this point of view?

Something that should be a subject of a different thread. I'm interested in seeing where Phoenix goes with the ring approach... your post has nothing to do with that (or, as far as I can tell, anything else in the thread, except you use a couple of the same words)

[phoenixthoth]

if there is a universal set, is this a metric space with a nontrivial metric? or a topological space with a topology other than the trivial ones?

in other words, a function d from UxU to R>=0 such that
1. d(x,y)=0 iff x=y
2. d(x,y)=d(y,x)
3. d(x,y)<=d(x,z)+d(z,y)

or at least some kind of pseudo metric space in which one can "mod out" by d=0 in the sense that we can define an equivalence relation such that x ~ y iff d(x,y)=0 and then take as the metric space U/~.

if i can define a function from U to R>=0, call it | |, then i can use the ring structure to say that maybe d(x,y):=|x-y| where this is either the ring difference, ie x+y, or set difference.

i'm guessing that what i'm looking for is a bounded metric where the "distance" between U and any set that isn't U is the upper bound of this metric.

this work has probably been done before in the context of pseudo-universal sets and powersets but i'm just not sure which key words to search for.

potential axioms on | | are as follows:
|Ø|=0
|U|=1
|x|&isin;[0,1] for all sets x

note that if i take the ring structure and define d(x,y)=|x-y| then that is |x+y| and so d(x,y)=0 iff |x+y|=0 if x=y if x=Ø. if x=y then x+y=Ø and so d(x,y)=0; hence d(x,y)=0 iff x=y. so i believe i need to focus on this | |. at first, i was thinking of relating |x| to the cardinality of x according to where it is on the following hiearchy:
A: subset of N
B: "power level" of N (which is any union of powersets of powersets of N)
C: "power level" of any set on level B
...

but there aren't even countably many of those levels so i gave up on defining |x| in terms of the cardinality of x.


if J={x in U | x!=U} then J is, i think, a maximal ideal and i think that U/J is isomorphic to the absolute infinite product of Z/2Z. perhaps i can mix these two ideas together though [0,1] is a subset of a characteristic 0 field and U/J has characterisitic 2...

in max tegmark's paper on his ensemble theory of everything, it is postulated that mathematical existance is physical existence. i'm guessing that in order for an observer to exist in an abstract space, there must be a nontrivial metric on that space in which measuring of some kind can take place. if U has a nontrivial metric, then we can potentially have observers under this hypthesis. if only the trivial metric exists, then it seems all an observer can do is say "this is me" and "this is not me."

[Hurkyl]

Sets aren't topological (metric) spaces; sets equiped with a topology (a metric) are topological (metric) spaces.

I think you know that, but your first paragraph was a little vague on this point.



Anyways, I'm wondering if the surreal numbers should come into play here; they're the only "normal" number system I know that is "big enough" to include the ordinals and cardinals. Maybe you want | | to be surreal-valued? Be warned, though, that surreal numbers are very difficult things to manipulate.


J={x in U | x!=U}

This can't be an ideal; it's not closed under addition.

In particular, for most A, A and (U+A) are both in J, but A+(U+A)=U.

[phoenixthoth]

i was abusing the notation in a way which i thought was standard you know such as referring to R as a metric space when technically, it can be equipped with metrics or (R,d) is the metric space.

if the range of d is not a nonnegative real number, then whatever is in question is not a metric or a metric space.

it's good to know that that J is not an ideal. thanks.

[Hurkyl]

Right; it is a standard abuse of notation. Can't hurt being a little extra cautious about the details, though, especially given the circumstances. [;)]


While a metric space must have a real valued metric, one can make metric-like spaces using other ranges for the "metric"; I was just wondering if the surreals might be more appropriate. A "surmetric space" might even have structure kind of like the halos from nonstandard analysis that might let you modulus parts of it into a normal metric space.

[matt grime]

Having read back a couple of posts, can I make the following observation without repeating someone else?

If you are going to have a universal set which has a metric (or topology) then the collection of all metric spaces will inherit that metric (topology) as a subset (subspace), and thus you come up against Russell's paradox straight away.

This is even something that physicists are finding in string theory and spin foam models.

Please point out if I'm way off topic, and I'll delete this straight away.

[phoenixthoth]

it's not off topic in my opinion. seems like the only way to equip U with a metric is with the trivial one. i'm not seeing how russell's paradox would be involved in the metric space aspect of it, though it is run up against in the main aspect of the existence of U. that all metric spaces would inherit the metric from U's metric leads me to suspect that the only metric definable on U is a trivial one in which
d(x,y)=1 if x!=y, else d(x,x)=0. and that would lead me to believe that the only topology on U that makes sense is for every set in U to be open.

russell's paradox is handled in the main treatment of U, as well as cantor's diagonal argument (somewhere near theorems about P(U) not being bigger than U). cantor's diagonal set boils down to russell's set in a certain situation.

incidental note: in quinne's (quine?) new foundations theory with a universal set, he somehow manages to avoid russell's paradox though without using three valued logic and with the axiom of choice being false for some reason. i'm guessing that's why not everyone has heard of it: no axiom of choice means no zorn's lemma and many things crumble in various fields (pun not intended). so far, i've been unable to find a free copy of his works online though a book in 1995 with his ideas is only $35. i would buy it to look for more theorems about U to prove.

[Russell E. Rierson]

The goal is to eliminate paradox, while maintaining an all inclusive principle of "comprehension"[semantics], yo, where an infinitely expanding chain of "sets[containment principles]" and concepts, such as "proper set", "ordinal" and "cardinal" are relativised to context, which would take care of paradox at all levels, except for the "top", which naturally does not exist, of course! So it becomes an infinite chain or composition of ever more inclusive situated sets[semantics] with an interesting informational - topological dynamic. So it comes full circle, and the poetic verses explaining Beingness and Nothingness become a unifying dialectic, and a new synthesis. It just needs to be put into a rigorous mathematical framework[syntax].

