**20**

votes

**0**answers

770 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 ...

**11**

votes

**0**answers

247 views

### How to measure the strength of Zermelo over bounded Zermelo?

Bounded Zermelo is Zermelo set theory with only bounded separation. It has the same strength as simple type theory or MacLane set theory or ETCS. It is a finitely axiomatized fragment of Zermelo, so ...

**9**

votes

**0**answers

144 views

### Reverse mathematics of meromorphic functions on Riemann surfaces

Various sources touch briefly on the reverse mathematics of measure theory and complex analysis. But I have found none on the uniformization theorem for Riemann surfaces or the existence of ...

**7**

votes

**0**answers

270 views

### “Hard” separation results in reverse mathematics (or similar)

This is a fairly broad question. In particular, I specify 5 questions (Q1, Q2.1, Q2.2, Q3, Q4) which for me all fall under one umbrella. Since this is unreasonably broad, I'm really interested in an ...

**7**

votes

**0**answers

305 views

### Can second order arithmetic make $\aleph_1^L$ countable?

Simpson's book Subsystems of Second Order Arithmetic shows $Z_2$ can interpret some fragments of ZF strong enough to give good theories of constructible sets and formalize statements like "there is a ...

**6**

votes

**0**answers

128 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 ...

**6**

votes

**0**answers

153 views

### When is a reduction not a reduction?

Every mathematician understands the concept of reducing a complicated problem to a simpler problem. "Without loss of generality, we may assume…" However, I've noticed that some kinds of ...

**4**

votes

**0**answers

64 views

### $n$th order arithmetic with predicates for orders

Two papers I have looked at lately axiomatize $n$-th order arithmetic in a single sorted language with predicates $Z_1,\dots,Z_n$ and axioms like $\forall x(Z_1(x)\vee\dots\vee Z_n(x))$ to say ...

**4**

votes

**0**answers

78 views

### Stabilization of recursive approximation in $PA^-+I\Sigma_1^0$

Over any model M of $PA^-+I\Sigma_1^0$. Suppose $A\in [T]$ where $T$ is a $\Delta_2^0$-tree and $A$ is one isolated path. Further, $A$ is regular, i.e. $\forall n A\upharpoonright n$ has a code in ...

**4**

votes

**0**answers

107 views

### cut-elimination for infinitary logic

Takeuti (1987, 223) deduces a cut-elimination theorem for infinitary logic from the corresponding soundness-and-completeness theorems. However, is there a way to adapt the basic Gentzen-style ...

**3**

votes

**0**answers

179 views

### The substitution theorem in first order logic (finitely many variables)

We consider the language ${\cal L}=\{\in\}$ with an arbitrary set of variables $V$. Specifically we define ${\bf P}(V)$ as the free algebra of type $\{\bot,\to\}\cup\{\forall x:x\in V\}$ (with the ...

**3**

votes

**0**answers

102 views

### Why the choice of pairing function in Subsystems of Second Order Arithmetic?

Simpson's book uses a pairing function $\langle i,j\rangle = (i+j)^2+j$. Is that choice of function simply unimportant, or does it have expository advantages over the Cantor pairing, or does it have ...

**2**

votes

**0**answers

69 views

### Seeking name for an order raising operator in Higher Order Arithmetic.

Any class $X$ of order $j$ in HOA is in bijection with the order $j+1$ class built up from singletons $\{x\}$ of natural numbers $x$ just the way that $X$ is built up from the numbers $x$. And of ...

**1**

vote

**0**answers

191 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 ...

**1**

vote

**0**answers

142 views

### Has the Ramified Theory of Types been applied to Predicative Set Theories?

Questions of predicativity are well-studied in the context of arithmetic. We have a base theory, first-order Peano arithmetic. Some people, like Edward Nelson (in chapter 1 of his book) and Charles ...

**0**

votes

**0**answers

97 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 ...

**0**

votes

**0**answers

76 views

### Beatty proof adaptation ?

If $m$ and $n$ are coprime integers, prove that each of these $m+n-2$ fractions: $$\frac{m+n}{m},\frac{2(m+n)}{m},\frac{3(m+n)}{m},...,\frac{(m-1)(m+n)}{m},$$ ...

**0**

votes

**0**answers

170 views

### subformula property (anchored proofs)

Hello,
I would like to ask for some explanation on some property of propositional sequent calculus.
The sequent calculus that I use here follows that of Stephen Cook, in "Logical Foundations of ...

**0**

votes

**0**answers

123 views

### Implications of complex solutions of Matiyasevich / Chaitin diophantine polynomials.

This is a shot in the dark: In twf:202, an isomorphism $T\cong T^{7}$ between binary trees $T$ and seven tuples of binary trees T^{7} is mentioned. The argument for this isomorphism starts with the ...