Is there a One True Set Theory?

Question

From the description of Category Theory in nlab:

Category theory is a structural approach to mathematics that can (through such methods as Lawvere's ETCS) provide foundations of mathematics and (through algebraic set theory) reproduce all the different axiomatic set theories; it does not need the concept of set to be formulated. Set theory is an analytic approach (element-wise) and can reproduce category theory by simply defining all the concepts in the usual way, as long as one include a technique to handle large categories (for instance by using classes instead of sets, or by including as an axiom that an uncountable inaccessible cardinal exists or even that Grothendieck universes exist).

That is the structural approach includes the analytic, and the analytic includes the structural. A little reflection shows that each theory includes images of itself and the other infinitely - somewhat like the mandelbrot set reproduces itself internally.

Given there are now two approaches to foundations, is it arguable that the 'true' Set Theory is something that is only represented in the two approaches? In the same way say that the number '9' is represented as nine or 1001?

Or is this an indication that there are in fact more than one Set Theory in the same way that denying the parallel postulate in non-euclidean geometry resulted in several different geometries: elliptic and spherical with the euclidean geometry occupying a special place because it is flat.

Certainly there is a similar picture in Topos Theory as a categorification of Set Theory (this is different to what is discussed above) where there are many Toposes but again the category of Sets occupy a unique place (I forget the characterisation of its uniqueness).

Answer 1

Or is this an indication that there are in fact more than one Set Theory in the same way that denying the parallel postulate in non-euclidean geometry resulted in several different geometries: elliptic and spherical with the euclidean geometry occupying a special place because it is flat

Yes, something like that: there are many set theories being studied, and each depends on what you want the "set" to be.

The problem with sets is that we use them in so many different contexts. So the question is: is the given set theory T appropriate for the given context? What we call "set" in one occasion might not have (almost) anything in common with the use in another occasion. Now, mathematics is about building systems that are (hopefully) internally consistent, without having to make external sense or correspond to any actual part of the nature (universe).

But as philosophers, we might ask ourselves: if this thing called set here and that thing called set there have so much in common (containing elements and such), might they be the same thing? If you're a Platonist (or at least math Platonist), then by Occam's razor, it seems reasonable to assume that this "set" thing is the one and the same. There might be some issues with talking about sets (the logics that we know of are full of problems, like second-order logic's incompleteness, and all our theories are based on those logics), but the idea you mentioned - representing of the same thing through different views - comes to rescue (different views give different ideas, perhaps sometimes even "untrue", but the essence is reflected in all views).

Personally, I'm not Platonist, and I don't think that sets exist in any form independent of us, and I do believe that in different contexts, "set" means different things, and I think there can hardly be any connection between those usages.

Answer 2

You have two questions here: one about the uniqueness of set theory, and another about foundations of mathematics.

• There is one naive set theory. Elements, unions, power sets, infinities and other operations and lots of small and big theorems, pretty much exactly what you want out of a set theory. But it doesn't hold up under even a little scrutiny (the collection of sets that aren't members of themselves, what are these things?) . And once you start to axiomatize it, you get questionable axioms (like the axiom of choice) and multiple semantic interpretations. The first situation is, as you note, kind of like Euclidean geometry being one of many geometry-like things. You choose which theory you want in order to help you with the things you want to talk about (from second order arithmetic all the way down to ultrafinitism) by either removing of modifying any of the usual axioms for ZFC. You can get a reasonable system AF (Anti-Foundation), which is ZFC with the negation of the axiom of Foundation. It gets you set-like things but with some other

• as to foundations, Set Theory and Category Theory are not the same kind of foundations. Set Theory is a foundations for provability for the sweep of mathematics, for how you know for any particular content (group theory or algebraic geometry) that you can reduce proofs in those areas to proofs in set theory. Category theory, on the other hand, is a foundation for concepts, for how concepts in one area can (or may not) look like those in another area. You can 'do' Set Theory in Category theory and vice versa, but the point is different in the different systems.

Answer 3

Given there are now two approaches to foundations, is it arguable that the 'true' Set Theory is something that is only represented in the two approaches? In the same way say that the number '9' is represented as nine or 1001?

Sticking with your example, we easily understand that differents representations of an object have differents properties. Thus, your three representations of the "same object" use respectively 1, 3 and 2 disnctints symbols.

Of course you don't want to mix up the object and its representations, but when you think about "truth of Foundations of mathematics" and "isomorphism between different foundation systems" you are manipulating representations. And you want to know if different representations provide equivalent proof system. That is if you can construct a representational object of an actual object in one system, you can also construct a representation of the same actual object in the other representation system, and that all propreties that you are interested in and that you can induce from one representation, you can also induce it from the supposed isomorph representation.

So your question could be reformulated like this:

• can we prove that all foundations of mathematics are isomorphics?
• if yes, can we call this isomorphic structure a "One True Theory"?

For the first question, one may documente on Gödel's incompleteness theorems which are "widely, but not universally, interpreted as showing [that finding] a complete and consistent set of axioms for all mathematics is impossible".

For the second, it's a matter of the personal criteria you apply to define truth.

I wasn't able to provide all relevant links I wanted to add, because I currently doesn't have enough "credit", but here are some useful keyword one interested on this question should seek for :

• Structural induction
• Proof system
• Consistency
• Completeness
• Soundness
• Gödel's incompleteness theorems

Answer 4

No, there is no true set theory (except for the finite parts). The most important and interesting feature of set theory is the existence of uncountable sets. But this existence is in contradiction with the axioms.

It can be proven that all definable elements of sets belong to a countable set. Therefore most elements of uncountable sets are undefinable. On the other hand Zermelo's axiom of extensionality says: "Axiom 1: ... . Or briefly: Every set is determined by its elements. (Axiom of determinatedness)" Since uncountable sets contain undefinable elements, the sets are undefinable too, hence not distinguishable and not existing.

But there is an even simpler contradiction of the actual infinity required for set theory. It is based on an argument applied frequently in set theory, already by Cantor in his first application of transfinite induction:

"If there were exceptions, then one of them was the smallest, call it a, such that the theorem was valid for all x < a but not for x =< a, in contradiction with the proof." [G. Cantor: "Beiträge zur Begründung der transfiniten Mengenlehre 2", Math. Annalen 49 (1897) pp. 207-246, § 18]

Why not apply it to the fact that no natural number is sufficient to make the set |N actually infinite?

Theorem: The sequence of natural numbers is not actually infinite.

Proof: The natural numbers 1, 2, ..., n do not produce an actually infinite set. If there were natural numbers capable of producing an actually infinite set, then one of them was the smallest, call it a, such that the theorem was valid for all x < a but not for x =< a. Contradiction.

Of course the natural numbers are potentially infinite. This cannot be disproved. With respect to this fact, proofs like the present one would be hilarious – and they have frequently been called so. But when the critics were silenced, then the meaning of infinity would be changed on the quiet and unnoticed from potential to actual. – A really perfid procedure.

That's why set theorists refuse to understand the difference between potential and actual infinity. Their standard procedure would become obvious.

There are many more contradictions collected in chapter VI of https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf