**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

257 views

### Do all linear orders in this class have computable copies?

This is a question which has been bothering me now for quite some time. I've talked to a number of people about it, and we've shown that a few basic ideas can't work, but other than that haven't made ...

**11**

votes

**0**answers

142 views

### Savings property: A transformation which turns nonnegative martingales into uniformly integrable ones

Background
I work in a subfield of computability theory called algorithmic randomness. We have been using martingales as long as probability theory (going back to work of von Mises). However, since ...

**11**

votes

**0**answers

507 views

### Minimal resources for Undecidability of First-Order Logic: the number of variables

It is well-known that First-Order Logic (FO) with a full vocabulary (i.e., a countable numbers of unary predicate symbols, a countable number of binary predicate symbols, etc.) is undecidable. And it ...

**10**

votes

**0**answers

462 views

### Groups generated by 3 involutions

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$.
Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition
$\tau_{r_1(m_1),r_2(m_2)}$ be the ...

**10**

votes

**0**answers

444 views

### Diagonal lemma from recursion theorem?

Does Gödel's diagonal lemma follow from Kleene's recursion theorem? I believe the converse is true, by an argument like the following.
Let e ↦ θe be a bijection between ω and ...

**9**

votes

**0**answers

241 views

### Various definitions of recursion from ordinal machines

Background: I'm trying to get an intuitive understanding of α-recursion and related concepts in higher recursion theory. Once nice book is Peter Hinman's Recursion-Theoretic Hierarchies, available ...

**8**

votes

**0**answers

248 views

### Automorphism group of the Turing degrees

It is conjectured that the automorphism group of the Turing degrees, $Aut(\mathcal{D})$, is trivial. However, to the best of my knowledge, the current state-of-the-art is that $Aut(\mathcal{D})$ is ...

**7**

votes

**0**answers

133 views

### Martin-Löf randomness relative to a $\Delta^0_2$-representation of a real

I have a question which I already asked on a more specialized site (http://logicblogfrontend.hoelzl.fr/), but perhaps M.O. will allow me to reach a wider range of experts.
Suppose that $X$ is ...

**7**

votes

**0**answers

271 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 ...

**6**

votes

**0**answers

165 views

### $\omega$-models of $\mathbf{\Sigma^1_1}-DC$ and $\mathbf{\Delta^1_1}-CA$

So what is needed to demonstrate something (say like $L_{\omega_1^{CK}}[a]$ is a $\omega$-models of $\mathbf{\Sigma^1_1}$-$DC$ or $\mathbf{\Delta^1_1}$-$CA$? It's not like I don't understand what the ...

**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

198 views

### About “natural proof” of Razborov and Rudich

The famous "Natural Proof" paper ,http://www.cs.umd.edu/~gasarch/BLOGPAPERS/natural.pdf , of Razborov and Rudich gives a barrier for any proof that try to separate P and NP. It mainly shows that if ...

**4**

votes

**0**answers

210 views

### Difference between lambda-calculus with well-formed formulas vs properly-formed formulas

In S.C. Kleene's 1935 paper "$\lambda$-definability and recursiveness," he proves that all $\lambda$-definable functions are general recursive in the Herbrand-Godel sense and vice-versa. However, the ...

**3**

votes

**0**answers

86 views

### Weak classes of diophantine functions

From a well-known work(s) by Putnam, Davis, Robinson and Matiyasevich, we know that every partially recursive function is diophantine.
Now it seems a natural question to ask: can we say something ...

**3**

votes

**0**answers

138 views

### Alternate proof of van de Wiele's theorem in E-recursion

Hello, all
I'm currently trying to understand $E$-recursion theory, which is a generalization of classical recursion theory to arbitrary sets. One of the difficulties I'm having with understanding ...

**3**

votes

**0**answers

90 views

### Is a parametric family which is universally consistent for multiple quantiles impossible?

Suppose I am dead-set on using Bayesian inference on independent and identically distributed data, but I'm lazy and insist on using a parametric likelihood function come what may. I'd be reassured to ...

**2**

votes

**0**answers

86 views

### Comparing two non-deterministic Turing equivalents as basis for Logic, request for references

I am designing a logic, that is simpler than FOL + PA. And I like to know if there already exists something in this direction.
First of all a non-deterministic Turing equivalent is defined by ...

**2**

votes

**0**answers

69 views

### Equivalence of LOOP (primitive recursive functions) and of SRL (reversible transformations) programs

This is a question about the decidability of program equivalence.
Primitive recursive functions correspond exactly to the functions
that can be implemented on a specific register machine usually
...

**2**

votes

**0**answers

68 views

### Is there any track for proving $D=NP$, besides showing that $D$ has polynomial-bounded universal quantifiers?

Background
By the MRDP theorem, every for every recursively enumerable set $S$, there exists a Diophantine polynomial $p$ such that
$$x \in S \iff \exists y_1, \dots, y_n \in \mathbb{N} \text{ such ...

**2**

votes

**0**answers

203 views

### A polytime feasible subuniverse of the Effective Topos

The effective topos is a well known universe of sets suitable for abstract computability, as it is build "from the ground up" via the classical notion of realisability by Kleene.
I have found a few ...

**1**

vote

**0**answers

98 views

### Is there an easy decision algorithm for the inhabitation problem for simple types?

Consider the basic system of simple types usually known as $TA_\lambda$. One can prove that (as a consequence of the Subject Reduction Property and the fact that any typable term is strongly ...

**1**

vote

**0**answers

128 views

### Reference for original paper (but translated to English) of Matiyasevich's proof of Fibonacci relation being Diophantine?

Hello. I am a maths undergraduate. I am doing a project about history of mathematics. I am looking for the original solution to Hilbert's 10th problem, or at least the theorems that is accessible to ...

**1**

vote

**0**answers

143 views

### Constructing hard inputs for the complement of bounded halting

If there is always a hard input for the complement of bounded halting, can that input be constructed?
More precisely, suppose that for any $M$ accepting coBHP={$\langle N,x,1^t\rangle|\langle ...

**0**

votes

**0**answers

96 views

### Recursive relation using successor function

What is the recursive relation for
H(m)=2^(m^2)
using successor function
recursive relation for multiplication:
mult(x,0)=0;
mult(x,S(y))=add(x,mult(x,y))
recursive relation for addition:
add(x,0)=x;
...