Functional equations vs. Mathematical ZFC models

Question

Is the solutions of a functional equation uniquely determined by ZFC axioms, or could it requires adittional axioms?

I refer to differential and integro-differential equations principally

Answer 1

The question doesn't really make sense. Our physical equations are simply equations - nothing more - and they are a model of the world, not the world itself. It just doesn't make sense to ask whether non-mathematical results are independent of the mathematical model: it's like asking whether the question of "what did I last eat?" is independent of ZFC.

Usually we say a statement is independent of ZFC if there is a model of ZFC in which the axiom is true, and a model in which it is false. OK, let's try and ask whether the Schrödinger equation (SE)'s truth is independent of ZFC. Well, is there a model of ZFC in which the SE is true? Er… I don't know how to answer that. It's like asking whether there's a model of ZFC in which I am five years old. It's not clear that I even exist as an entity in ZFC.

As another angle on your question, it is possible to construct much of mathematics - including real analysis - in systems other than ZFC. Real analysis can be developed using much weaker choice principles than the full power of the Axiom of Choice; and I seem to recall that the real numbers may be constructed in New Foundations (which is very much not ZFC). Additionally, you can relax quite a lot of the Axiom of Replacement, because real analysis occurs at a very shallow level of the von Neumann hierarchy. Therefore, you don't have to use ZFC if you want to state the Schrödinger equation.

Of course, just stating the SE doesn't help you at all in saying things about the real world. If you want to say things about the real world, you need to state the SE and then make and test predictions about your model. ZFC can't help you there.