This post is partially about opinions and partially about more precise mathematical questions. Most of this post is not as formal as a precise mathematical question. However, I hope that most readers will understand this post and the nature of the question.
I will first try to explain what I would call Realistic Mathematics. Let us say that mathematics is about the formalization, organization and expression of thought.
At the same time one could have the feeling that thought is usually trying to capture some aspect of physical reality; and let that be anything from a feeling, an impression of something, to experimental data of an experiment, or the observed geometric properties of lines and points in two-dimensional space. Of course thought itself (as described above) can be about anything and hence anything axiomatizable could then be seen as mathematics (that is David Hilbert's point of view). At the other hand if thought is primarily about physical reality then the focus of mathematics should be more restrictive. (I remember that Arnold argued in favor of this view; also von Neumann.) Let us call this restricted part of mathematics for the moment Realistic Mathematics. I am not saying that such a restriction of focus would be good or necessary; and I do not want to start a discussion about this. I just want to find out and discuss, whether mathematicians could agree on what we could call Realistic Mathematics. Let us suppose for a moment that we made some sense of the concept of Realistic Mathematics, and observed that it is a science that is about some part or aspect of physical reality. My naive question is now:
Question: What is Realistic Mathematics about from a mathematical (or model theoretic) perspective?
or
Question: Is there any mathematical structure which serves as the part of mathematics which is about physical reality?
Just to give some examples: I consider anything related to finite-dimensional geometry (manifolds, simplicial complexes, convex sets etc.), number theory, operators or algebras of operators on separable Hilbert spaces, differential equations, discrete geometry, combinatorics (under countability assumptions) etc. as being part of observed physical reality or potentially useful for the study of physical reality. At the other side, existence of large cardinals, non-measurable subsets of the reals, etc. are not (immediately) useful for such a study. In particular, my view is that the Axiom of Choice does not add anything to the understanding of physical reality. It produces highly counter-intuitive statements (not observed in nature) about subsets of finite-dimensional euclidean space and has its merits (in Realistic Mathematics) only through short proofs and the knowledge that many statements are provable in ZFC if and only they are provable with more realistic assumptions.)
What about $L(R)$? (See Wikipedia for definitions.) That would be a concrete model and maybe a realistic mathematician just studies properties of $L(R)$? Maybe a realistic mathematician studies what can be proved using $ZF + DC$? Is there any other canonical candidate which arises? My question here is mainly about opinions or some sort of vision which explains why this or that model or object of study arises naturally.
Question: Is there any mathematical application to the study of physical reality which is not captured by the study of the model $L(R)$?
More specifically: What about concrete statements which are undecidable in $ZF$? Does the Continuum Hypothesis belong to the statements we want to be true in Realistic Mathematics? What about the Open Coloring Axiom? Here, I am also asking for opinions or some consistent perspective on the realistic part of mathematics which captures my imprecise way of describing it.
Best Answer
When Solovay showed that ZF + DC + "all sets of reals are Lebesgue measurable" is consistent (assuming ZFC + "there is an inaccessible cardinal" is consistent), there was an expectation among set-theorists that analysts (and others doing what you call realistic mathematics) would adopt ZF + DC + "all sets of reals are Lebesgue measurable" as their preferred foundational framework. There would be no more worries about "pathological" phenomena (like the Banach-Tarski paradox), no more tedious verification that some function is measurable in order to apply Fubini's theorem, and no more of various other headaches. But that expectation wasn't realized at all; analysts still work in ZFC. Why? I don't know, but I can imagine three reasons.
First, the axiom of choice is clearly true for the (nowadays) intended meaning of "set". Solovay's model consists of certain "definable" sets. Although there's considerable flexibility in this sort of definability (e.g., any countable sequence of ordinal numbers can be used as a parameter in such a definition), it's still not quite so natural as the general notion of "arbitrary set." So by adopting the new framework, people would be committing themselves to a limited notion of set, and that might well produce some discomfort.
Second, it's important that Solovay's theory, though it doesn't include the full axiom of choice, does include the axiom of dependent choice (DC). Much of (non-pathological) analysis relies on DC or at least on the (weaker) axiom of countable choice. (For example, countable additivity of Lebesgue measure is not provable in ZF alone.) So to work in Solovay's theory, one would have to keep in mind the distinction between "good" uses of choice (countable choice or DC) and "bad" uses (of the sort involved in the construction of Vitali sets or the Banach-Tarski paradox). The distinction is quite clear to set-theorists but analysts might not want to get near such subtleties.
Third, in ZF + DC + "all sets of reals are Lebesgue measurable," one lacks some theorems that analysts like, for example Tychonoff's theorem (even for compact Hausdorff spaces, where it's weaker than full choice). I suspect (though I haven't actually studied this) that the particular uses of Tychonoff's theorem needed in "realistic mathematics" may well be provable in ZF + DC + "all sets of reals are Lebesgue measurable" (or even in just ZF + DC). But again, analysts may feel uncomfortable with the need to distinguish the "available" cases of Tychonoff's theorem from the more general cases.
The bottom line here seems to be that there's a reasonable way to do realistic mathematics without the axiom of choice, but adopting it would require some work, and people have generally not been willing to do that work.