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In what sense is set theory a foundation for mathematics? To my mind (for what that is worth), there are at least three (somewhat) distinct senses in which set 'theory' (I put "theory" in scare quotes to indicate that the notion of being able to reduce the language of mathematics to the language of sets may not imply that there is a background theory of sets for mathematics) might be regarded as a 'foundation' for mathematics:

i) as a 'universal language' in which all of mathematics may be expressed (i.e. to which all of mathematics may be reduced), and, having that

ii) the class of all true sentences of that language (on the order of "true elementary number theory")

iii) a formal axiomatic theory such as ZFC, Kelly-Morse, etc.

By the incompleteness theorems, iii) seems not to be a viable candidate for such a foundation, and by Tarski's theorem regarding the definition of truth in formalized languages, ii) seems not to be a viable candidate, either. That seems to leave i) as the viable candidate, but on the other hand, that seems too little (after all, naive set theory, being inconsistent, derives all well-formed formulas (of its formal language) so the unrestricted Axiom of Comprehension may more properly be considered a formation rule, but then the naive set theory known as ideal set theory would properly consist of just one axiom, Extensionality, and that seems too little, too).

So, to ask in another, slightly different, way; in what sense should set theory be considered a foundation for mathematics?

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I don't understand what you mean in (ii) and (iii). What is the basis for truth in (ii)? In what sense do the incompleteness theorems make (iii) not viable? –  François G. Dorais May 25 '13 at 14:26
The incompleteness theorems are a red herring here: using set theory as a foundation for mathematics doesn't require having a complete axiomatization of set theory. In practice everyone means (i) and (iii), with no assumption of completeness. If you're a Platonist then (ii) would be nice, but of course we can't actually achieve that. –  Henry Cohn May 25 '13 at 14:32
I think I see where you come from. Perhaps you need to ask yourself a background question first: What do foundations have to do with truth? (Bringing set theory into the picture just confuses the subject matter.) –  François G. Dorais May 25 '13 at 14:33
If you're a Platonist, then you should already believe in one of the other truth values of CH. If you're not a Platonist, why does independence bother you? We can develop mathematics in any model of ZFC, that's the beauty of it. Much like you can say that $2$ has zero, one, or three cubic roots, depending on your field -- then which frame deserves to be called "numbers", if something like this is so... unprovable from a basic system of numbers (read: a field)? –  Asaf Karagila May 25 '13 at 21:12
Meta discussion - tea.mathoverflow.net/discussion/1596 –  François G. Dorais May 25 '13 at 21:41
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2 Answers

I think your object is misplaced. Set Theory is no more a foundation for mathematics than musical notation is for music. You can use set theory as a basis for formalizing notions and mechanizing certain fragments of mathematics, just as you can use notation and certain semantics behind it to communicate certain compositions. The creation, performance, and communication of mathematics (and of music) is a human activity that is more than just the data and collection of knowledge amassed from that activity. The choice of set theory or bars and staffs to aid in the communication is an arbitrary choice of medium, as is topoi or waveforms/.wav files.

