**85**

votes

**42**answers

17k views

### What are the most attractive Turing undecidable problems in mathematics?

What are the most attractive Turing undecidable problems in mathematics?
There are thousands of examples, so please post here only the most attractive, best examples. Some examples already appear on ...

**62**

votes

**2**answers

2k views

### How feasible is it to prove Kazhdan's property (T) by a computer?

Recently, I have proved that Kazhdan's property (T) is theoretically provable
by computers (arXiv:1312.5431,
explained below), but I'm quite lame with computers and have
no idea what they actually ...

**55**

votes

**3**answers

5k views

### Can a group be a universal Turing machine?

This question was inspired by this blog post of Jordan Ellenberg.
Define a "computable group" to be an at most countable group $G$ whose elements can be represented by finite binary strings, with the ...

**45**

votes

**8**answers

5k views

### Succinctly naming big numbers: ZFC versus Busy-Beaver

Years ago, I wrote an essay called Who Can Name the Bigger Number?, which posed the following challenge:
You have fifteen seconds. Using standard math notation, English words, or both, name a single ...

**41**

votes

**5**answers

4k views

### Which graphs are Cayley graphs?

Every group presentation determines the corresponding Cayley graph, which has a node for each group element, and arrows labeled with the generators to get from one group element to another.
My main ...

**29**

votes

**2**answers

938 views

### Does an existence of large cardinals have implications in number theory or combinatorics?

Does an existence of large cardinals have implications in more down-to-earth fields like number theory, finite combinatorics, graph theory, Ramsey theory or computability theory? Are there any ...

**27**

votes

**3**answers

1k views

### Are surjectivity and injectivity of polynomial functions from $\mathbb{Q}^n$ to $\mathbb{Q}$ algorithmically decidable?

Is there an algorithm which, given a polynomial $f \in \mathbb{Q}[x_1, \dots, x_n]$,
decides whether the mapping $f: \mathbb{Q}^n \rightarrow \mathbb{Q}$ is surjective,
respectively, injective? --
And ...

**26**

votes

**10**answers

2k views

### Can We Decide Whether Small Computer Programs Halt?

The undecidability of the halting problem states that there is no general procedure for deciding whether an arbitrary sufficiently complex computer program will halt or not.
Are there some large $n$ ...

**26**

votes

**3**answers

1k views

### “Simpler” statements equivalent to Con(PA) or Con(ZFC)?

Given any reasonable formal system F (e.g., Peano Arithmetic or ZFC), we all know that one can construct a Turing machine that runs forever iff F is consistent, by enumerating the theorems of F and ...

**25**

votes

**3**answers

1k views

### Is it decidable whether or not a collection of integer matrices generates a free group?

Suppose we have integer matrices $A_1,\ldots,A_n\in\operatorname{GL}(n,\mathbb Z)$. Define $\varphi:F_n\to\operatorname{GL}(n,\mathbb Z)$ by $x_i\mapsto A_i$. Is there an algorithm to decide whether ...

**24**

votes

**15**answers

4k views

### What's a magical theorem in logic?

Some theorems are magical: their hypotheses are easy to meet, and when invoked (as lemmas) in the midst of an otherwise routine proof, they deliver the desired conclusion more or less ...

**24**

votes

**1**answer

732 views

### Are sums of sequences decidable?

Suppose that $f,g$ are rational functions with integer coefficients such that $\sum_{n=0}^{\infty}f(n)$ and $\sum_{n=0}^{\infty}g(n)$ both converge. Is it decidable whether
...

**24**

votes

**1**answer

818 views

### Can a string's sophistication be defined in an unsophisticated way?

This question is about sophistication, a way of measuring the amount of "interesting, non-random information" in a binary string, which was proposed by Kolmogorov and others in the 1980s. I'll define ...

**23**

votes

**2**answers

1k views

### Given a polynomial-time algorithm, can we compute an explicit polynomial time bound just from the program?

Question. Given a Turing-machine program $e$, which
is guaranteed to run in polynomial time, can we computably
find such a polynomial?
In other words, is there a
computable function $e\mapsto p_e$, ...

**23**

votes

**4**answers

18k views

### How large is TREE(3)?

Friedman, in http://www.math.osu.edu/~friedman.8/pdf/EnormousInt112201.pdf, shows that TREE(3) is much larger than n(4), itself bounded below by $A^{A(187195)}(3)$ (where $A$ is the Ackerman ...

