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}<br /> A &amp; B &amp; \symbol{126}A &amp; A\vee B &amp; A\wedge B &amp; A\rightarrow B &amp; <br /> A\leftrightarrow B &amp; \left( A\wedge \left( A\rightarrow B\right) \right)<br /> \rightarrow B \\ <br /> T &amp; T &amp; F &amp; T &amp; T &amp; T &amp; T &amp; T \\ <br /> T &amp; M &amp; F &amp; T &amp; M &amp; M &amp; M &amp; M \\ <br /> T &amp; F &amp; F &amp; T &amp; F &amp; F &amp; F &amp; T \\ <br /> M &amp; T &amp; M &amp; T &amp; M &amp; T &amp; M &amp; T \\ <br /> M &amp; M &amp; M &amp; M &amp; M &amp; M &amp; M &amp; M \\ <br /> M &amp; F &amp; M &amp; M &amp; F &amp; M &amp; M &amp; M \\ <br /> F &amp; T &amp; T &amp; T &amp; F &amp; T &amp; F &amp; T \\ <br /> F &amp; M &amp; T &amp; M &amp; F &amp; T &amp; M &amp; T \\ <br /> F &amp; F &amp; T &amp; F &amp; F &amp; T &amp; T &amp; T<br /> \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)<br /> =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<br /> 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(<br /> y\right) \right)

axiom of subsets version 2:

V\left( \exists x\forall yV\left( y\in x\leftrightarrow y\in a\wedge<br /> 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<br /> 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<br /> z\right) \right)

axiom of the sum set version 2:

V\left( \exists x\forall y\left( V\left( y\in x\leftrightarrow<br /> \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<br /> x\right) \right)

axiom of the power set version 2:

V\left( \forall x\exists y\left( \forall zV\left( z\in<br /> 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)<br /> \right)

axiom of infinity version 2:

V\left( \exists x\left( V\left( \O \in x\wedge \forall y\in<br /> 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<br /> \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(<br /> y,z\right) \rightarrow \exists z\in xA\left( y,z\right) \right) =T\right)<br /> \right) =T



10. axiom of foundation/regularity:

\exists xA\left( x\right) \rightarrow \exists x\left( A\left( x\right)<br /> \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(<br /> V\left( A\left( x\right) \wedge \forall y\in x\left( !A\left( y\right)<br /> \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<br /> 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<br /> \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]

self-aware structures

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:

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]

changes

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.

 

[PFanalog57]

[ 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.

Langauge 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.

 

[PFanalog57]

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 possesses 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]

restatement plus a bit more

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 <br /> 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:

<br /> \begin{array}{cccccccc}<br /> A &amp; B &amp; A\rightarrow B &amp; A\rightarrow _{T}B &amp; A\leftrightarrow B &amp; <br /> A\leftrightarrow _{T}B &amp; A\leftrightarrow _{=}B &amp; A\leftrightarrow _{\neq F}B<br /> \\ <br /> T &amp; T &amp; T &amp; T &amp; T &amp; T &amp; T &amp; T \\ <br /> T &amp; M &amp; M &amp; F &amp; M &amp; F &amp; F &amp; T \\ <br /> T &amp; F &amp; F &amp; F &amp; F &amp; F &amp; F &amp; F \\ <br /> M &amp; T &amp; T &amp; T &amp; M &amp; F &amp; F &amp; T \\ <br /> M &amp; M &amp; M &amp; F &amp; M &amp; F &amp; T &amp; T \\ <br /> M &amp; F &amp; M &amp; F &amp; M &amp; F &amp; F &amp; T \\ <br /> F &amp; T &amp; T &amp; T &amp; F &amp; F &amp; F &amp; F \\ <br /> F &amp; M &amp; T &amp; T &amp; M &amp; F &amp; F &amp; T \\ <br /> F &amp; F &amp; T &amp; T &amp; T &amp; T &amp; T &amp; T<br /> \end{array}<br />

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<br /> 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<br /> 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<br /> _{T}z\subset x\right)

axiom of the power set version 3:
\forall x\exists y\forall z\left( z\in y\leftrightarrow<br /> _{=}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)<br /> \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(<br /> 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<br /> _{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\{<br /> 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<br /> _{?}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.

 

[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 don't know 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]

another attempt

second truth table (REPEAT):

<br /> \begin{array}{cccccccc}<br /> A &amp; B &amp; A\rightarrow B &amp; A\rightarrow _{T}B &amp; A\leftrightarrow B &amp; <br /> A\leftrightarrow _{T}B &amp; A\leftrightarrow _{=}B &amp; A\leftrightarrow _{\neq F}B<br /> \\ <br /> T &amp; T &amp; T &amp; T &amp; T &amp; T &amp; T &amp; T \\ <br /> T &amp; M &amp; M &amp; F &amp; M &amp; F &amp; F &amp; T \\ <br /> T &amp; F &amp; F &amp; F &amp; F &amp; F &amp; F &amp; F \\ <br /> M &amp; T &amp; T &amp; T &amp; M &amp; F &amp; F &amp; T \\ <br /> M &amp; M &amp; M &amp; F &amp; M &amp; F &amp; T &amp; T \\ <br /> M &amp; F &amp; M &amp; F &amp; M &amp; F &amp; F &amp; T \\ <br /> F &amp; T &amp; T &amp; T &amp; F &amp; F &amp; F &amp; F \\ <br /> F &amp; M &amp; T &amp; T &amp; M &amp; F &amp; F &amp; T \\ <br /> F &amp; F &amp; T &amp; T &amp; T &amp; T &amp; T &amp; T<br /> \end{array}<br />
define a new conditional:
<br /> \begin{array}{ccccc}<br /> A &amp; B &amp; A\rightarrow _{+}B &amp; A\leftrightarrow _{+}B &amp; A\leftrightarrow _{+}<br /> \symbol{126}A \\ <br /> T &amp; T &amp; T &amp; T &amp; F \\ <br /> T &amp; M &amp; T &amp; T &amp; F \\ <br /> T &amp; F &amp; F &amp; F &amp; F \\ <br /> M &amp; T &amp; T &amp; T &amp; T \\ <br /> M &amp; M &amp; T &amp; T &amp; T \\ <br /> M &amp; F &amp; T &amp; T &amp; T \\ <br /> F &amp; T &amp; T &amp; F &amp; F \\ <br /> F &amp; M &amp; T &amp; T &amp; F \\ <br /> F &amp; F &amp; T &amp; T &amp; F<br /> \end{array}<br />

the axioms of subsets would read this:

axiom of subsets version 6:

\exists x\forall y\left( y\in _{?}x\leftrightarrow _{+}y\in<br /> _{?}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:

<br /> \begin{array}{ccc}<br /> S\in _{?}S &amp; S\notin S &amp; S\in _{?}S\leftrightarrow _{+}S\notin S \\ <br /> T &amp; F &amp; F \\ <br /> M &amp; F &amp; T \\ <br /> F &amp; T &amp; F<br /> \end{array}<br />

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 <br /> 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]

summary of current work

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.