# Tagged Questions

**6**

votes

**1**answer

231 views

### Does the totality of Ackermann's function prove the consistency of $\Sigma_1$-induction?

It is well known that Ackermann's function is not primitive recursive. Therefore, the theories of primitive recursive arithmetic (PRA) and of $\Sigma_1$-induction ($I\Sigma_1$) cannot prove the ...

**8**

votes

**3**answers

570 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 ...

**0**

votes

**0**answers

103 views

### What references cover finitary systems of Ramified Analysis with transfinite levels?

The ramified theory of types, invented by Bertrand Russell, is a way of dealing with impredicativity by breaking the comprehension schema of second-order logic into levels. The comprehension schema ...

**14**

votes

**1**answer

516 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 ...

**9**

votes

**2**answers

385 views

### Reverse mathematics below RCA

I'm sure this is a fairly basic question, but I can't seem to find a solid answer:
My primary question is: Is there a reasonably nice subsystem of second-order arithmetic corresponding essentially to ...

**20**

votes

**0**answers

795 views

### Godel on recursion-theoretic hierarchies

At the end of his excellent article, "The Emergence of Descriptive Set Theory" (http://math.bu.edu/people/aki/2.pdf), Kanamori writes:
"Another mathematical eternal return: Toward the end of his ...

**8**

votes

**5**answers

939 views

### Understanding the nature and structure of proofs; Reverse Mathematics and Proof Theory. Prerequisites? Good introductory texts?

I'm still studying maths at undergraduate level, but intend to continue exploring topics in pure maths after I have graduated, so am thinking already about what directions I'd like to persue now, (as ...

**4**

votes

**1**answer

222 views

### History of provably total functions of a theory

Provably total functions of an arithmetical theory is one of the tools used in proof theoretic analysis of theories.
I am looking for early history of its development. In particular,
Where was ...

**7**

votes

**1**answer

326 views

### Looking for papers and articles on the Tarskian Möglichkeit

Some background: Łukasiewicz many-valued logics were intended as modal logics, and Łukasiewicz gave an extensional definition of the modal operator: $\Diamond A =_{def} \neg A \to A$ (which he ...

**1**

vote

**2**answers

601 views

### Equational logic

I'm a beginner to this. Can anyone please point me to any resources for studying about equational logic, preferably with some example proofs to wet my feet in?
Thanks in advance!

**3**

votes

**1**answer

530 views

### Derivability conditions for Robinson arithmetic

Two pieces of hearsay I have encountered about Robinson's Q:
Q fails to satisfy the Löb derivability conditions;
Pudlák criticised the Löb derivability conditions and suggested rival, weaker ...

**6**

votes

**1**answer

472 views

### How to locate the paper that established Robinson Arithmetic?

If I'm not mistaken, it was in his seminal paper “An Essentially Undecidable Axiom System”, published in
Proceedings of the International Congress of Mathematics (1950), 729–730,
where R.M. ...

**6**

votes

**6**answers

1k views

### Do you know any good introductory resource on sequent calculus?

I'm looking for a good introductory resource on sequent calculus suitable for someone who has studied natural deduction before. Books and online resources are both OK, as long as each rule of ...