**23**

votes

**2**answers

1k views

### Are any natural examples of Gödel speed-up known?

In 1936 Gödel announced a theorem to the effect that proofs of certain theorems $T_1,T_2,\ldots$ become dramatically shorter when one passes from a formal system, such as Peano arithmetic PA, to a ...

**23**

votes

**0**answers

869 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 ...

**22**

votes

**10**answers

3k views

### Physics and Church–Turing Thesis

Is there constructed some set of physical laws from which we can logically obtain that any function that can be implemented in some device is Turing computable?
EDIT
I believe that if we restrict ...

**22**

votes

**3**answers

721 views

### Does the exact pair phenomenon for partial orders occur in your area of mathematics?

Suppose that I have a partial order P and an increasing sequence $a_0< a_1<a_2<\cdots$ of elements of $P$. A pair of elements (b,c) from P is said to be an exact pair for this sequence, if
...

**22**

votes

**0**answers

657 views

### On certain representations of algebraic numbers in terms of trigonometric functions

Let's say that a real number has a simple trigonometric representation, if it can be represented as a product of zero or more rational powers of positive integers and zero or more (positive or ...

**20**

votes

**3**answers

1k views

### Is the theory of categories decidable?

There are a lot of theorems in basic homological algebra, such as the five lemma or the snake lemma, that seem like they'd be more easily proven by computer than by hand. This led me to consider the ...

**20**

votes

**4**answers

1k views

### Algorithmically unsolvable problems in topology

This question is inspired by a paper by B. Poonen that appeared on the arxiv some time ago: http://arxiv.org/abs/1204.0299. The paper gives a sample of algorithmically unsolvable problems from various ...

**20**

votes

**1**answer

507 views

### Busy Beaver modulo 2

There is well-known Rado's "Busy Beaver" sequence — the maximal number of marks which a halting Turing machine with n states, 2 symbols (blank, mark) can produce onto an initially blank two-way ...

**19**

votes

**3**answers

1k views

### Inverse Ackermann - primitive recursive or not?

I wanted to put this originally on math.stackexchange, since I considered it to be a straightforward question and probably a fairly known fact. After I failed to solve the problem, I browsed through ...

**19**

votes

**3**answers

1k views

### Is deciding whether a Turing machine *provably* runs forever equivalent to the halting problem?

Assume for this question that ZF set theory is sound.
Now consider the language "PROVELOOP," which consists of all descriptions of Turing machines M, for which there exists a ZF proof that M runs ...

**19**

votes

**1**answer

717 views

### Why isn't this a computable description of the ordinal of ZF?

In a previous MO question, I was told by several commenters that
(a) it's known that there exists a computable ordinal $\alpha_{ZF}$ that "encodes the strength of ZF set theory" (i.e., a least ...

**19**

votes

**1**answer

832 views

### Word problem for fundamental group of submanifolds of the 4-sphere

Given any finitely-presented group $G$, there are a few equivalent techniques for constructing smooth/PL 4-manifolds $M$ such that $\pi_1 M$ is isomorphic to $G$. For most constructions of these ...

**18**

votes

**5**answers

2k views

### Why relativization can't solve NP !=P?

If this problem is really stupid, please close it. But I really wanna get some answer for it. And I learnt computational complexity by reading books only.
When I learnt to the topic of relativization ...

**18**

votes

**4**answers

836 views

### Kolmogorov complexity is the strongest noncomputable function

Yury I. Manin says that Kolmogorov complexity (in some nontrivial sense) is the strongest noncomputable function ("Колмогоровская сложность... невычислима... она во многих интересных смыслах ...

**18**

votes

**1**answer

825 views

### Looking for a copy of Leo Harrington's unpublished notes on the first nonprojectible ordinal

Sometime around 1975, Leo Harrington wrote a set of notes, apparently 13 pages long, entitled Kolmogorov's $R$-operator and the first nonprojectible ordinal. I do not know how widely they were ...

**17**

votes

**5**answers

4k views

### The problem of finding the first digit in Graham's number

Motivation
In this BBC video about infinity they mention Graham's number. In the second part, Graham mentions that "maybe no one will ever know what [the first] digit is". This made me think: Could ...

**16**

votes

**3**answers

1k views

### Is there a computable model of ZFC?

Background
Assuming ZFC is consistent, then by downward Löwenheim–Skolem, there is a countable model (M,$\in$) of ZFC. Since the universe M is countable, we may as well think of it as actually being ...