Barwise Situation Theory?:

http://www.cs.bilkent.edu.tr/~akman/jour-papers/sigart/node1.html

[Russell E. Rierson]

Interesting:

http://www.cs.bilkent.edu.tr/~akman/jour-papers/air/node13.html



Barwise defined the operation M (to model situations with sets) taking values in hypersets and satisfying (cf Note 12):


if b is not a situation or state of affairs, then , M(b) = b

if , rho = <R,a,i> then M(rho) = <R,b,i> (which is called a state model), where b is a function on the domain of "a" , satisfying ,
b(x) = M(a(x))

if s is a situation, then M(s) ={M(rho) : s |= rho}.

Using this operation, Barwise then proves some theorems, including the one which states that there is no largest situation (corresponding to the absence of a universal set in ZF).

[matt grime]

If you are going to have a universal set, and if the collection of *all* metric spaces is a subset of this set, then a metric on the universal set would imply a metric on this set of all metric spaces, and you have a set that contains itself. The way to sidestep it would be to say that the universal set contains only *a* set of metric spaces, not _the_ set of all metric spaces. Personally I adhere to the Grothendieck school and just ignore these issues.

As for Quine's Universal set theory, I can see how the assumption that Zorn's Lemma is false would be used. Without it, P(U) might be empty.

[phoenixthoth]

my gut tells me that deduction operators like |= wouldn't apply the same way in ternary logic.

Originally posted by matt grime
If you are going to have a universal set, and if the collection of *all* metric spaces is a subset of this set, then a metric on the universal set would imply a metric on this set of all metric spaces, and you have a set that contains itself. The way to sidestep it would be to say that the universal set contains only *a* set of metric spaces, not _the_ set of all metric spaces. Personally I adhere to the Grothendieck school and just ignore these issues.

As for Quine's Universal set theory, I can see how the assumption that Zorn's Lemma is false would be used. Without it, P(U) might be empty.

this doesn't seem to be a problem because every set can be equipped with a trivial metric. one way to say that the reason for this could be that this is the only way to metrize U in which case all sets can be equipped with a metric inherited from U, being the trivial metric that basically says each point "knows" if a second point in question is "me" or "not me."

by changing the foundation axiom, i have a statement that wouldn't apply if there were no universal set and when there is one, it is a member of itself.

as far as the powerset of U goes, P(U)=U. furthermore, i have a couple of other theorems in my paper which go something like this:
1. P(x)=U iff x=U.
2. if x is crisp then there is a map from x onto P(x) if and only if x=U.
3. if P(x)=x then either x is fuzzy or x=U.

the first statement implies that U will never be built from below by powerset operations, which is something already known. i believe stronger versions of 2 and 3 are lurking out there that would say:
2'. there is a map from x onto P(x) iff x=U
3'. P(x)=x iff x=U.
i believe those are out there because in my theory, the powerset ignores partial membership and is somewhat "forgetful" with respect to elements that are only partial members.

cantor's diagonal argument which would normally rule out 2 and 2' i believe i have dealt with by adapting the subsets axiom in a way that
a. accomodates ternary logic
b. in a way that extends the situation for binary logic so that
if all one wanted to do was regular set theory with the two new axioms then nothing should be different.

[Russell E. Rierson]

http://www.cs.bilkent.edu.tr/~akman/conf-papers/Tueb/node6.html



One can assert facts that a situation will support. For example, if s1 supports the fact that Bob is a young person, this can be defined in the current situation s as:
s: (|= s1 (young Bob)). Note that the syntax is similar to that of Lisp and the fact is in the form of a predicate. The supports relation, !=, is situated so that whether a situation supports a fact depends on where the query is made.






s1 supports relation A, defined as situation s

s: |= s1


U[]U[]U][]U[]U[]U...

"like points on a line" ...?

[phoenixthoth]

i was suggesting that the following looks like a line in which U is like a point on that line:
...U&isin;U&isin;U&isin;U&isin;U&isin;U...
yet it's kind of a hyperline in that those six dots mean a lot of stuff (transfinite membership strings and such).

[phoenixthoth]

U as a pseudo-pseudo metric space:
define d to be a function from UxU to U (not just R) such that if x and y are sets, then
d(x,y)=x+y.

then
1. d(x,y)=0=Ø iff x=y
2. Ø<=d(x,y) where <= can be taken to mean either "is a subset of" or "can be injected into"
3. d(x,y)=d(y,x)
4. d(x,y)=d(x,z)+d(z,y).

then i'm wondering about limits and open balls.
an open ball would be something like this where delta is a set:
B(x,delta)={y in U | x+y<=delta}.

limn->Uxn=x means that for all nonempty e, there is a set N such that xn+x<=e whenever n<=N. i haven't decided which of these <='s it would be best to have as subset and which as "can be injected into." i'll have to do some examples like [/sub]xn=U for all n and [/sub]xn={0,1,2,...,n} or =(0,1/n) or =[0,1/n) or =[0,1/n] modulo some adjustments because n can be any set. the adjustment might look something like this: [/sub]xn={0,1,...,n} if n&isin;N and [/sub]xn=N (the set of natural numbers), or some infinite set, otherwise.

anyways, this "metric" is kind of neat in a way. if you visualize two sets and consider + to be xor (symmetric difference) then the more the sets have in common, the closer to empty this "metric" is; ie the closer the two sets are. the less that have in common or the larger they are, the further away from empty (0) the "metric" is.

if cosine can be extended to hyperreal numbers, can it be extended to arbitary sets? perhaps an angle between sets, which would be a set like all other angles just not a real one, is possible. i'm guessing that the "angle" between sets will be super in multiplicity and not constant modulo some ideal. hmmm... maybe the angle should be some coset x+J where cos(J)=1=U and cos(x)=1. arccos can be defined in terms of logs which can potentially be defined as an inverse to the powerset operation. thought for food...

[Hurkyl]

The hyperreal numbers have an extremely special relationship to the reals that enable you to transfer real functions to hyperreal functions; I don't think you can do anything here. It feels wierd wondering about the angle between two sets as well, instead of looking at dot products...


Have you considered having d(x, y) mean |x + y|? I.E. the cardinality of the set (x + y)?


Also, what about multisets? We can make a module of multiset-like things over the cardinals, and then we can make a dot product out of the metric, and given a dot product we can define the angle between multisets.


We might be able to do the same with ordinary sets, but the base field would be Z_2, which puts our metric living in the "wrong" domain.

[phoenixthoth]

Originally posted by Hurkyl
The hyperreal numbers have an extremely special relationship to the reals that enable you to transfer real functions to hyperreal functions; I don't think you can do anything here. It feels wierd wondering about the angle between two sets as well, instead of looking at dot products...


Have you considered having d(x, y) mean |x + y|? I.E. the cardinality of the set (x + y)?


Also, what about multisets? We can make a module of multiset-like things over the cardinals, and then we can make a dot product out of the metric, and given a dot product we can define the angle between multisets.


We might be able to do the same with ordinary sets, but the base field would be Z_2, which puts our metric living in the "wrong" domain.

interesting...

for |x+y|, how would you prove the triangle inequality? i tried a little but got stuck.

how would you make a dot product out of the metric? seems like you might have this for the angle:
cos t = (a.b)/|a||b| but we'd have to divide cardinals wouldn't we?

all one has to do for a dot product is define |a| cuz then since |a|^2=a.a and a.b=((a+b).(a+b)-a.a-b.b)/2, we get a.b=(|a+b|^2-|a|^2-|b|^2)/2. if those are all infinite cardinal numbers, we get a.b=|a+b|-|a|-|b| though showing that that satisfies the definition of dot product is probably not possible. but it's kind of pseudo-dot like.

thanks for the suggestions.

[Hurkyl]

Well, it's clear that |x U y| <= |x| + |y|, right? And we have x + y is a subset of x U y, so we have |x + y| <= |x| + |y|


The cardinal arithmetic is messy; I dunno if anything can be done with it. [:(]

[phoenixthoth]

limn->Uxn=x means that for all nonempty e, there is a set N such that xn+x<=e whenever n<=N.
made a mistake. i should have said N<=n. but i'll use the letter y or n' because of the possible confusion with the set of natural numbers.

btw: xn would be a function from U to U.

i'm wondering why e has to be nonempty. if it is "for all nonempty e" then i can get limits whose symmetric difference is either empty (implying equality [a)]), a singleton, or a doubleton.

if i change that to "for all e" then i can get limits to be unique.

working on a cauchy completeness type of business now now that uniqueness is in order. also working on giving one darn example that's not trivial.

[phoenixthoth]

zorn's lemma:
let S be a nonempty partially ordered set (ie we are given a relation x<=y on S which is reflexive and transitive and such that x<=y and y<=x together imply that x=y). a subset T of S is a chain if either x<=y or y<=x for every pair of elements of x,y in T (ie every pair of elements of T are comparable). Then Zorn's lemma may be stated as follows: if every chain T of S has an upper bound in S (ie if there is an x&isin;S such taht t<=x for all t&isin;T) then S has at least one maximal element.

consider the relation <= given by x<=y iff there is a 1-1 map from x into y. then can zorn's lemma be strengthened so that instead of x<=y and y<=x implying x=y, it just imples that they are isomorphic (as sets)?

let's leave it the way it is and suppose that f is a map from A to B where A and B are subsets of U. let f[A] denote the image of A under f, ie f[A]={f(a)&isin;B: a&isin;A}.

suppose that f has the property that n<=m iff f(n) is a subset of f(m). i will abbreviate this by writing f(n)$f(m). also suppose that there is an M such that f(n)$M for all n&isin;A. one can assume that M is not U to get a stronger result.

what i want to show is that f "converges" to some limit in this universal limit sense. i believe i can show that limn->Uf(n)=L iff there is an n' such that for all n, if n>=n' then f(n)=L.

let S be f[A] u {M}. by the assumption on f, every element of S is comparable. i claim that any chain T in S has an upper bound in S. let T be a chain in S. then T's elements are comparable as all elements in S are comparable. every element in T has an upper bound in S: namely M. then S has at least one maximal element L.

either L=M or L&isin;f[A]. if L&isin;f[A], then i claim that f U-converges to L, ie that limn->Uf(n)=L. we know that f(n)$L for all n&isin;A as L is a maximal element of S. as L&isin;T, f(n')=L for some n'&isin;A. now suppose n>=n'. by assumption on f, f(n) contains f(n')=L; hence f(n)=L. by the lemma i haven't proved here, this is sufficient to prove the claim.

if M is not in f[A] the i claim f does not converge. i haven't worked out the details.

i'm trying to also show that U is a noetherian ring, ie a ring which satisfies the ascending chain condition. i want to use this result and take f(n) to be a n-th ideal of some kind which form an asencing chain. then M might be the union of the f(n)'s or something.

is it known that boolean rings are noetherian? some, all, none?

[phoenixthoth]

this is the current incarnation of it: http://www.alephnulldimension.net/matharticles/tuzfcver8-1-0.pdf (103kb/12 pages)

i'm contemplating publishing this somewhere but i'd appreciate feedback before i try to do so so i don't make a complete a** out of myself.
any thoughts?

[matt grime]

The proof of theorem 2B is wrong.

just because something is an element of the power set P(x) does not imply it is an element of x.

