Comment by andriy_koval
9 hours ago
> Goedel Incompleteness -- the proof that the the axiomatic itself cannot be proven, like using ZFC to prove ZFC, but that's another topic.
Godel theorems are for systems with basic arithmetic, zfc doesn't include arithmetic, thus are not object of Godel theorems.
If you start with "I'm not a strong expert" maybe you should stop continuing saying wrong stuff. What you just wrote is completely wrong.
support your point with explanation or be ignored :-)
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
11 replies →
> Moreover, Robinson arithmetic can be interpreted in general set theory, a small fragment of ZFC.
https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...
> interpreted
its hard to me to tell what this means formally(as I said I am not expert). There is no "interpret" operator in zfc. I believe what it says if you add some robinson axioms + some logical rules on top of zfc, you can carry your results.
ZFC has greater consistency strength than PA.
If we take ZFC (or some other set theory) as our meta theory, we can easily see that the axiom of infinity (of ZFC) gives a set of natural numbers (using the von Neumann encoding), which, when equipped with the successor function, is a model of the natural numbers.
zfc doesn't have functions, so you are building something new on top of it.
Also, I am not sure successor function is enough for PA.
That is wildly wrong.