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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).

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    There might be a one true set theory, but it has the problem that it doesn't exist. This is similar to infinity, which often doesn't exist either. Mar 13, 2013 at 7:39
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    I've never been able to understand why people took seriously the project determining whether e.g. the continuum hypothesis is "really really" true, given that it's been proven independent of ZFC. If one cares about ZFC and also CH at all, it's obvious that there is more than one set theory. Perhaps one is interested then in exploring alternative set theories, and that's fine. More interestingly, perhaps eventually people will change their mind about what should qualify as a set theory. But until that happens, I don't know what it would mean for there to be a One True Set Theory. Mar 13, 2013 at 10:57
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    @NieldeBeaudrap "If one cares about ZFC and also CH at all, it's obvious that there is more than one set theory." It's obvious that there is more than one set theory, but not obvious that there is more than one _true_/correct set theory. You could take the failure of ZFC to establish (or refute) CH to be evidence that our axiomatization of ZFC isn't complete.
    – Dennis
    Mar 13, 2013 at 13:13
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    @NieldeBeaudrap:I don't think there is a One True Set Theory for similar reasons as you've outlined as well as the reasons in the question. Though I admit it may be a motivating factor in research. Joel Hamkins pointed out in an answer in Math.Overflow that there was a large contingent of Set Theorists that do hold this view. But what exactly they mean by this I'm not sure. It seems to have something to do with the Large Cardinal hierarchy. Mar 13, 2013 at 16:09
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    You might also look at Penelope Maddy's two part article "Believing in the Axioms" (part 1 and part 2). In these papers she gives a number of reasons to believe in the axioms (many drawing from remarks Godel made).
    – Dennis
    Mar 13, 2013 at 17:44

6 Answers 6

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Since this question was recently bumped up to the top page, here is my 2¢. When one mentions "set theory" one typically refers to the Zermelo-Fraenkel (ZF) axioms for set theory. If any set theory is going to be proposed as "the" set theory, it would likely be ZF. Now an important point often overlooked by adherents of such a view is that Abraham (formerly Adolf) Fraenkel himself went to great length in his set theory textbooks to provide numerous alternative possibilities for axiomatic foundations for the discipline, and never referred to ZF as "the" set theory.

In the 1960s, Fraenkel's student Abraham Robinson developed mathematical foundations for infinitesimal analysis, and Fraenkel in his memoirs expressed admiration for this accomplishment (see Ehrlich's appreciation here). In the 1970s, Ed Nelson developed an axiomatic foundation enriching ZFC which remains conservative over it, called IST (Internal Set Theory). This foundation provides a more faithful formalisation of the historical discipline of mathematical analysis because it includes a formalisation of Leibniz's distinction between assignable and inassignable quantities, crucial for Leibnizian infinitesimal analysis.

Arguably, IST is a better foundation than ZFC; at any rate, recent developements in category theory, as well as developments in constructive mathematics, have pretty much done away with the idea that there should be a unique set-theoretic foundation for mathematics, and provided support for foundational pluralism.

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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.

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If we use negative thinking, then we can see that set theory is actually 0-category theory; set theory is decategorified category theory. In this sense, there is no circularity; sets are special cases of categories.

To answer your title question more directly, there is no single best model of set theory, by the Löwenheim–Skolem theorem.

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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.

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  • Before you axiomatize it you have Russel's paradox, so there is not one naive set theory either, there are zero consistent versions of naive set theory.
    – user9166
    Feb 23, 2018 at 21:37
  • You get questionable assumptions and multiple meanings in naive set theory already, and paradoxes on top of that. Russell's and Zermelo's ways of resolving paradoxes are so different that it makes more sense to say that there are different naive conceptions of sets already. For instance, platonist one, where elements are given in advance, and intensionalist one, where they can be added indefinitely according to formation rules. Indeed, the axiom of choice controversy was generated by preconceptions about what sets are supposed to be, not by formalization, Zermelo's original proof was informal.
    – Conifold
    Feb 23, 2018 at 22:50
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Suppose that monism and pluralism are theories on the appropriate level of use of the word "theories," here. Now, if monism is not true, then it seems as if pluralism is true. But if pluralism is true, and is a theory, then it is a true theory. But then even if pluralism is true, so is monism (since monism is a theory and pluralism would have it that all theories are true). Then suppose instead that pluralism is false; then it seems as if monism is true, which is the same conclusion, it seems, that we just came to on the opposite assumption. Since monism seems to be true even if assumed false at first, then even if pluralism is true, it is not absolutely true. Alternatively, when pressed too hard, the distinction between monism and pluralism about set theory evaporates, "and we are done!"

Now, all that is rather simple-minded reasoning, however. The more interesting reality is that the "one true set theory" as such is trivial; but there are many nontrivial, and true, set theories. What this means is that there is a pluralism of questions with nontrivial answers, answers provided by a plurality of sets/types of axioms, and there are almost no nontrivial, privileged axioms at all, for there are almost no privileged questions, here, at all, either, but we are free to ask what questions we please (e.g. "What happens if there are only two measurable cardinals?" or, "What happens if the Continuum is forced to be the fourth aleph?").


Pro-pluralism references:

  1. Hamkins[11], "The set-theoretic multiverse"
  2. Hamkins[12], "Is the dream solution to the Continuum Hypothesis attainable?"
  3. Hamkins[22], "Nonlinearity and ill-foundedness in the hierarchy of large cardinal consistency strength"
  4. Zalta[23], "Mathematical pluralism"

Pro-monism references:

  1. Koellner[09], "Truth in Mathematics: The Question of Pluralism" (not absolutely monistic!)
  2. Koellner[13], "Large Cardinals and Determinacy" (specifically §1.3)
  3. Levy[??], "The price of mathematical skepticism"

See also Barton[21], "Indeterminateness and ‘The’ Universe of Sets: Multiversism, Potentialism, and Pluralism."

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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
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    I feel you don't take OP's main theme into account here, involving one theory being expressed within another. OP's main question has the problem of talking about true set theory, which e.g. implies that statements about infinite sets can be true and this alone is debatable. So a) OP's question might not have an answer, while b) your interpretation of it can be answered negatively: Depending on the underlying formal logics of two set theories (and there are more than one logic) these theories will not agree on truth - even if they are strong enough to express the other logical calculus.
    – Nikolaj-K
    Jul 11, 2013 at 10:46
  • Call it metalogic if you want, but all logics have some common attribute. It's always about expliciting premisses and rules and how you combine them to reach some statement. Jul 11, 2013 at 13:23
  • What do you want to say? Different logics still make certain statements either true or not and so there is a difference and they can't be isomorphic.
    – Nikolaj-K
    Jul 11, 2013 at 13:42
  • I didn't mean to give "my interpretation", but pointing to relevant concepts the OP may use to feed its reflection. The answer will largely depend on what meaning is given to "true". Logics, whichever you chose, at best will tell if a statement is valid according to the chosen logic. Whether the statement is true is depending on whether a valid statement of this logic is considered a true statement. Jul 11, 2013 at 14:56
  • Exactly, and this is why two foundational theories, one build on classical logic and the other say on intuitionistic logic, are not isomorphic: While they might both be able to tell you what the other theory will say about a given statement, they don't themselves agree on certain given statements, and so this answers your first question in the negative.
    – Nikolaj-K
    Jul 11, 2013 at 15:19

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