Godel proved that any system expressive enough to produce an arithmetic is incomplete. He initially proved it for the peano axioms but then it got generalized. ZFC can produce an arithmetic. Also, before being arrogant and demanding explanations, you should give them first for your claims

> expressive enough to produce

you understand that "expressive enough to produce" are not obvious elements of zfc, that's some average consumer napkin math and not strict formalization.

why should they be obvious? they are derived and have been thoroughly proven.

looks like we are in disagreement

That increases the likelihood that they are right.

> support your point with explanation or be ignored :-)

Anyone who says "Godel theorems are for systems with basic arithmetic, zfc doesn't include arithmetic, thus are not object of Godel theorems" and isn't joking warrants a permanent ignore.

https://math.stackexchange.com/questions/1366560/why-does-g%...

https://math.stackexchange.com/questions/1090437/how-to-prov...

imo, those two links are example of rather low quality weird math discussions, but you can keep your opinion

> imo, those two links are example of rather low quality weird math discussions, but you can keep your opinion

I've seen a lot of bad faith on this site, but none exceeding that.

A quick google search shows different proof assistants have been used to obtain the Peano axioms from ZFC, such as Isabelle/ZF and Metamath. I think you're just wrong

you are entitled to have your opinion :-)

and you are entitled to talk about maths while rejecting maths

coming back to your argument about peano being obtained from zfc, you obviously can't prove that it happened using purely zfc, and not some logical framework embedded into those proof assistants.

I said I am not expert, I am indeed not expert in zfc and godel theorems, but I am an expert (phd) in actual formalization theory. Formal theory is very simple concept: its alphabet, set of formulas on top of this alphabet, and function which translates one formula to another.

ZFC can't "obtain" peano, simply because it doesn't have say * operator defined. You need to do something on top of it. Additionally, zfc itself looks like loosely formalized say in wikipedia (and I am not sure if there is any strict formalization anywhere), we take it as common sense that it can utilize some simple logical rules (e.g. modus ponens), but what are exactly rules, which could be separate topic of research, this detail is skipped.

Eh? Any first course in set theory will present ZFC as a one-sorted theory with ten axioms (/schemas) in first order logic (inheriting an equality symbol, forall, implies etc) with one binary predicate (namely set membership), or will present a theory that is equiconsistent with a usual ZFC presentation. Honestly I’m not sure how you simultaneously claim to be a PhD in formalisation and also not be aware of the existence of Isabelle/ZF, for example.

> Honestly I’m not sure how you simultaneously claim to be a PhD in formalisation and also not be aware of the existence of Isabelle/ZF, for example.

I am aware, also I am not sure why you wrote all of this. Your unknown to me "first course" claims to be some authority of formalization purity?

Because you wrote:

> what are exactly rules, which could be separate topic of research, this detail is skipped

I am now confident you’re a troll, though, so I am going to bow out.

I referred to specific definition in wikipedia. Your "first course notes" are irrelevant here, they can't be reviewed, they not proofread and unlikely can be considered as any reasonable quality if we are talking about real formalization of math.

What are you nerds fighting about please explain