In fact, it might be that, if we can form the set of sets not equal to U that its power set is U (if it isn't U already). I didn't look at the ternary logic enough to state that for certain.


Also the proof that no proper subset of U is in bijection with U is wrong - as you are using ZFC with U, you have the axiom of infinity, which assures that there is an inductive set, hence U contains the sets used to define the infinite set, and thus there is s trivail bijection from U to U\{{}} that is U omitting the set containing the empty set. This is a constructive proof, so ternary logic doesn't enter into it.

[phoenixthoth]

...{a} is an element of P (x) implies that {a} is a subset of x. this implis that a is an element of x. since a was arbitrary, by the uniqueness of U, x=U.

"there is s trivail bijection from U to U\{{}} "
can you please prove that it is a bijection because i don't see that.

[matt grime]

First one: {a} in P(x) implies that a is a subset of x, it does not imply a is an element of x.

proof x is an element of P(x) but for an arbitrary set x is not in x.

there was no other constraint placed on x other than it be a set whose power set was the universal set.


second. if you have all the ZF axioms then you have a collection of sets labelled by the integers - the elements in the inductive set, send the set labelled by 1 to2, by 2 to 3 etc. and define the map to be the identity elsewhere. this is clearly a bijection onto a proper subset, and it works for any set containing an infinite number of elements, and it is constructive.

[phoenixthoth]

again, i wrote this:
...{a} is an element of P (x) implies that {a} is a subset of x. this implis that a is an element of x. since a was arbitrary, by the uniqueness of U, x=U.

not this:
...{a} is an element of P (x) implies that a is a subset of x. this implis that a is an element of x. since a was arbitrary, by the uniqueness of U, x=U.

oh, i get it now. in general Z in P(x) implies Z is a subset of x, no? let Z={a}.

anyways don't you expect this theorem to be true anyway because it implies that U can not be arrived at by power-setting a smaller set.

"second. if you have all the ZF axioms then you have a collection of sets labelled by the integers - the elements in the inductive set, send the set labelled by 1 to2, by 2 to 3 etc. and define the map to be the identity elsewhere. this is clearly a bijection onto a proper subset, and it works for any set containing an infinite number of elements, and it is constructive."

i'm not understanding the relevance. is that a map that is a bijection from U onto a PROPER subset of itself, U\{{}}? how can a map from a set to a proper subset of itself be injective? oh i see, in the case of potentially infinite sets of course! duh. but still, i'm not seeing an explicit example of a bijection between U and a proper subset of U which WOULD violate something in my paper. intuitively, you're stirring around the elements in U not mapping U bijectively to a proper subset of U. i'm just not understanding you but please please be patient with me.

[matt grime]

Erm, in what way did you not understand the counter example to the 'proof' you've got? (the unversal thing isn't at issue here, just the assertion that as {a} is in P(X), that a must be
*an* element of X. a is a collection of elements of X is all that you can deduce. X is an element of P(X) yet X is not in general an element of X!


Second. You what? By construction the map is injective, find distinct x and y with f(x)=f(y) for f the function defined in my last post. It's elementary to show that it is injective, unless you aer going to argue that I cannot split the universal set into those elements in the inductive set and those not.


A map from a set to a proper subset can easily be injective if a set is infinite as you yourself allude to in the paper round about that theorem on there being no bijection from U to a proper subset of itself. Remove the axiom of infinity and this example goes away.

[phoenixthoth]

i edited my post but it's not of real consequence now.

Originally posted by matt grime
Erm, in what way did you not understand the counter example to the 'proof' you've got? (the unversal thing isn't at issue here, just the assertion that as {a} is in P(X), that a must be
*an* element of X. a is a collection of elements of X is all that you can deduce. X is an element of P(X) yet X is not in general an element of X!
ok. let z be any set and consider P(x). suppose z is in P(x). is z a subset of x or not? suppose it is. then that means all elements of z are in x. ok? now take z={a}. since the assumption is that P(x) is the universal set, {a} is in P(x). then all elements of {a} are in x. that means a is in x.



Second. You what? By construction the map is injective, find distinct x and y with f(x)=f(y) for f the function defined in my last post. It's elementary to show that it is injective, unless you aer going to argue that I cannot split the universal set into those elements in the inductive set and those not.
let me get this straight. is this an equivalent example:
U=N union (U\N).
let f be a self mapping of U such that
f(x)=x+1 for x in N and
f(x)=x for x in U\N.
is that your example?

thanks for helping me correct my paper, btw.

[matt grime]

Originally posted by phoenixthoth
i edited my post but it's not of real consequence now.


ok. let z be any set and consider P(x). suppose z is in P(x). is z a subset of x or not? suppose it is. then that means all elements of z are in x. ok? now take z={a}. since the assumption is that P(x) is the universal set, {a} is in P(x). then all elements of {a} are in x. that means a is in x.

No, the assumption about the unversality of P(X) does not come into it. Firstly, {a} is a set with one element, a, that a is also a set is misleading. You cannot conclude that all the elements of {a} are in x. Secondly, the last line exactly states the objection that a is only a subset of x. You are confusing 'is a subset of a set' with 'is an element of a set of sets'



let me get this straight. is this an equivalent example:
U=N union (U\N).
let f be a self mapping of U such that
f(x)=x+1 for x in N and
f(x)=x for x in U\N.
is that your example?

thanks for helping me correct my paper, btw.

Not quite, I want a collection of sets labelled by N, not the set N itself. The axiom of infinity means that such must exist in any model we're looking at. The sets are:

0 - {} the empty set
1 -{{}} the set containing the empty set
2 -{{},{{}}} the set containing the two previoius sets.


I can just shift the labels by one here and leave all other sets unchanged.

[matt grime]

Acutally ignore the universal set power set unique thing for now, maybe some light has just come on in my head.

