For question in Proof Theory, where "proofs" themselves are the object of mathematical investigation. It is not to be used to request a proof of some result.

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11
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1answer
389 views

Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions?

Let the proof theoretic ordinal $\alpha$ of a theory $T$ be the least recursive ordinal such that $T$ does not prove that $\alpha$ is well-founded. This ordinal is intended to quantify in some sense ...
12
votes
1answer
240 views

What metatheory proves $\mathsf{ACA}_0$ conservative over PA?

Simpson's book shows $\mathsf{ACA}_0$ is conservative over $\mathsf{PA}$ in the natural way by model theory using definable subsets. Of course, $\mathsf{ACA}_0$ being conservative over PA is ...
4
votes
1answer
231 views

Can the Burgess-Hazen analysis of Predicative Arithmetic be extended to Transfinite Types?

Around page 300 of his book "Mathematical Thought and its Objects", Charles Parsons discusses the work of Edward Nelson, who believes that mathematical induction is impredicative, because it can be ...
13
votes
2answers
2k views

Clarification of Gödel's second incompleteness theorem

I am sorry for the following question, because the actual answer to this question is in the beautiful works of Feferman and Jeroslow, but, unfortunately, I havn't any time to go into that specific ...
8
votes
3answers
595 views

Consistency of Analysis (second order arithmetic)

Is there a proof of the consistency of Analysis (second order arithmetic), which is similar to Gentzen's proof of the consistency of arithmetic? Update: Which (different) methods can be used to ...
3
votes
1answer
487 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 ...
10
votes
3answers
725 views

Reducing ACA₀ proof to First Order PA

According to the Wikipedia ACA0 is a conservative extension of First Order logic + PA. http://en.wikipedia.org/wiki/Reverse_Mathematics First of all I have a few questions about the proof: a - What ...
1
vote
0answers
278 views

What is the role of the (formalized) omega rule in Ramified Analysis?

In the 1960's, Feferman and Schutte did groundbreaking proof-theoretic work to find out the strength of predicative systems of second-order arithmetic. They used the ramified theory of types, a ...
51
votes
14answers
4k views

When are two proofs of the same theorem really different proofs

Many well-known theorems have lots of "different" proofs. Often new proofs of a theorem arise surprisingly from other branches of mathematics than the theorem itself. When are two proofs really the ...
23
votes
15answers
4k views

What's a magical theorem in logic?

Some theorems are magical: their hypotheses are easy to meet, and when invoked (as lemmas) in the midst of an otherwise routine proof, they deliver the desired conclusion more or less ...
17
votes
2answers
802 views

Deep theorems and long proofs

I ran across this discussion by Daniel Shanks, "Is the quadratic reciprocity law a deep theorem?." Solved and Unsolved Problems in Number Theory. Vol. 297. AMS, 2001. p.64ff. which made me ...
13
votes
1answer
968 views

Proof theoretic ordinal

In Ordinal Analysis, Proof-theoretic Ordinal of a theory is thought as measure of a consistency strength and computational power. Is it always the case? I. e. are there some general results about ...
7
votes
3answers
1k views

Most general formulation of Gödel's incompleteness theorems

Modern statements of Gödel's incompleteness theorems are usually in terms of first-order predicate logic. However, I've often read the claim that they extend to arbitrary formal systems that can prove ...
3
votes
2answers
2k views

Independence of PA implies independence of PA union all true $\Pi_1$ statements

Prove that if a statement is independent of Peano Arithmetic (PA), then it's also independent of PA$_1$, where PA$_1$ is the union of the set of axioms in PA and the set of all true $\Pi_1$ ...
14
votes
1answer
518 views

Goodstein's theorem without transfinite induction

Goodstein's theorem is an example of a theorem that is not provable from first order arithmetic. All proofs of the theorem seem to deploy transfinite induction and I've wondered if one could prove the ...
4
votes
2answers
366 views

Did Gödel prove that the Ramified Theory of Types collapses at $\omega_1$?

Second-Order Arithmetic is considered impredicative, because the comprehension scheme allows formulas with bound second-order variables that range over all sets of natural numbers, including the set ...
2
votes
1answer
152 views

what are the proof-theoretic ordinals of second-order arithmetic and ZFC? [duplicate]

are they still smaller than omega-1-CK?what are the notations of them?
1
vote
2answers
268 views

Embedding of consistent subset in first order logic (finitely many variables)

I am looking at FOL with no equality, no constant, no function symbol and the unique binary predicate $\in$ with variables in arbitrary sets $V$ or $W$. Specifically we define ${\bf P}(V)$ as the free ...