As we know since Gödel's incompleteness theorems, there can never be a "One True Set Theory", since such a system could not be both complete and consistent. So whenever there is an independent statement, we can decide whether we want it to be true or false, and introduce it (or its negation) as an axiom. Such as we did with the axiom of choice. Doing this merely increases the precision of our definition of sets.

Correct me if I'm wrong, but AFAIK one of the assumptions of the Gödel incompleteness theorem is that you have a finite number of axioms or axiom schemes. Which makes sense given the fact that any axiom system we can actually handle will be finite in this way. However it doesn't preclude that there exists an infinite (possibly even uncountable infinite) set of axioms that are both consistent and complete. That would not be an axiom system in the classical sense (and nothing we could actually handle), but there's nothing inherently contradictory in it.

If there happens to be (up to equivalence) only one such generalized axiom set that includes ZFC, this would mean that there exists only one "true" set theory, Of course we will never know because we simply can't handle an infinite (let alone uncountable) set of axioms.

My point is that it is merely a matter of definition what we put in a set theory, so there cannot a "single true one", but there are infinitely many variations, each based on preference.

But it also becomes messy when we need an infinite amount of axioms, since a proof can never be infinite, so it can only ever use a finite amount of axioms. Given that you also cannot produce and use an infinite amount of proofs, you can only ever use a finite amount of axioms. So it can never be required for usability that there is an infinite amount of axioms. Things are different when it comes to semantic consequence (in contrast to syntactic consequence), but that one concerns model theory (in contrast to proof theory), which aims to describe models / theories (not proofs) in the first place.

@xamidi, you're missing the larger point that this is a question of philosophy. You believe so completely in the multiverse philosophy that you're rejecting the possibility of existence of any other philosophy.

Peter Taylor‭ My comments were neither philosophical, nor do they have anything to do with physics. Not sure where you see a connection to the multiverse theory. Logical systems, just like formal systems in general, are not per se related to physical objects.

xamidi‭, physics has nothing to do with this discussion, but the question and the answer are both about philosophy of mathematics, so if your comments aren't philosophical then they're definitely missing the point. Multiverse here is a term taken from the paper which the question asks about.

Peter Taylor‭ Technical remarks do not miss the point but are required to substantiate any philosophical view on a formal topic. In this case, I used technical arguments to show that certain philosophical standpoints are either (a) a mere lack of understanding in the subject, or (b) an intended mystification of formal methods. Either way, it is unfounded. There cannot be a valid argument as to why prefer which formal system over all others when all of them work for their intended purpose, and neither can there be a correct argument that none of the others work correctly (which is demonstrably false).

(Off-topic:) Regarding this bug report, Peter did you get a notification for this comment?

Monica Cellio‭, yes, I did.

Debugging: Peter Taylor‭ did this comment notify you? (I apologize to the other people following this thread. I'm trying to minimize disruptions but we have a weird bug that is specifically happening here. I'm sorry for the disruption and will clean it up ASAP.)