The search for absolute infinity

[Organic]

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

 

[phoenixthoth]

an odd result that makes sense

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→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→Q
M,T: P
--------
T: Q

This notation means that we know V(P→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→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→Q
T: P
--------
M: Q

or more generally, the deduction schema

X: P→Q
T: P
--------
X: Q

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

And other familiar rules, like:

T: P→Q
T: Q→R
--------
T: P→R

M: P→Q
M: Q→R
--------
M: P→R

T: P→Q
M: Q→R
--------
M,T: P→R

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

X: A→~A
--------
X: ~A

T: A→F
--------
F: A

T: P→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:

<br /> \forall a \exists x \forall y:<br /> (y \in_? x \leftrightarrow_+ (y \in_? a \wedge A(y)))<br />

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:

<br /> \begin{array}{c|c|c}<br /> A &amp; B &amp; A \circ B \\<br /> \hline<br /> T &amp; T &amp; T \\<br /> T &amp; M &amp; X \\<br /> T &amp; F &amp; X \\<br /> M &amp; T &amp; T \\<br /> M &amp; M &amp; T \\<br /> M &amp; F &amp; T \\<br /> F &amp; T &amp; T \\<br /> F &amp; M &amp; T \\<br /> F &amp; F &amp; T \\<br /> \end{array}<br />

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:

<br /> \begin{array}{ll}<br /> T: &amp; A \rightarrow \neg A \\<br /> \hline<br /> T: &amp; \neg A<br /> \end{array}<br />

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

<br /> \begin{array}{c|c|c|c}<br /> A &amp; \neg A &amp; A \rightarrow \neg A &amp; (A \rightarrow \neg A) \circ \neg A \\<br /> \hline<br /> T &amp; F &amp; F &amp; T \\<br /> M &amp; M &amp; M &amp; T \\<br /> F &amp; T &amp; T &amp; T \\<br /> \end{array}<br />

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):

<br /> \begin{array}{c|c|c|c|c}<br /> A &amp; B &amp; A \circ B &amp; B \circ A &amp; (A \circ B) \wedge (B \circ A) \\<br /> \hline<br /> T &amp; T &amp; T &amp; T &amp; T \\<br /> T &amp; M &amp; X &amp; T &amp; X \\<br /> T &amp; F &amp; X &amp; T &amp; X \\<br /> M &amp; T &amp; T &amp; X &amp; X \\<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; X &amp; X \\<br /> F &amp; M &amp; T &amp; T &amp; T \\<br /> F &amp; F &amp; T &amp; T &amp; T \\<br /> \end{array}<br />

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

<br /> \begin{array}{ll}<br /> T: &amp; (A \circ B) \wedge (B \circ A) \\<br /> T: &amp; A \\<br /> \hline<br /> T: &amp; B<br /> \end{array}<br />

<br /> \begin{array}{ll}<br /> T: &amp; (A \circ B) \wedge (B \circ A) \\<br /> F,M: &amp; A \\<br /> \hline<br /> F,M: &amp; B<br /> \end{array}<br />

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:

<br /> \forall a \exists x \forall y:<br /> (y \in_? x \leftrightarrow_+ (y \in_? a \wedge A(y)))<br />

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}<br /> y\in _{?}x &amp; y\in _{?}a\wedge A\left( y\right) &amp; y\in _{?}x\leftrightarrow_{+}y\in _{?}a\wedge A\left( y\right) \\ <br /> T &amp; T &amp; T \\ <br /> T &amp; M &amp; F \\ <br /> T &amp; F &amp; F \\ <br /> M &amp; T &amp; F \\ <br /> M &amp; M &amp; T \\ <br /> M &amp; F &amp; T \\ <br /> F &amp; T &amp; F \\ <br /> F &amp; M &amp; F \\ <br /> F &amp; F &amp; T<br /> \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.

 

[PFanalog57]

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.

 

[PFanalog57]

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]

nonstandard/universal set theory applied to set theory

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 existence 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.

 

[PFanalog57]

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

 

[PFanalog57]

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. accommodates 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.

 

[PFanalog57]

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}.

lim_n->Ux_n=x means that for all nonempty e, there is a set N such that x_n+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]x_n=U for all n and [/sub]x_n={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]x_n={0,1,...,n} if n&isin;N and [/sub]x_n=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 weird 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 weird 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| because 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 don't know if anything can be done with it.

 

[phoenixthoth]

lim_n->Ux_n=x means that for all nonempty e, there is a set N such that x_n+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: x_n 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 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

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 lim_n->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 lim_n->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 believe 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.