As written your proof could do with explanation, well, as written the 'proof' is wrong or at least the assertion needs more explaining, but the result might hold.

I beleive you want to consider the set which contains the set which contains a, for then {a} is a subset of X, but it contains one element, then a is in X - too few braces used

[phoenixthoth]

"Not quite, I want a collection of sets labelled by N, not the set N itself. The axiom of infinity means that such must exist in any model we're looking at. The sets are:

0 - {} the empty set
1 -{{}} the set containing the empty set
2 -{{},{{}}} the set containing the two previoius sets.


I can just shift the labels by one here and leave all other sets unchanged."

i'm only detecting the essence of what you're saying but i'm not getting it just quite yet. the claim, your claim, is that there is a bijection between U and a proper subset of U. are we discussing this corollary: if x is a proper subset of U, then there is no 1-1 map from U to x? i've lost track because i'm having a brain fart and you shot off two points before i could handle the first one. overload! so, if we're discussing that corollary, then your example should indicate (hopefully as explicitly as i need it to be if possible) a set x that isn't U such that there is a 1-1 map from U to x. i think your claim is that there is a 1-1 map from U to U\{{}}. i do want to keep the axiom of infinity (especially since i think it might be a consequence of the universal set axiom and so i must keep it), so i want to exactly pinpoint my error. well, i admire you if this example is 'trivial'.

[matt grime]

First, the P(X) thing can be corrected, as hopefully you saw above:given any set Z, {{Z}} is in P(X)

so the set with one element {Z} is a subset of X, so Z is an element of X.



The second.

Yes, it is your corollary that there is no 1-1 map form U to a proper subset of U, and the statement that for all sets A, U\{A} is not in bijection with U. Well, that isn't true if you have the axiom of infinity.

[phoenixthoth]

doesn't {{Z}} is in P(X) imply that {{Z}} is a subset of x?

[matt grime]

Originally posted by phoenixthoth
doesn't {{Z}} is in P(X) imply that {{Z}} is a subset of x?

Yes. It implies that {{Z}} is a set of some things in X, right? ie that {Z} is some set of elements of X, but {Z} has only one element, Z, so Z must be an element of X. Careful with your bracketing.

[phoenixthoth]

"Yes."

so if {{Z}} is a subset of x then the following conditional holds for all sets y:
if y in {{Z}} then y in x. ok?

suppose y in {{Z}}.

first of all, that means y in x.

second of all, that means that y={Z}. this works the other way: {Z} is in {{Z}}.

hence, {Z} in x.

now go back to {a} is a subset of x (which follows from {a} is in P(x)):
for all sets y, if y in {a} then y in x. ok?

suppose y is in {a}.
1. y=a
2. y in x.
3. therefore, a in x.

it's ok as it is.

[matt grime]

Maybe it's me that's got his braces wrong, it's a headache, but i agree the result is true. I don't dispute the result, and I increasingly think I agree with your original proof.

I stand by the second issue though, about proper subsets

[phoenixthoth]

i will heed your advice about being careful though.

i'd still like you to be more specific and concrete with me on your counterexample because that would seriously damage the paper. i mean to be as detailed as possible because i'm not an expert in set theory so i can't understand your sketch.

[matt grime]

By the ZF axioms there is the empty set {}, then there is the set containing it { {} }, and then the set containing those { {} , {{}} }, and so on, each of these can be labelled by an element of N correspeonding to the cardinality. The axiom of infinity states, that when I say 'and so on' that actually there is an infinite number of sets created inductively (this apparently does not follow from all the other axioms) with the labels from all the natural numbers. If you have this, your universal set cannot be finite, and must contain these sets.


Since there is a bijection from N to a proper subset of itself (n to n+1) then there is a bijection from those sets to a proper subset of the sets, and defining it to be the identity for all other sets gives a contradiction to your corollary.

[phoenixthoth]

thank you.

now another question.
that corollary was meant to be the contrapositive to the statement: if f is a 1-1 function from U to x, then U = x. what's the contrapositive of that? cuz if the contrapositive is wrong that means the theorem it draws its energy from is wrong.

[phoenixthoth]

what are "those sets " you mention in paragraph 2? they're indexed by N right? but what are they? are they N, P(N), P(P(N)), etc., which can be indexed by 0, 1, 2, ...? just for reference, here is the axiom of infinity:
\exists x\left( \emptyset \in x\wedge \forall y\in x\left( y\cup \left\{ y\right\} \in x\right) \right)

[matt grime]

I said above

0 ---{} the empty set
1---{ {} } the set containing the empty set
2 ---{ {} , { {} }} the set containg the previous two sets

3 is the set continaing the previous 3 sets, and so on n is the set containing all the previous n-1 sets constructed. This is often called omega, a quick google for axiom of infinity wil provide you with some good links.

[phoenixthoth]

omega is the ordinal number for the set you're describing, which is N.

0 is defined to equal Ø.
1 is defined to equal {Ø}.
2 is defined to equal {0,1}
3 is defined to equal {0,1,2}.
N is defined to equal {0,1,2,...} which exists by the infinity axiom.
do a search on that yourself. see enderton's "elements of set theory," et al.

so your map really is this:
f(x)=x+1 for x in N
f(x)=x for x in U\N.
that's what i thought.

[phoenixthoth]

a point of clarification: when we write U\N or any relative complement, do we mean the set of elements in U that are not in N or the set of elements such that it is not true that they are in N?

ie, U\backslash N=\left\{ x\in U:x\notin N\right\} or
U\backslash N=\left\{ x\in U:\lnot \left( x\in N\right) \right\} =\left\{ x\in U:x\in _{M}N\vee x\notin N\right\} ?

