The search for absolute infinity

[matt grime]

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


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

 

[phoenixthoth]

thank you.

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

 

[phoenixthoth]

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

 

[matt grime]

I said above

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

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

 

[phoenixthoth]

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

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

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

 

[phoenixthoth]

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

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

 

[phoenixthoth]

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

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

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

 

[matt grime]

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

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

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

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

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

 

[phoenixthoth]

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

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

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

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

thanks again for your patient feedback.

 

[matt grime]

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

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

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

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

 

[phoenixthoth]

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

well, so much for theorem 3!

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

 

[matt grime]

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

 

[phoenixthoth]

LOL

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

 

[matt grime]

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

 

[phoenixthoth]

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

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

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

 

[matt grime]

Not hankering for a co-authorship in the slightest.

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

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

 

[phoenixthoth]

of course! cantor-schroeder!

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

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

 

[phoenixthoth]

version 9

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

any thoughts?

 

[matt grime]

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

 

[phoenixthoth]

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

in bijection with.

 

[matt grime]

Here is some stuff for you then:

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

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

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

 

[phoenixthoth]

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

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

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

in tuzfc, U passes this test for setness because of theorem 3 on page 5.

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

i want to reiterate that I'm not saying ZF allows for U. I'm saying what you know, i think, which is that one must change things around in order to have a U. also, since P(x)=U iff x=U implies that U cannot be arrived at from a lessor cardinal, proving that it is inaccessible in some sense. hence certain other axioms about inaccessible cardinals may be derivable from my system. however, proofs of relative consistency are way beyond my current understanding; for one thing, it would have to involve passage from a binary logic to ternary logic and/or a combination of the two.

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

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

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

 

[matt grime]

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

 

[phoenixthoth]

appearances can be decieving! ;P

 

[phoenixthoth]

any thoughts on how U might interact with ultrafilters?

 

[phoenixthoth]

i have a new version 10 with the following theorem:

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

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

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

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

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

 

[phoenixthoth]

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

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

the addition is this:

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

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

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

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

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

 

[phoenixthoth]

current version

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

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

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

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

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

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

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

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

 

[matt grime]

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

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

 

[phoenixthoth]

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

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

 

[PFanalog57]

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

Dx < h/Dp + C*Dp

Which points towards a type of "discrete" spacetime?


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

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

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

S' = S_m + A/4

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

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

A definition of "Algorithm":

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

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

2^x = x is a recursion

2^x = x

then

2^[2^x] = x

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

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

etc.

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

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

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

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

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

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

The universal set.

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

 

[phoenixthoth]

not this again!

a quote from a random passerby reading my paper:

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

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

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

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

 

[Hurkyl]

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