Tagged Questions

16
votes
3answers
631 views

Is the Collatz conjecture in $\Sigma_1 / \Pi_1$?

Prompted by some of the comments on this question, I'm wondering if anything is known about the place of the Collatz Conjecture in the arithmetic hierarchy. More specifically, is Collatz known to be ...
13
votes
1answer
1k views

Is chess Turing-complete?

Is there a set of rules that translates any program into a configuration of finite pieces on an infinite board, such that if black and white plays only legal moves, the game ends in finite time iff ...
11
votes
6answers
974 views

What philosophical consequence of Goedel's incompleteness theorems?

I want to write a philosophical essay centered about Goedel's incompleteness theorem. However I cannot find any real philosophical consequences that I can write more than half a page about. I read the ...
10
votes
7answers
1k views

“Proof” that ZFC is inconsistent using Turing machines

I came across the following "proof" for the inconsistency of ZFC and can't find the flaw in it (if there is one...): Construct a Turing machine A which sequentially runs on all proofs in ZFC and ...
10
votes
2answers
140 views

How does Borelness overlap with definability, computability, or constructiveness?

Background: I am writing a short paper aimed at math undergrads and focused as narrowly as possible on Borel equivalence relations. So, e.g., I am not assuming familiarity with recursion theory and am ...
8
votes
2answers
92 views

A homogeneous set of some kind

Let $f : \mathbb{N}^k \to \mathbb{N}$ be a computable total function such that $f (\vec{x}) > \max \vec{x}$ for all $\vec{x}$. Question. Why is there a decidable set $A$ such that ...
7
votes
3answers
187 views

From lightface $\Sigma^1_1$ to boldface $\mathbf\Delta^1_1$

Fix some standard Polish space, e.g. Baire's space. It's a simple observation that every $\Delta^1_1$ is also $\mathbf\Delta^1_1$. It is the same observation that $\Sigma^1_1$ becomes ...
7
votes
2answers
155 views

Complexity of the set of computable ordinals

According to http://en.wikipedia.org/wiki/Analytical_hierarchy The set of all natural numbers which are indices of computable ordinals is a $\Pi^1_1$ set which is not $\Sigma^1_1$. However, "the ...
6
votes
3answers
226 views

Is the set of all deducible formulas decidable?

Consider any standard, "sufficiently expressive" first-order theory (say, $ZFC$ or Peano arithmetic) so that all the usual arithmetization and incompleteness results hold. The set $D$ of deducible ...
6
votes
1answer
241 views

What are $\Sigma _n^i$, $\Pi _n^i$ and $\Delta _n^i$?

Sometimes reading on wikipedia or in this site (and in very different context like topology, arithmetic and logic) I have found these symbols $\Sigma _n^i$, $\Pi _n^i$ and $\Delta _n^i$. They are ...
6
votes
3answers
329 views

Why are $\Delta_1$ sentences of arithmetic called recursive?

The arithmetic hierarchy defines the $\Pi_1$ formulae of arithmetic to be formulae that are provably equivalent to a formula in prenex normal form that only has universal quantifiers, and $\Sigma_1$ ...
6
votes
1answer
90 views

Primitive recursive function which isn't $\Delta_0$

What is the simplest/cutest example (and/or example with the most student-friendly proof that it is an example) of a primitive recursive function which isn't representable by a $\Delta_0$ wff?
6
votes
0answers
54 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 ...
5
votes
7answers
532 views

Is there at least one irrational number with the property that it cannot be defined by a finite string of information?

Ok, so maybe that wasn't the best way of phrasing the question, but I think it's specific enough. Let me explain myself a bit more below in case I am wrong. So I'm assuming (although I've never ...
5
votes
2answers
345 views

Example of an UnDecidable Logical Theory which is an extension of a Logical Decidable Theory?

Let $T_1$ and $T_2$ be two first-order logical theories (over the same signature) such that $T_1 \subseteq T_2$ and both are recursively axiomatized. My question is the following: is it possible that ...
5
votes
3answers
205 views

Recursive function that outputs its own code

This problem is probably a rather trivial one, since I have the impression, that it is a textbook-style one, but nonetheless somehow it won't give in. Here it is: I have to show that there exists a ...
5
votes
3answers
200 views

What does the concept of computation actually mean?

