The search for absolute infinity

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The discussion focuses on the quest to axiomatize a universal set without contradictions, particularly addressing Russell's paradox. Two potential approaches are proposed: modifying the subsets axiom or adopting ternary logic to redefine axioms. The use of ternary logic suggests that certain tautologies in classical logic do not hold, allowing for a re-examination of foundational axioms like extensionality and subsets. The conversation also explores the implications of self-awareness in mathematical structures, particularly regarding a universal set, and how this relates to concepts of awareness and existence in a mathematical context. Ultimately, the thread examines the intersection of set theory, logic, and the nature of infinity.
  • #61
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.
 
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  • #62
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∪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∈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
 
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  • #63
  • #64
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.
 
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  • #65
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?
 

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  • #66
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)
 
  • #67
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."
 
  • #68
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.
 
  • #69
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.
 
  • #70
Right; it is a standard abuse of notation. Can't hurt being a little extra cautious about the details, though, especially given the circumstances. :wink:


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.
 
  • #71
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.
 
  • #72
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.
 
  • #73
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
 
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  • #74
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).

 
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  • #75
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.
 
  • #76
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.
 
  • #77
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" ...?
 
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  • #78
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).
 
  • #79
U as a pseudo-pseudo metric space:
define d to be a function from UxU to U (not just R) such that if x and y are sets, then
d(x,y)=x+y.

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

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

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

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

if cosine can be extended to hyperreal numbers, can it be extended to arbitary sets? perhaps an angle between sets, which would be a set like all other angles just not a real one, is possible. I'm guessing that the "angle" between sets will be super in multiplicity and not constant modulo some ideal. hmmm... maybe the angle should be some coset x+J where cos(J)=1=U and cos(x)=1. arccos can be defined in terms of logs which can potentially be defined as an inverse to the powerset operation. thought for food...
 
  • #80
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.
 
  • #81
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.
 
  • #82
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. :frown:
 
  • #83
limn->Uxn=x means that for all nonempty e, there is a set N such that xn+x<=e whenever n<=N.
made a mistake. i should have said N<=n. but i'll use the letter y or n' because of the possible confusion with the set of natural numbers.

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

i'm wondering why e has to be nonempty. if it is "for all nonempty e" then i can get limits whose symmetric difference is either empty (implying equality ), a 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.
 
  • #84
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 limn->Uf(n)=L iff there is an n' such that for all n, if n>=n' then f(n)=L.

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

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

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

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

is it known that boolean rings are noetherian? some, all, none?
 
  • #85
  • #86
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.
 
  • #87
...{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.
 
  • #88
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.
 
  • #89
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.
 
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  • #90
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.
 

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