Type Theory: It's the New Set Theory (Zoom for Thought 02/09/21)

[geek]

With the axioms of ZF (no need for C or its negation), you can prove that the set of all natural numbers satisfies the axioms of Peano Arithmetic. Thus, if ZF is consistent, then so too is Peano Arithmetic. (Due to incompleteness, all of these sorts of results are "If ZF is consistent, then so too is ___".) I'm not sure if that's what the speaker was asking about with regards to whether ZF can prove Peano Arithmetic.

[MrRyanroberson]

32:00 the better explanation would simply be:
A function is a subset of the pairs of elements (x, y) where for every value x there is exactly one pair containing x. The function's domain is the set of all X values, and its range is the set of all Y values. The axiom of choice, therefore, is the assertion that a function exists with its domain being the set of all proofs that sets are nonempty, and its range being the set of all elements of sets.

[mageover]

Types being a type, while also being of type type, isn't that weird. In fact Type being the type type, while also being of type type isn't that weird at all. Now you should understand how the type keyword works in python.

[willyh.r.]

So now, am wondering if I deserved my PhD in abstract mathematics. Maybe I have to review my whole thesis after learning this type theory. Cool.

[lamalamalex]

If this is correct I would be able to tell by:
No analytic-synthetic dichotomy. Which is false btw.
No multivalued logic, i.e., no violation of the law of identity.

If the law of identity is violated, it has no ground to stand on.

[Xayuap]

why don't just talks about Booleans instead of props?

[riccardoplati]

What a pretentious talk. Set Theory really flourished only with Cohen's forcing, while the talk carefully avoids mentioning anything about it or the questions it allowed to answer.

[blacknick]

I am sorry to say that, it was wasted my time to watch this video, doesn't help me more understander about type theory.