Gödel's Incompleteness Theorems

The second theorem tells us that we can't prove the consistency of set theory without using an even more powerful system. This doesn't mean that set theory is inconsistent, it just means that we can't prove its consistency in any meaningful way. So yes, Gödel indeed shows that there are no 'ultimate axioms' that contain everything of interest in mathematics.

So maths isn't axiomatic, and can be fully proven - Godel was wrong?

No system is powerful enough to prove all of the facts about natural numbers, right?

Yeah, Cantors theorem is a theorem written in the language of ZFC set theory. In other axiom systems it may not be true. But you can also say that about literally every theorem beyond, like, modus ponens. Is that the point you are trying to make?

"The mathematical statements discussed below are provably independent of ZFC (the canonical axiomatic set theory of contemporary mathematics, consisting of the Zermelo–Fraenkel axioms plus the axiom of choice), assuming that ZFC is consistent. A statement is independent of ZFC (sometimes phrased "undecidable in ZFC") if it can neither be proven nor disproven from the axioms of ZFC."- TFA

Isn't this just a special case of one of Godel's incompleteness theorems?Roughly, the incompleteness theorems state that, in any sufficiently powerful, self-consistent, decidable mathematical axiom system, there are statements which are true, but cannot be proved. And "sufficiently powerful" means basically "including addition and multiplication." The high school axioms look like they're at least that powerful, so wouldn't it just be a direct application of Godel that such statements exist?

"The most comprehensive formal systems yet set up are, on the one hand, the system of Principia Mathematica and, on the other, the axiom system for set theory of Zermelo-Fraenkel (later extended by J. v. Neumann). These two systems are so extensive that all methods of proof used in mathematics today have been formalized in them, i.e. reduced to a few axioms and rules of inference. It may therefore be surmised that these axioms and rules of inference are also sufficient to decide all mat

These are the implications of Godel's incompleteness theorems. No formal system expressive enough to encode arithmetic can simultaneously be both complete and prove its own consistency, because there will always be true propositions expressible in that system that cannot be proven in that system.This is why Hilbert's program to finitely axiomatize mathematics can't be completed. The "escape hatch" here is simply that not all propositions are actually interesting, so f

Probability is relative to an axiom systems.A more powerful system can prove statements that are true but not provable in a weaker system.For example "This sentence is not provable." (Godel's statement, written informally) is true, but not provable, in first order arithmetic logic.https://en.m.wikipedia.org/wiki/Diagonal_lemmaIn constructivist mathematics

Science doesn’t prove axioms... Gödel already parsed this problem :)