Err? Peano Arithmetic is provably consistent in ZFC, but it is not in itself (if PA is consistent). Therefore if PA is consistent it is not equivalent to ZFC (regardless of whether ZFC is consistent or not)
Even before I started the video, I had a feeling it was going to lead to a kind of "introspective" mathematics that can reason about its own reasoning. I was not disappointed, thank you.
It's also deep, it goes all the way to the bottom.
> The world of mathematics is both broad and deep, and we need birds and frogs working together to explore it. -- Freeman Dyson
Weird Things Happen When Math Gets Too Expressive
https://www.youtube.com/watch?v=EVwQsvof7Hw
Peano arithmetic is sufficiently expressive enough to be equivalent to any possible future theory of mathematics.
Err? Peano Arithmetic is provably consistent in ZFC, but it is not in itself (if PA is consistent). Therefore if PA is consistent it is not equivalent to ZFC (regardless of whether ZFC is consistent or not)
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Even before I started the video, I had a feeling it was going to lead to a kind of "introspective" mathematics that can reason about its own reasoning. I was not disappointed, thank you.
Physics, Topology, Logic and Computation: A Rosetta Stone - https://arxiv.org/abs/0903.0340
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