It seems as though many consider ZF to be the foundational set of axioms for all of mathematics (or at least, a crucial part of the foundations); when a theorem is found to be independent of ZF, it's generally accepted that there will never be a proof of the theorem one way or another. My question is, why is this? It seems as though ZF is flawed in a number of ways, since propositions like the axiom of choice and the continuum hypothesis are independent of it. Shouldn't our axioms of set theory be able to give us firm answers to questions like these? Yes, the incompleteness theorems say that we'll never develop a perfect set of axioms, and many of the theorems independent of our axioms will probably be quite interesting, but is ZF really the best we can do? Is there hard evidence that ZF is the "best" set theory we can come up with, or is it merely a philosophical argument that ZF is what set theory "should" look like?
[Math] The Importance of ZF
mathematical-philosophyset-theorysoft-question
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I would like to question two statements you make because they paint an oversimplified picture, which unfortunately is alluring to mathematicians who do not want to think about foundations (and they should not be blamed for it anymore than I should be blamed for not wanting to think about PDEs).
"Most mathematicians accept as given the ZFC (or at least ZF) axioms for sets." This is what mathematicians say, but most cannot even tell you what ZFC is. Mathematicians work at a more intuitive and informal manner. High party officials once declared that ZFC was being used by everyone, so it has become the party line. But if you read a random text of mathematics, it will be equally easy to interpret it in other kinds of foundations, such as type theory, bounded Zermelo set theory, etc. They do not use the language of ZFC. The language of ZFC is completely unusable for the working mathematician, as it only has a single relation symbol $\in$. As soon as you allow in abbreviations, your exposition becomes expressible more naturally in other formal systems that actually handle abbreviations formally. Informal mathematics is informal, and thankfully, it does not require any foundation to function, just like people do not need an ideology to think. If you doubt that, you have to doubt all mathematics that happened before late 19th century.
"They [logicians] realize that in order to talk of logic they don't need the full power of set theory, so they take logic as God-given instead." I do not know of any logicians, and I know many, who would say that logic is "God-given", or anything like that. I do not think logicians are born into a life rich with the "full power of set theory" which they throw away in order to become ascetic first-order logicians. That is a nice philosophical story detached from reality. The logicians I know are usually quite careful, skeptical, and inquisitive about foundational issues, reflect carefully on their own experiences, and almost never give you a straight answer when you ask "where does logic come from?" Your view is naive and inaccurate, if not slightly demeaning.
If I understand your question correctly, you are asking whether there is a difference between the following two views:
We start with naive set theory and on top of it we formalize set theory.
We start with first-order logic and immediately formalize set theory.
Well, we are proceeding from two different meta-theories. The first one allows us a wide spectrum of semantic methods. We can refer to "the standard model of Peano arithmetic" because we "believe in natural numbers", and we can invent Tarskian model theory without worrying where it came from.
The second method is more restricted. It will lead to syntactic and proof-theoretic methods, since the only thing we have given ourselves initially are syntactic in nature, namely first-order theories. There will be careful analysis of syntax. For advanced methods, however, we will typically resort to at least some amount of "naive mathematics". Ordinals will come into play, it will be hard to live without completeness theorems (which involve semantics), etc.
However, this is not how real life works. The dilemma you present is not really there. A working mathematician does not concern himself with these issues, anyhow, while a logician will likely refuse to be categorized as one or the other breed.
That is my guess, based on the experience that my fellow logicians are complicated animals and it is hard to get to the bottom of their foundational guts.
For the philosophical point encapsulated in (*) in the question, it seems that corollaries of the second incompleteness theorem are more relevant than the theorem itself. If we had doubts about the consistency of ZFC, then a proof of Con(ZFC) carried out in ZFC would indeed be of little use. But a proof of Con(ZFC) carried out in a more reliable system, like Peano arithmetic or primitive recursive arithmetic, would (before Gödel) have been useful, and I think this is what Hilbert was hoping for. Gödel's second incompleteness theorem tells us that this sort of thing can't happen (unless even the more reliable system is inconsistent).
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Very few mathematicions these days wish to base their mathematics on ZF without the axiom of choice, as your question seems to imply. Yes, there is an intuitionist school about which I don't know a whole lot, but they seem to be quite the minority. So let's consider ZFC. I think the main philosophical argument for it is that the axioms seem obviuosly true, if you are willing to believe that such things as infinite sets exist in some fashion, and that nobody has been able to come up with an equally obviously “true” statement about sets that is not a consequence of these axioms. The independence of the continuum hypothesis doesn't seem to bother people much, since to most of us it doesn't seem either obviously true or obviously false. Though some people are working on settling it by finding other axioms that will at least be considered likely true or at least useful. There was a recent article in the Notices about these efforts, in fact. But is there “hard evidence” that ZFC is the best we can come up with? Not by most mathematicians' standards I think, but there seems to be plenty of soft evidence.
I might add that many working mathematicians (I am talking here mostly about analysts, since they are the people I know best) don't care one whit about these questions, but happily go about their business using Zorn's lemma whenever it seems necessary and never let it bother them. And there are some who would rather base all mathematics on category theory rather than ZFC set theory, but that is not my cup of tea, so I will leave it unstirred.