> Mathematical reality is somewhat different from the physical reality ("our universe").

I think in general when we're referring to "reality" we're being synonymous with physical reality. Include the reality beyond observable (like String Theory predicts) if you want. With your usage I honestly have no concept of what you mean by "reality" that I can differentiate from any other abstract concept. If the apple in my head is part of reality (excluding the literal physical state of my brain analogous to bit state of computer memory) then I feel this discussion is rather moot and we can go ahead and conclude that nothing is invented and everything is discovered. But then why not discuss with the definition of reality I'm using (and what I'm assuming is fairly status quo)?

And at what point is math "real?" Is any axiomatic system real? If so we can create arbitrary reality and go back to the above point. If not, then not all math is real. I don't see how we can have something that is self consistent here. Unless you're saying that "math" is whatever rules the universe operates on and not these other inaccurate set theories/systems that we use to approximate this thing." But then again the latter is what we call math and I guess we need to come up with a new definition for many things that exist within recreational mathematics. Sorry happy numbers, idk what you are but I guess not math?

> You are asking about ZF, ZFC, etc. Try asking yourself about XYZQJJX (it's language, after all, which you are free to invent, right?) and you won't be getting anywhere.

Sorry, you're going to have to define it if you want me to determine if it goes anywhere or not.

I'm referring to Zermelo–Fraenkel set theory and Zermelo–Fraenkel Choice set theory respectively[0]. (I figured it wasn't hard to Google, but you're right, I should clarify) These are for all intents and purposes different "maths." They have different axioms. Different rules. Different conclusions.

I'm unconvinced that utility has anything to do with something being real or not. Your usage suggests you aren't considering language real but it's one of the most useful things animals have invented (especially spoken and written language). Nor is the an arbitrary computer image "real" in the same sense. There's a lot of things that belong to this same categorical set.

And while we're at it, just give me anything, 1 or more, that wouldn't be considered real. I'll represent them with a set and I can do math with them... After all, math is the study of relationships between arbitrary objects. See Poincare or ... Category Theory

To be clear, we can create arbitrary set theories that will go "nowhere" and not have utility in physics or other common things. So if you don't want to call this math (I'm not sure who would say that. It looks like math, acts like math, and even mathematicians call it math) what would you call it?

Here, I'll provide an example:

∀ u(u ⊂ X≡u ⊂ Y)⇒X≠Y. : (Axiom of non-Extensionality) When x and y have the same members they are not the same set

Idk we can have the rest of ZF if we want here but it isn't really going to make sense because we destroyed uniqueness. But we can still do math from here. This is a more extreme version of what I was suggesting with Finite ZF vs ZF vs others. They have different axioms and they lead to different things. ZFC is in contention specifically because the axiom of choice leads to nonsensical things (again, see Banach-Taraki paradox). Are these both math? Back to paragraph 1.

[0] https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...