My question is very general, and the kind of answer I look for would be as low level as it could be. I think I may illustrate my query more succinctly with an example. In propositional logic, you ...
5
votes
2answers
112 views

Effective cardinality

Consider $X,Y \subseteq \mathbb{N}$. We say that $X \equiv Y$ iff there exists a bijection between $X$ and $Y$. We say that $X \equiv_c Y$ iff there exist a bijective computable function between ...
5
votes
2answers
320 views

Does every infinite $\Sigma^1_1$ set have an infinite $\Delta^1_1$ subset?

The question is exactly that in the title: Does every infinite $\Sigma^1_1$ set of natural numbers have an infinite $\Delta^1_1$ subset? Some background: The lower-level analog of this question, Does ...
5
votes
1answer
136 views

The Permitting Method

Define the term late permitting in the following way: $C$ late permits an element $x$ to enter $A_{s+1}$ if for a fixed computable function $f$ with $f(n)>n$, there exists $y\leq x$ such that $y\in ...
4
votes
4answers
189 views

Examples of partial functions outside recursive function theory?

My math background is very narrow. I've mostly read logic, recursive function theory, and set theory. In recursive function theory one studies partial functions on the set of natural numbers. Are ...
4
votes
3answers
627 views

Prove Gödel's incompleteness theorem using halting problem

How can you prove Gödel's incompleteness theorem from the halting problem? Is it really possible to prove the full theorem? If so, what are the differences between original proof and proof by ...
4
votes
1answer
201 views

What is the power of a recursive language vs. that of one that is recursively enumerable?

I am simply wondering, as the title states, what the central differences are between recursive and recursively enumerable languages? If I am not mistaken a recursive language is a is Turing decidable ...
4
votes
2answers
230 views

What is the relationship between ZFC and Turing machine?

I did not learn Logic properly but so far I understand that proof systems can be viewed as a kind of machine. For proof system, ZFC seems to be the most powerful one that we use so far. Similarly, for ...
4
votes
1answer
79 views

Why is computable function in $\Delta_{1}^0$?

I am not sure why computable functions are in $\Delta_{1}^0$. Can anyone explain this?
4
votes
1answer
77 views

$\Sigma^0_n$ complete sets

Does anyone know of a way of showing that a $\Sigma^0_n$-complete set is not $\Pi^0_n$ without having to appeal to $\Sigma^0_n$-universal sets? For instance a more direct diagonalization argument ...
4
votes
2answers
590 views

Is this function a primitive recursive function?

Let $t \in \mathbb{N}$ and consider the function $f: \mathbb{N} \rightarrow \mathbb{N}$, defined by $f_t (m)= 2 \uparrow^{m} t$, where "$\uparrow$" is Knuth's up-arrow notation (which can be ...
4
votes
1answer
75 views

Notation in Sacks' 'Higher Recursion Theory'

I'm having trouble with the notation in Sacks' Higher Recursion Theory. I've asked specific questions from page 4. Instead of reading my question in detail and trying to understand my confusion (which ...
4
votes
1answer
104 views

A technique for deciding satisfiability in fragments of first-order logic

By Goedels completeness theorem satisfiability in first-order logic is $\Pi_1$. So to obtain decidability in some fragment, it is enough to show that satisfiability is $\Sigma_1$ in this fragment. I ...
4
votes
1answer
103 views

Question about $\Sigma_n$-soundness

According to wikipedia (http://en.wikipedia.org/wiki/%CE%A9-consistent_theory#Definition): "$\Sigma_n$-soundness has the following computational interpretation: if the theory proves that a program C ...
4
votes
1answer
140 views

Projecting onto (lightface) Borel sets

Suppose $A \subseteq \omega^{\omega} \times \omega^{\omega}$ is Borel. If we project $A$ onto $\omega^\omega$, we get a $\mathbf{\Sigma^{1}_{1}}$ set $\{y: \exists x (y,x) \in A\}$. What if we project ...
3
votes
3answers
138 views

A non-arithmetical set?

A set is called arithmetical if it can be defined by a first-order formula in Peano arithmetic. I first encountered these sets when exploring the arithmetical hierarchy in the context of ...
3
votes
3answers
225 views

Proving that $\{x|\varphi_x \; \text{is extendible to a total computable function} \} \neq \omega$

The problem that I'm working on is to prove that $$ Ext=\{ x \ | \ \varphi_x \text{is extendible to a total computable function}\} $$ is not equal to $\omega$. Here $\varphi_x$ is the $x$-th partial ...
3
votes
3answers
128 views

Definition of computability of real numbers?

