3
votes
1answer
93 views

Undefinable Real Numbers

Disclamer: I'm sure my definition of "definable" may be different than the/a established mathematical one, I am more than interested in learning why/how this is so, but that is not my question Part ...
3
votes
2answers
86 views

Distinguishing sets according to more fine-grained notions than cardinality.

I'm interested in distinguishing sets according to more fine-grained notions than cardinality. Now I don't know a thing about computability theory, but it seems to me that considering sets up to ...
0
votes
0answers
67 views

A Question Regarding Ordinal Turing Machines

Consider the following theorem of Koepke: 'A set x of ordinals is ordinal computable from a finite set of ordinal parameters if and only if it is an element of the constructible universe L". Taking ...
11
votes
1answer
142 views

Consistency strength of Turing measurability

This is probably well-known to recursion theorists, but as google didn't help me, I'll ask it here. Convention: All sets of reals in the following discussion are assumed to be closed under Turing ...
2
votes
1answer
66 views

Possible Turing degrees of countable models of ZFC

Let $M$ be a countable model in a signature $\Sigma$. We assume $\Sigma$ is finite, and (for convenience) has no function or constant symbols. Without loss of generality, we can assume that $M$'s ...
7
votes
1answer
116 views

Does an undecidable decision problem have a ZFC-independent instance?

Is it true that every undecidable decision problem has an instance whose solution is independent of ZFC? For example, let $G$ be a finitely-presented group with undecidable word problem. Does there ...
3
votes
2answers
174 views

Random reals and Martin-Löf randomness

My questions are about the relationship between the following notions of randomness: A real $r$ is random over the model $M$ if $r\notin B$ for every null Borel set $B$ coded in $M$. A real $r$ is ...
6
votes
1answer
83 views

A question on non-standard ordinals in $\alpha-$recursion

Let $M$ be an admissible set, namely, $M\models KP$ where KP stands for axioms of Kripke–Platek set theory. Denote $\beta=M\cap ORD$ where $ORD$ is the class of ordinals. I wanted to prove ...
1
vote
3answers
73 views

a finite algorithm mapping from $\omega \times \omega$ to $\omega$ possible?

We know that $\omega \times \omega$ is isomorphic to $\omega$, but I am not sure if there would exist a finite algorithm mapping from $\omega \times \omega$ to $\omega$. An algorithm would of course ...
0
votes
1answer
90 views

Computability function - how to express it in set theory/arithmetic hierarchy

Let's say that $f$ is computable function such that for particular inputs $x$ and $y$, $f(x) = 0$ and $f(y) = 0$. If we want to express this in logical form (arithmetic hierarchy formula), what would ...
1
vote
1answer
115 views

How does one prove that 1-generic set is not computable?

Without resorting to diagonalization proof of halting problem, how does one prove that 1-generic set is not computable?
2
votes
2answers
90 views

$\omega_1^{CK} - \omega$ - infinite or finite set? And boundary

I am curious whether $\omega_1^{CK} - \omega$ would result in a finite set or infinite set. Does anyone know what happens? Edit: OK, let me add one more question: Suppose that we take $\omega \cdot ...
1
vote
1answer
48 views

analytical hierarchy and individual variable quantifiers

For analytical hierarchy, $\Sigma^1_0$ is usually defined as the class of formula that does not have any set quantifier - but does this mean that there can be any number of quantifiers for individual ...
3
votes
0answers
201 views

How to derive Church-Kleene ordinal

Crossing-out: (How does one prove the existence of Church-Kleene ordinal? Also, why is it labeled as $\omega_1^{CK}$? And why is it first ordinal not hyperarithmetical, and is the first admissible ...
4
votes
2answers
151 views

Ordinals definable over $L_\kappa$

Suppose $\kappa$ is an uncountable cardinal, with $L_\kappa$ an admissible set (i.e. a model of Kripke–Platek set theory). Let $<_\gamma \subseteq \kappa \times \kappa$ denote a wellordering of ...
5
votes
2answers
290 views

Is there any generalization of the hyperarithmetical hierarchy using the analytical hierarchy to formulas belonging to third-order logic and above?

As I understand, hyperarithmetical sets are defined according to the analytical hierarchy, that is, second-order-logic formulas. There is a generalization of hyperarithmetic theory named α-recursion ...
2
votes
0answers
109 views

constructive ordinal and $\Delta^1_1$ predicate

Everything I know on this subject comes from Sacks book : "Higher recursion theory" Let $\mathcal{O^Y}$ be the set of codes for ordinals constructive in $Y$. We should have the result that $A ...
4
votes
2answers
461 views

Non-recursive subset which is recursively enumerable

What is an example of recursively enumerable subset of the natural numbers which is not recursive?
3
votes
1answer
321 views

is a semirecursive (recursively semidecidable) set enumerable?

I'm getting confused. According to Computability and Logic fifth edition, semirecursive = recursively semidecidable, and according to wikipedia http://en.wikipedia.org/wiki/Recursive_set , ...
10
votes
1answer
180 views

Is there a dense subset of $\mathbb{R}^2$ with all distances being incommensurable?

Is there a set $S$ of points on the real plane $\mathbb{R}^2$ such that: there is a point belonging to $S$ in any neighborhood of every point of $\mathbb{R}^2$ (so, $S$ is dense) and ratio of any ...
3
votes
1answer
291 views

Properties of computable numbers

If we enumerate* all the computable numbers, those for which there exist a turing machine that outputs its digits to arbitrary precision. What is known about the asymptotic density of rationals, ...
2
votes
2answers
126 views

Undecidable countable structure built on decidable relation?

My question is, is there a relation $R$ on the integers that's decidable (i.e. the function ${\mathbb Z}^2 \to \lbrace \text{true},\text{false} \rbrace, \ (i,j) \mapsto i R j$ is computable) , but ...
8
votes
4answers
457 views

Consequences of solving the Halting problem

What impact would a device (ie super-computer or relativistic computer or other method) that solves the halting problem have on math? Would there be any mathematical problems left to solve? What ...