1
vote
3answers
479 views

Applicability of Deduction theorem to Primitive recursive arithmetic [closed]

Hello. I already asked the question here. The main point is that I tried to prove in Primitive recursive arithmetic (PRA) the totality of the Ackerman function, and I found, that the single thing ...
3
votes
1answer
460 views

Feferman's extensional and intensional applications of the method of arithmetization

At the very beginning of Feferman's Arithmetization of metamathematics in a general setting it can be read: The method of arithmetization, as developed by Gödel[10], exploits the possibility of ...
18
votes
8answers
3k views

Proofs of Gödel's theorem

I am interested in different contexts in which Gödel's incompleteness theorems arise. Besides traditional Gödelian proof via arithmetization and formalization of liar paradox it may also be obtained ...
4
votes
5answers
833 views

“Let $x \in A$”, beginning a proof of “$\forall x \in A$ …”, if A were empty [closed]

I work at a four-year teaching school, where we pride ourselves on teaching pure math, proof, and a rather obsessive carefulness of work. Recently I have been criticized for saying that "Let $x \in ...
5
votes
1answer
931 views

Is there a known way to formalise notion that certain theorems are essential ones?

Suppose You ask a question beginning from "Why some structure is..." or "Why some object has property..." and several answers arises. Which criteria do You use to qualify which answer is correct? ...
9
votes
3answers
3k views

Bourbaki's epsilon-calculus notation

Bourbaki used a very very strange notation for the epsilon-calculus consisting of $\tau$s and $\blacksquare$. In fact, that box should not be filled in, but for some reason, I can't produce a \Box. ...
18
votes
11answers
1k views

Are there any good nonconstructive “existential metatheorems”?

Are there any good examples of theorems in reasonably expressive theories (like Peano arithmetic) for which it is substantially easier to prove (in a metatheory) that a proof exists than it is ...
51
votes
28answers
5k views

Can infinity shorten proofs a lot?

I've just been asked for a good example of a situation in maths where using infinity can greatly shorten an argument. The person who wants the example wants it as part of a presentation to the general ...