[phoenixthoth]

Originally posted by phoenixthoth
a point of clarification: when we write U\N or any relative complement, do we mean the set of elements in U that are not in N or the set of elements such that it is not true that they are in N?

ie, U\backslash N=\left\{ x\in U:x\notin N\right\} or
U\backslash N=\left\{ x\in U:\lnot \left( x\in N\right) \right\} =\left\{ x\in U:x\in _{M}N\vee x\notin N\right\} ?

which of these is true, if any:
\left\{ x\in U:x\in _{M}N\vee x\notin N\right\} \cup N=U and/or
\left\{ x\in U:x\notin N\right\} \cup N=U

[matt grime]

Originally posted by phoenixthoth
omega is the ordinal number for the set you're describing, which is N.

0 is defined to equal Ø.
1 is defined to equal {Ø}.
2 is defined to equal {0,1}
3 is defined to equal {0,1,2}.
N is defined to equal {0,1,2,...} which exists by the infinity axiom.
do a search on that yourself. see enderton's "elements of set theory," et al.

so your map really is this:
f(x)=x+1 for x in N
f(x)=x for x in U\N.
that's what i thought.

Some people label the set omega, some N, whatever. It is just a label. Personally, I would never say a number is *equal* to a set, but then I don't define my numbers as sets, cos I don't do set theory.

It's your set theory, I don't know what your definition of complement is. As said originally, perhaps you've axiomatized these issues away. Perhaps there are things which you do not know to be subsets, this is your three valued logic system, you ought to know what is what. My comments are pointing out where there are possible issues. I owuld suggest that I'm defining f(x)=x for all sets X where the value of X in N is NOT T, which I naively assume to be F OR M. Presumably it is not possible for a statement to be simultaneously T AND M, that is T AND M is F.

[phoenixthoth]

"It's your set theory, I don't know what your definition of complement is. As said originally, perhaps you've axiomatized these issues away. Perhaps there are things which you do not know to be subsets, this is your three valued logic system, you ought to know what is what. My comments are pointing out where there are possible issues. I owuld suggest that I'm defining f(x)=x for all sets X where the value of X in N is NOT T, which I naively assume to be F OR M. Presumably it is not possible for a statement to be simultaneously T AND M, that is T AND M is F."

i guess what i was asking was what do you think is the best way to define complements to cover all the loose ends. you're right i should know that and i'll look into it. i am not sure that certain things we're talking about are sets anyway so it may not matter, or they may be fuzzy sets so it may be important to recognize that. i believe N is a crisp set and so how one defines complement doesn't matter. that's my gut feeling. hmm... your naivete is in your mind for i did say that veracity thingy's are functions which entails that they can't have simultaneously values T and M which is to say that T AND M is F. a veracity relation would be a whole different story and wouldn't resemble normal set theory at all in any way shape or form.

brain fart and for objective verification: what is the contrapositive of the statement "if f is a 1-1 function from U to x, then x=U?"

second: that is theorem 3. do you care to point out the error in theorem 3? i just want to pin down exactly where it's incorrect and see if it's repairable. now the whole theory wouldn't crumble if theorem 3 had to be torn out but i found it rather nice to have it there because it meant nothing i could think of was bigger than U. hmm... well, one doesn't need theorem 3 for that i see now. you know what? i had spotted a major error in my paper a while back and wondered if anyone else would notice it but this wasn't it. however, the major error was a consequence of theorem 3. good thing 3 isn't essential. but it will mean another push is ahead of me. i'm actually glad to see this now because i really hated what this major error had to say anyway. the major error implied that ALL sets are FUNCTIONS which is FALSE! that was the major error i wanted to see if others noticed. it is, in fact, corollary 4 to theorem 3. you almost got to it as you noticed a problem with theorem 3 in corollary 2. sigh... ok. it can still work though i will have to change some things around.

thanks again for your patient feedback.

[matt grime]

Well, my issue so far is with the part of the proof the theorem three that states

g: D to R ... if g^-1(y).....

well, g is only injective, therefore one can only define the inverse for the y in the image, that is y cannot be arbitrary, and here D and R are any sets and G any injection. So you may not define the preimage of elements not in the image.

I didn't read as far as the corollary you found to be in error.

[phoenixthoth]

i always had issues with that proof. i just didn't believe it 100%.

well, so much for theorem 3!

now i got to try to refit the carpet into the room...

[matt grime]

Just use a pair of Banach-Tarski scissors, it'll always fit

[phoenixthoth]

LOL

if this ever gets into a publishable form, i will definitely thank you and hurkyl for the feedback in it. i've been wondering who i'd dedicate it to. i think i may dedicate it to the letter M.

[matt grime]

Here's a thought. Seeing as you are axiomatically assuming a universal set anyway, part of the definition of its universality ought to be that if f is any injection from U to S, some set S, then f must be a bijection. I think the problem here is that you are attempting to prove what an axiom without using the relevant axioms. There is no harm in assuming this as an explicit axiom, and it might be that with a little thinking there is some way round this. Here I would suggest that the issue is resovable, by saying IF f is an injection, then S must be 'at least as big' as U, as U is universal then if f is not surjective, you have some problem (phrase it in your own preferred manner of thinking). Because really, the issue is that if f is a bijection from U to S that S is equal to U. Note that the set of natural numbers is bijective with the rationals but they are not EQUAL. Perhaps now the problems vanish a little.

[phoenixthoth]

are you trying to get on the co-authors list now? you're more than welcome to! so you want a tenneson number of 1, huh? maybe one day it will be time to stamp "three truth values are sufficient" on our stationary LOL.

hmm... i'd hate to add another axiom but i will if i must.

are you saying that U in bijection with x SHOULD imply U=x or SHOULD NOT imply U=x? i no longer think it should. i do think that if f is an injection from U to x then there exists a bijection between U and x.

[matt grime]

Not hankering for a co-authorship in the slightest.

