Foundation for analysis without axiom of choice?
An interesting question, that would take many pages to begin to answer! We make a small disjointed series of comments.
In the last few years, there has been a systematic program, initiated by Friedman and usually called Reverse Mathematics, to discover precisely how much we need to prove various theorems. The rough answer is that for many important things, we need very much less than ZFC. For many things, full ZF is vast overkill. Small fragments of second-order arithmetic, together with very limited versions of AC, are often enough.
About the Axiom of Choice, for a fair bit of basic analysis, it is pleasant to have Countable Choice, or Dependent Choice, at least for some kinds of sets. We really want, for example, sequential continuity in the reals to be equivalent to continuity. One could do this without full DC, but DC sounds not unreasonable to many people who have some discomfort with the full AC. This was amusingly illustrated in the early $20$-th century. A number of mathematicians who had publicly objected to AC turned out to have unwittingly used some form of AC in their published work.
Next, bases. For finite dimensional vector spaces, there is no problem, we do not need any form of AC (though amusingly we do to prove that the Dedekind definition of finiteness is equivalent to the usual definition.)
For some infinite dimensional vector spaces, we cannot prove the existence of a basis in ZF (I guess I have to add the usual caveat "if ZF is consistent"). However, an algebraic basis is not usually what we need in analysis. For example, we often express nice functions as $\sum_0^\infty a_nx^n$. This is an infinite "sum." The same remark can be made about Fourier series. True, we would use an algebraic basis for $\mathbb{R}$ over $\mathbb{Q}$ to show that there are strange solutions to the functional equation $f(x+y)=f(x)+f(y)$. But are these strange solutions of any conceivable use in Physics?
Finally, why should the Banach-Tarski result be unacceptable to a physicist as physicist? It is easy to show that the sets in the decomposition cannot be all measurable. In mathematical models of physical situations, do non-measurable sets of points in $\mathbb{R}^3$ ever appear?
Blass's theorem is a very strong one indeed. If the axiom of choice does not hold then there is a vector space without a basis. It is unusual to be able and tell which vector space it is (unless assuming more, or constructing the model directly).
In particular, finite dimensional vector spaces always have a basis, since such basis is finite and thus completely definable in the universe.
Most of the basic analysis would require the axiom of countable choice, or the axiom of dependent choice. Both would be enough for almost every theorem you learn in basic calculus class - but neither is enough for Banach-Tarski. You may wish to add something like the ultrafilter lemma, however once there is a free ultrafilter over $\mathbb N$ there are unmeasurable sets - if that would bother you.
In general to prove that a space has a basis may require some choice, for example $\mathbb R$ as a vector space over $\mathbb Q$ requires choice. If however your interest is in finitely dimensional vector spaces then you can relax, since those would be fine regardless to the axiom of choice. There are infinitely dimensional spaces which have explicit basis as well, for example all the infinite sequences which are eventually zero.
Once you go beyond that it becomes harder and harder to produce a basis without the axiom of choice, but your needs might not go that far.
By Solovay's theorem I suppose you mean his model in which every set is measurable, and such. This is irrelevant, and in fact it holds a horrible secret:
In Solovay's model we can cut $\mathbb R$ into more parts then it has elements. Namely, we can cut $\mathbb R$ into non-empty parts and have more parts than real numbers. This sort of partition might sound very bizarre and pathological, much like the Banach-Tarski paradox. However such partitions can be a handful in some parts of set theory.
You may want to think that you're screwed basing yourself on ZF either way, but the problem is that mathematics almost always have this way to sting you in the back, no matter how you put it. You can simply put "new limitations" (e.g. limit yourself to measurable sets, which still gives you a rich and fulfilling world) and just use mathematics as you first wanted.
One of the least known facts about choice is that the definitions of continuity: $\epsilon-\delta$ and sequential continuity are not equivalent without some choice. If you have used this before, you have used the axiom of choice.
The point above is that the axiom of choice simply allows us to control many infinitary processes in a very simple way. While physics itself does not really talk about infinite processes (at least not as far as I know) you should be able to get away from that if you ditch the axiom of choice. However you may want to keep enough of it to ensure that what you have approximated with finite parts is continuously carried over to the limit point. This is in its essence the principle of dependent choice (and to a lesser extent the axiom of countable choice).
Something to Read:
- Finite dimensional subspaces of a linear space
- Finite choice without AC
- Why worry about the axiom of choice?
- Does the Axiom of Choice (or any other “optional” set theory axiom) have real-world consequences? [closed]
This answer consists basically of two remarks.
First: If you do find Banach-Tarski unacceptable and want to be sure that you live in a world without such decompositions, dropping the axiom of choice is not enough. Since the axiom of choice is consistent, you cannot prove that there is no Banach-Tarski decomposition lurking somewhere. What you need is a theory in which you assume something that blatantly contradicts the axiom of choice.
There are axiom systems that do that and give you a number of "pleasant" consequences. For example assuming ZF+Dependent Choice+Every set of real numbers has the Baire property, leads to several convenient results. For example, you can then show that every two complete norms on the same vector space give you the same topology. You can find more such results in this wonderful book.
The downside of making such an assumption is that it is ot so clear how to view the corresponding set theoretic universe. The axiom of choice seems to be a natural consequence of the notion of an "arbitrary set". The problem might be how you embed your physical theory in set theory, not with set theory itself. Which leads to my second remark.
Second: Being really careful that the mathematical objects you are working with have physical meaning, should lead you not to meddling in the foundations of mathematics but toembracing the theory of measurement, where you have explicit theory of what is meaningful. You can find a readable first introduction here and here.