**16**

votes

**1**answer

629 views

### Polynomial-time algorithm to compare numbers in Conway chained arrow notation

I am looking for a polynomial-time algorithm which, given a character string containing two numbers in Conway's chained arrow notation for large numbers, indicates whether the first number is less ...

**15**

votes

**5**answers

1k views

### Intermediate value theorem on computable reals

Wikipedia says that the intermediate value theorem “depends on (and is actually equivalent to) the completeness of the real numbers.” It then offers a simple counterexample to the ...

**15**

votes

**7**answers

1k views

### Between mu- and primitive recursion

It is well known that primitive recursion is not powerful enough
to express all functions, Ackermann function being probably the best
known example.
Now, in the logic courses (that I have had look ...

**15**

votes

**3**answers

605 views

### Which distributions can you sample if you can sample a Gaussian?

Explicitly: You have a computer that is able to pick a real number at random according to the normal distribution: $\mathcal{N}(0,1) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$. Which distributions can this ...

**15**

votes

**1**answer

706 views

### Undecidable theories easier than $Q$

Most proofs of undecidability for various theories (pure logic with binary relation, group theory, etc.) show that the natural numbers and Robinson's $Q$, in one form or another, can be encoded ...

**15**

votes

**1**answer

869 views

### Lawvere's fixed point theorem and the Recursion Theorem

Building off of Qiaochu's comment on my answer to a previous mathoverflow question, I would like to know: can the Recursion Theorem, $$\forall e\exists k[\Phi_e\text{ is total }\implies ...

**15**

votes

**3**answers

553 views

### Do decidable properties of finitely presented groups depend only on the profinitization?

This is a just-for-fun question inspired by this one. Let $P$ be a property of finitely presentable groups. Suppose that
The truth of $P(G)$ only depends on the isomorphism class of $G$.
Given a ...

**15**

votes

**2**answers

1k views

### Structure theorems for Turing-decidable languages?

Languages decidable by weak models of computation often have certain necessary characteristics, e.g. the pumping lemma for regular languages or the pumping lemma for context-free languages. Such ...

**15**

votes

**2**answers

609 views

### Effectively closed computable functions

I've recently been interested in the following type of functions. A total computable function f:N→N is effectively closed if there is a computable function p such that f[N \ We] = N \ Wp(e), ...

**15**

votes

**0**answers

273 views

### Computability of Brauer groups

A friend of mine and I were talking about computable algebra, and this question came up. The answer may already be known, but I couldn't find it with Google:
Suppose I have a countable field, $k$. ...

**14**

votes

**5**answers

1k views

### Are the two meanings of “undecidable” related?

I am usually confused by questions of the type "could such and such a problem be undecidable", because as far as I know there are two distinct possible meanings of "undecidable". I regard the ...

**14**

votes

**2**answers

661 views

### Are the axioms for higher category-theory effectively computable?

I ask this, although I don't conduct any research in the area, or even plan to. -- There seems to be general agreement that the axioms for higher categories grow very rapidly in complexity as the ...

**13**

votes

**6**answers

1k views

### Non-constructive proofs of decidability?

Are there examples of sets of natural numbers that are proven to be decidable but by non-constructive proofs only?

**13**

votes

**7**answers

1k views

### Finding the largest integer describable with a string of symbols of predefined length

(This question is motivated by the reading of the article Large numbers and unprovable theorems by Joel Spencer, which can be found at ...

**13**

votes

**2**answers

1k views

### What proofs cannot be relativized

I am afraid this post may show my naivety. At a recent conference, someone told me that there are some arguments in computability theory that don't relativize. Unfortunately, this person (who I ...

**13**

votes

**2**answers

1k views

### Why is Kleene's notion of computability better than Banach-Mazur's?

In this post about the difference between the recursive and effective topos, Andrej Bauer said:
If you are looking for a deeper explanation, then perhaps it is fair to say that the Recursive Topos ...

**13**

votes

**1**answer

660 views

### Are wild problems related to undecidable ones?

In representation theory, there is a well-known notion of a wild classification problem (such problems have been discussed often on this forum, for example, here). In logic, there is a notion of an ...

**13**

votes

**2**answers

894 views

### Can randomness add computability?

I have been looking at Church's Thesis, which asserts that all intuitively computable functions are recursive. The definition of recursion does not allow for randomness, and some people have suggested ...