It seems that there is some lattitude in what one means be universal here, and the definition of complement may allow for it.

Clearly if F:U to S is injective then, since the natrual inclusion S to U is injective, it is a 'proof from the book' that U and S are bijective, at least in ordinary binary logic.

[phoenixthoth]

of course! cantor-schroeder!

i was just trying to prove that theorem and i was like banging my head against the wall. almost got it (NOT) but i decided to see if you had the easier way and you did. my teacher always told me to not get the office next to the library because you have to think about it rather than look it up and i cheated and cheat by asking you. how real mathematicians go it alone is unimaginable ;).

no seriously, you can be a co-author if you want. in fact if you have more letters after your name than myself then that could add credibility to it. not saying it's publishable now or that anyone would care to read it but one day it will be doable. one day soon (like three months max i suspect, depending on when i get around to re-vamping it which i'm now a lot more eager to do than i was half an hour ago)...

[phoenixthoth]

here it is:
http://www.alephnulldimension.net/matharticles/tuzfcver9.pdf

any thoughts?

[matt grime]

What does the wavy equals sign mean here? (Yes, theorem 3!)

[phoenixthoth]

i should probably specify it in the document in the next draft...

in bijection with.

[matt grime]

Here is some stuff for you then:

as collection is not a set if given any cardinal A, there is a subset of cardinality greater than A.

this is why there is no set of all sets in ZF - if there were it would contain a set of card A for any A, but then it must contain P(A) which has card strictly greater than A. or if you like there is no maximal cardinal. you should probably look up weakly/strongly unreachable cardinals.

as it is what you appear to be axiomatizing is that the class of all things is called a 'set' because you aren't going to require it to satisfy any pesky things like the other axioms. (defining P(X) for all X but the universal set.

[phoenixthoth]

thank you for your insights. some questions/comments...

Originally posted by matt grime
Here is some stuff for you then:

as collection is not a set if given any cardinal A, there is a subset of cardinality greater than A.
in tuzfc, U passes this test for setness because of theorem 3 on page 5.

this is why there is no set of all sets in ZF - if there were it would contain a set of card A for any A, but then it must contain P(A) which has card strictly greater than A. or if you like there is no maximal cardinal. you should probably look up weakly/strongly unreachable cardinals.
i want to reiterate that i'm not saying ZF allows for U. i'm saying what you know, i think, which is that one must change things around in order to have a U. also, since P(x)=U iff x=U implies that U cannot be arrived at from a lessor cardinal, proving that it is inaccessible in some sense. hence certain other axioms about inaccessible cardinals may be derivable from my system. however, proofs of relative consistency are way beyond my current understanding; for one thing, it would have to involve passage from a binary logic to ternary logic and/or a combination of the two.

keep in mind that whether or not you like this system, as long as it is consistent, and that remains to be fully seen, imo, it is mathematically sound. what the observer must decide is whether it is useful or not useful and/or interesting or not interesting. that is all. i think that the way i squeezed out the universal set by throwing together a definition of the circle connective would be totally repulsive to some people. but hey, it generalized the biconditional in binary logic and i'm free to generalize it any way i want. such generalizations are either useful or not.

as it is what you appear to be axiomatizing is that the class of all things is called a 'set' because you aren't going to require it to satisfy any pesky things like the other axioms. (defining P(X) for all X but the universal set.

i dont' see why the universal set can't have a powerset. it just so happens to be identical to itself, as theorem 2 on page 5 shows. then theorem 2B shows that if P(x)=U then x=U which i find very satisfying. i do see your point in terms of U not satisfying the foundations axiom but that is the exception and not the rule. for example, one can have a set like this {U} from the pair set axiom and one can take the sum set of U. one can have a choice set for U with no apparent contradiction. etc... i've thought about this a bit and i think the infinity axiom is independent of the universal set axiom; i used to suspect that it was derivable from the unviersal set axiom but i no longer think it is. i need to learn more about forcing and such to really accomplish something non-elementary, i think. if i could prove the relative consistency of tuzfc and zfc then i would be most pleased...

[matt grime]

None of the things I've said should be construed as a 'this is absolutely wrong' as a theory, because I@ve not had time to read and understand it. they are observations about style, points about proofs, and some of the possible problems I can imagine going wrong - but you appear to have thought them through already.

[phoenixthoth]

appearances can be decieving! ;P

[phoenixthoth]

any thoughts on how U might interact with ultrafilters?

[phoenixthoth]

i have a new version 10 with the following theorem:

if f maps x onto P(x) then P(x) contains at least one fuzzy element, ie, x contains at least one fuzzy subset.

the contrapositive of this is that if x does not contain at least one fuzzy subset then there is no map from x onto P(x).

U=P(U) does contain a fuzzy element, namely the S in russell's 'paradox'.

a set is called fuzzy if there is a set whose membership value in that set is neither true nor false. otherwise the set is called crisp.

if you want version 10, then use the same link but replace the 9 with a 10.

[phoenixthoth]

i have version 10-2 which is the current version here:
http://www.alephnulldimension.net/matharticles/

it's not necessarily the first on the list but here's where each update will go. version 10-2 is 103kb and is pdf format.

the addition is this:

1. define [x] to be the set of z in U such that z~x (where z~x means there is a bijection from z to x).

2. the theorem is that for all nonempty x, [x]~U.

eg, there are an equal number of the following three things:
1. sets
2. sets that are singletons
3. sets in bijection with U

it seems like everything that once was a proper class, intuitively, is just now in bijection with U.

is it true that all proper classes are of the same size? well, since the cardinal number of each nonempty set is a proper class, that means i showed that all proper classes are in bijection with U (i think). i'm basing that on the assumption that if a class is in bijection with a set then the class is a set.