Gerhard "Medium And Message Matter Minimally" Paseman, 2013.05.25

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When talking about “foundations of mathematics” you usually do not include all aspects of mathematical intercourse, aesthetics etc. –  The User May 25 '13 at 21:19
Indeed, but neither the title nor question use 'Foundations Of Mathematics', a phrase which over the years has come to represent the search for a philosophically justifiable presentation which gives as much satisfaction as mechanization and repeatability. I am addressing the question about foundations for mathematics. Leaving aesthetics and other aspects aside, my point remains that choosing such foundations is arbitrary, but also influential. Gerhard "Like The Choice Of Title" Paseman, 2013.05.25 –  Gerhard Paseman May 25 '13 at 21:27
Formal methods provide an essence of what is regarded “mathematical”. If your arguments do not fit into a formal framework, it is philosophy, informal speculation, heuristic calculations for physicists, but it does not belong to the essence of mathematics. Contrarily, musical notations are definitively not essential to music, in no way. Sounds are essential for music and (semi)formal proofs are essential for mathematics. –  The User May 25 '13 at 21:36
I think I can argue that proofs are not essential for mathematics, but I don't think I can do so effectively in a comment box. (Or can I?) I suspect the best I can do at present is convince The User that we mean different things by the word 'Mathematics'. Gerhard "Proofs: Crutches For The Unimaginative" Paseman, 2013.05.25 –  Gerhard Paseman May 25 '13 at 21:53
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iii is very viable, in my opinion. Sometimes you might feel sad about the implications of the incompleteness theorem and it is not as perfect as the formalists expected it to be before Gödel. But nevertheless a formal system like predicate calculus+ZFC provides a viable foundation for mathematics. You state all your theorems in such a way that everybody will believe you that they could be translated into the formal language of ZFC and you give proofs such that the arguments used can be interpreted as applications of the axioms or rules of inference. I think that this conception is very important and mathematics without such foundations are very questionable. Mathematicians should have formal foundations in mind. It is an essential property of modern mathematics that there are formal rules defining what is a proof and what is not a proof. You do not have to care about the incompleteness theorems if you want to prove theorems.

Well, sometimes you get into trouble with your foundations. However, you can choose between many different foundations. You need some large cardinals/Grothendieck universes? You want to use the generalised continuum hypothesis? It is fine, just use it and mention which axiomatic system you are using. And the work you have done in a different system can be transferred using various theorems. Of course these theorems (like theorems regarding provability etc.) are metalingual theorems. However, also your metalanguage should be formal. If you want to prove advanced theorems in logic, you should do that in a formal system like ZFC for the same reason you are using this approach for other theorems: You do not want to do handwaving, but you want to have a well-defined notion of a proof. Thus set theory also provides foundations for metamathematics. You might want to argue that the common ground of all mathematics is a finitistic system allowing you to speak about proofs. However, this does not make set theory less important, it is just an additional, philosophically motivated layer.

ii is not viable in my opinion. What the hell is a “true sentence”? You cannot define that in any reasonable way. The essential notion is the notion of a proof, not the notion of truth. Truth gets important when doing model theory, but for building the foundations it is not needed at all.

i is just a very weak statement. A “universal language” is nice, but we want to have a better language than English. Thus we go for formal systems, leading to iii.

Notice, that I am not talking about set theory in particular. These considerations also include e. g. type theory, but you asked for set theory and set theoretical foundations are the most popular ones.

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@TheUser You asked "What the hell is a “true sentence”?". I can give some examples. "5 is a prime number" is a true sentence. "$\int_{-\infty}^\infty e^{-x^2}dx=\sqrt\pi$" is a true sentence. "There are arbitrarily long arithmetic progressions consisting of prime numbers" is a true sentence. If your axioms imply that these are false statements, then you axioms are false. –  Vladimir Reshetnikov May 25 '13 at 16:22
mathoverflow.net/questions/24350/… contains some very good answers. –  TauMu May 25 '13 at 16:31
@Vladimir But for doing mathematics it is only important that you can prove them. Then you also say that they are true, but you do not have to define truth to do mathematics. –  The User May 25 '13 at 16:32
@TheUser Selecting true axioms to work with is equally important for mathematics. Mathematics would be useless if it did not produce true results. Usually you do not have to select your axioms every day, so you can pretend for some period of time that you do not care about truth (or "do not know" what it is) and just do some string manipulation and even get some great results this way. But if somebody claims he has found an error in your string manipulations and you have to check if this claim is true or not, you immediately get back your understanding of what true means. –  Vladimir Reshetnikov May 25 '13 at 17:35
@TheUser You have to at least understand what is finitistic truth to check your proofs, to make sure your string manipulations comply with the rules. And sometimes your proofs are meta-proofs: you prove $\vdash A$ rather than $A$ itself and then rely on truth of $\vdash A$. And I do not think all your proofs are completely formalized - you rely on meaning of things you reason about and intuition to justify some steps - not to replace rigour with intuition, but to convince yourself and others that a rigorous proof of some step is possible in principle, but without actually producing the proof. –  Vladimir Reshetnikov May 26 '13 at 17:42
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