What exactly does it mean to say that a real number $x$ is computable? I can think of two reasonable definitions but I am not sure whether or not they are equivalent: 1) There is an algorithm which ...
3
votes
1answer
133 views

Is every recursively enumerable set $A \subseteq \mathbb{N}$ also recursive?

Is every recursively enumerable set $A \subseteq \mathbb{N}$ also recursive ? I'm not particularly interested in a detailed proof or counterexample, just a quick argument why this affirmation should ...
3
votes
3answers
236 views

Placing some sets in the arithmetic hierarchy

I'm working on the following problem: let $W_e$ be the computably enumerable set which is the domain of the $e$-th Turing program, and $K$ be the Halting problem, at which level of the arithmetic ...
3
votes
1answer
42 views

Determine whether two primitive recursive functions are equal

Is there an algorithm to determine whether two primitive recursive functions are equal (as mathematical functions)?
3
votes
2answers
171 views

delta-zero formula and power of 2

How can we show that $x=2^k$ for some $k$ is equivalent (in the Naturals) to a $\Delta_0$ formula? So, I'm stuck at showing that 'y divides x' and '2 divides y' are equivalent in the Naturals to ...
3
votes
2answers
86 views

How does second-order logic relate to lambda calculus?

How does second-order arithmetic/logic relate to lambda calculus? By lambda calculus, I mean both typed and untyped. And is there any relationship with recursive and recursively enumerable sets? ...
3
votes
1answer
94 views

The set of arithmetical numbers

Define $x\in\mathbb{R}$ to be arithmetical number if the set $\{\langle p, q \rangle \in \mathbb{Z}^2 : \frac{p}{q} < x\}$ is an arithmetical set. Define $x\in\mathbb{C}$ to be arithmetical number ...
3
votes
2answers
278 views

How can I prove that this set is recursively enumerable?

Let $g _c (x)$ be the output of a program that is encoded by $c \in \mathbb{N}$ for the given input $x$. $g_c$ can obviously be undefined, in case the program encoded by $c$ doesn't halt. If we define ...
3
votes
1answer
130 views

Non-universal Turing machines

Is it possible to have two (or more) non-universal Turing machines labeled $A_1$ and $A_2$, such that if $f(A_i)$ is the set of functions computable by $A_i$, and S={every computable function} then ...
3
votes
1answer
188 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 , ...
3
votes
2answers
125 views

Slick way to define p.c. $f$ so that $f(e) \in W_{e}$

Is there a slick way to define a partial computable function $f$ so that $f(e) \in W_{e}$ whenever $W_{e} \neq \emptyset$? (Here $W_{e}$ denotes the $e^{\text{th}}$ c.e. set.) My only solution is to ...
3
votes
1answer
147 views

Is there a decidable theory in propositional logic whose consequences are not decidable?

I want to know if there is a decidable theory in propositional logic whose consequences are not decidable. If there is, can we have a constructive example or we can only prove the existence of it? ...
3
votes
1answer
74 views

Limits of Computable sequences

Turing introduced the fact that the limit of a computable sequence is not necessarily computable, and the Specker sequence is a specific example of such a number (with supremum not computable). My ...
3
votes
2answers
92 views

Upper and Lower bounds on proof length

Given a First Order language say, for arithmetic $\langle 0, 1, +,\cdot ,^\wedge, S \rangle$, Can one establish any lower or upper bounds on the length of proofs from certain recursively enumerable ...
3
votes
1answer
58 views

Extending the recursive functions to higher classes in the aritmetical hierarchy

It is an important theorem that the recursive functions are exactly those which are definable by $\Delta^0_1$ formulas. We have just finished the part about incompleteness in a course I'm TA'ing, and ...
3
votes
1answer
54 views

Recursion schema and the arithmetical hierarchy

In computability we define the following basic functions, the zero function, the successor function, and the functions $I_{n,k}(x_1,\ldots,x_n)=x_k$ for $k\leq n$. Next we define three schemata for ...
3
votes
1answer
122 views

Members of (lightface) Borel sets

I'm fairly certain this question has a very simple answer, and that I've learned it before; I just can't seem to remember it. Suppose I have a nonempty lightface Borel set $X\subseteq 2^\omega$. What ...

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