[phoenixthoth]

here is the current version of the universal set article that i'm trying to eventually get published:

http://www.alephnulldimension.net/matharticles/tuzfcver13.pdf

i've trimmed this down now to the essentials. i submitted it to the american mathematical monthly but i sent them a zipped file and they wanted pdf only. before they replied, i spotted something i needed to change so i changed it and here's the version after the change.

here's what happened since last you may have checked it out:
1. x&isin;x iff x=U. thus, U is a hyperset and U is the only hyperset. this is theorem 0 on page 7.

2. the main thing is that i noticed that if there is a universal set U and a subsets axiom that has the form {y&isin;x : A(y)} then one can take x to be U to get something that looks like this: {y:A(y)}. this is discussed on pages 4-6.

3. on page 8 there is a corollary which states this: if P(x), the powerset of x, contains no fuzzy sets then there is no function f that maps x onto P(x). (this is the contrapositive of a theorem which states that if there is a function that maps x onto P(x) then P(x) contains at least one fuzzy set.)

if anyone were to write a paper based on this, there are two directions i see it going. one is that with statement 2 above in mind, a bunch of other axioms then follow from the modified subsets axiom. just take {y:y=y} to be the universal set so the universal set follows from the modified subsets axiom. the pair set and powerset axiom also follow, i think, as well as perhaps others. another direction is to look into class theory and see if there is no need for classes all together (see last post on cardinal numbers). finally, an investigation into fuzzy sets would be interesting. are there as many sets as there are fuzzy sets? it seems to me that fuzzy sets would be rare or some such...

so criticism i can use would be greatly appreciated as i try to prepare this for submission. if you've been published yourself, any advice on how to go about doing it would also be immensely useful... for matt grime and hurkyl, i want to add you to the aknowledgements list so send me your real name iff you want to be aknowledged. i suppose i'll also plug the PF itself but that's iff it is publishable! i hope i'm not deluding myself or nothin...

[matt grime]

firstly i do not wish any mention of my name in the acknowledgements; it wouldn't do you any good, nor me, I am not a set theorist, and have no desire for my name to appear anywhere in that area.

secondly, you talk about thing like is it true all proper classes are aof the same size, well they don't have a size, except bigger than any cardinal. There can be no bijections between them because there are no functions between them (in ZF) as that would require them to be a set. you'd have to define functions without reference to sets and that would no longer be in ZF

[phoenixthoth]

i didn't know that's what aknowledgements were for, to do someone good (me if i acknowledged a big time set theorist or the aknowledgee). it was just the expression of the intent to recognize your effort to correct the theory. thanks for your contributions.

second, why can't you have a "class function" whose "domain" and "range" are proper classes? this isn't a big deal though because the article doesn't talk about proper classes though it does state that there is a bijection from the set of all nonempty sets having the same cardinality and the universal set.

[Russell E. Rierson]

According to string theory, the uncertainty in position is given by:

Dx < h/Dp + C*Dp

Which points towards a type of "discrete" spacetime?


Dx and Dp are the uncertainties in position and momentum, represented as probabiliuty distributions; h is Planck's constant and C is another constant related to the Planck scale.

There is a minimum size that can be probed in string theory. An absolute limit to the precision that any object can be located in space. Ergo, according to M-theory, space cannot be continuous; an infinite amount of information cannot be packed into a finite volume of space.

According to conventional theories, the surface area of the horizon surrounding a black hole, measures its entropy, where entropy is defined as a measure of the number of internal states that the black hole can be in without looking different to an outside observer, who must measure only mass, rotation, and charge. Another theory states that the maximum entropy of any closed region of space can never exceed one quarter of the area of the circumscribing surface, with the entropy being the measure of the total information contained by the system.

S' = S_m + A/4

So the "black hole" theorists came to realize that the information associated with all phenomena in the three dimensional world, can be stored on a two dimensional boundary, analogous to the storing of a holographic image.

The set of all dogs is itself "not" a dog. It is not a member of itself. Sets that are not members of themselves leads to a contradiction in the construction of a universal set. The "set of all sets" cannot exist under these limiting conditions.

A definition of "Algorithm":

http://education.yahoo.com/search/ref?p=algorithm



A step-by-step problem-solving procedure, especially an established, recursive computational procedure for solving a problem in a finite number of steps.



2^x = x is a recursion

2^x = x

then

2^[2^x] = x

2^[2^[2^x]] = x

2^[2^[2^[2^x]]] = x

etc.

DNA is also defined as an algorithm. A finite set? of instructions, a step by step problem solving procedure.

The information contained in DNA can construct a carbon based life form.

So the "DNA" contains the life form analogously to the way a blueprint contains a house.

The life form contains the DNA in the topological sense, while the DNA contains the life form in the "abstract" sense.

The Universal Algorithm contains the Universe in the abstract sense, while the Universe contains the algorithm in the topological sense.

[<-[->[U]<-]->]

The universal set.

The abstract contains the concrete and the concrete contains the abstract.

[phoenixthoth]

a quote from a random passerby reading my paper:

"To do this seriously, we need to develop both a proof theory and a model theory for it, and show soundness and completeness. Only then would we be ready to move on to ZFC, and see what the consequences were, of the change in logic."

anyone know good links to *Online* references in proof theory and model theory?

apparently, the system i "invented" was developed previously by kleene and his three-valued logic system. so maybe i can steal -- i mean reference -- his ideas regarding a model theory and a proof theory.

the passerby wants to use this three valued logic, including the new connective that generalizes iff, to see if the continuum hypothesis is decidable. my wild guess is that CH is true for crisp sets and false for fuzzy sets and i guess what i mean by that is partly that there might be a fuzzy set might have cardinality less than c but equal to aleph1. something like that. my main interest is to provide another avenue towards an absolute infinity; if i just knew the results thus far are sound then i could delve more into it...

[Hurkyl]

Nifty. I can't say I know any references off hand...