78
votes

Accepted

### Why is uncomputability of the spectral decomposition not a problem?

The singular value decomposition, when applied to a real symmetric matrix $A = \sum_i \lambda_i(A) u_i(A) u_i(A)^T$, computes a stable mathematical object (spectral measure $\mu_A = \sum_i \delta_{\...

78
votes

Accepted

### Will this Turing machine find a proof of its halting?

It is a very nice question. The answer is yes, the machine will find a proof of its own halting nature, and it will halt when it does so.
I claim this is a consequence of Löb's theorem. Let $M$ be a ...

77
votes

### Will this Turing machine find a proof of its halting?

Build a second machine $N$. $N$ searches for a proof in ZFC of, "if $N$ halts then $M$ halts". If it finds one, it halts.
ZFC can argue as follows. "Suppose $N$ halts. Then it found a ...

55
votes

### On mathematical arguments against Quantum computing

Here are some references and following them a short answer.
A good reference for the current situation to start with is John Preskill's recent paper "Quantum Computing in the NISQ era and beyond" ...

Community wiki

53
votes

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

I think it is appropriate to let MO users know (the OP himself knows it well) that this question was recently solved: it is feasible to provide a computer based proof for property (T) using the Ozawa ...

46
votes

### On mathematical arguments against Quantum computing

Scott Aaronson has this list of Eleven Objections, involving both mathematics and physics arguments.
What I did is to write out every skeptical argument against the
possibility of quantum ...

Community wiki

44
votes

Accepted

### Decision problems for which it is unknown whether they are decidable

An integer linear recurrence sequence is a sequence $x_0, x_1, x_2, \ldots$ of integers that obeys a linear recurrence relation
$$x_n = a_1 x_{n-1} + a_2 x_{n-2} + \cdots + a_d x_{n-d}$$
for some ...

Community wiki

38
votes

Accepted

### Is it decidable to check if an element has finite order or not?

A finitely presented group with decidable word problem and undecidable order problem is in McCool, James
Unsolvable problems in groups with solvable word problem.
Canad. J. Math. 22 1970 836–838.

31
votes

### Using Busy Beavers to prove conjectures

Indeed, the second option is a problem: the BB($n$) cannot be computed in ZF for $n$ large (an explicit bound $n\ge 7910$ was given by Aaronson-Yedidia in their article A Relatively Small Turing ...

30
votes

### Decision problems for which it is unknown whether they are decidable

In Conway's Game of Life, the problem of deciding whether a given pattern with finitely many live cells is a Garden of Eden (i.e. whether it lacks a predecessor).
The main obstacle is that there could ...

Community wiki

28
votes

Accepted

### Any important consequences with presupposition of $\mathbf{P} \neq \mathbf{NP}$

Because there are natural computational problems involving many mathematical objects, there are a bunch of implications of complexity class separations like $\mathrm{P} \neq \mathrm{NP}$. I think the ...

Community wiki

28
votes

### On mathematical arguments against Quantum computing

The promise of quantum computing supremacy is bunk by Colin Earl references to
Polynomial Time and Extravagant Models by Leonid A. Levin
On Quantum Computing by Oded Goldreich
Note (d) Quantum ...

Community wiki

27
votes

Accepted

### Does "every" first-order theory have a finitely axiomatizable conservative extension?

Essentially, yes. An old result of Kleene [1], later strengthened by Craig and Vaught [2], shows that every recursively axiomatizable theory in first-order logic without identity, and every ...

27
votes

Accepted

### Is "almost-solvability" of Diophantine equations decidable?

A Diophantine equation is almost-satisfiable iff it is satisfiable over the ring $\widehat{\mathbb Z}$, the profinite completion of $\mathbb Z$ (also called by some the Prüfer ring), by a standard ...

Community wiki

25
votes

Accepted

### Languages beyond enumerable

Yes, for starters there is the arithmetical hierarchy, where enumerable = $\Sigma^0_1$ and it continues $\Pi^0_1$, $\Delta^0_2$, $\Sigma^0_2$ etc.
See also the Computability Menagerie.

25
votes

Accepted

### How (non-)computable is set theory?

The question is extremely interesting, and I have looked into this kind of thing with various colleagues (including Russell Miller and Kameryn Williams), although our investigation has not yet ...

25
votes

### Using Busy Beavers to prove conjectures

Although the other answers point out correctly that the exact value of $\text{BB}(n)$ is independent of ZF for large enough and even moderately sized values of $n$, nevertheless I should like to point ...

24
votes

Accepted

### The Lucas argument vs the theorem-provers -- who wins and why?

Yes, computers can infer that the Gödel sentence is true. This is performed in a meta-theory which is stronger than the object theory, as it has to be.
For example, Russell O'Connor formalized Gödel'...

24
votes

Accepted

### Do we expect that sufficiently large computable ordinals settle every question of arithmetic?

The question of whether a computable linear order is well-founded is $\Pi^1_1$-complete, so this is true in a sense:
There is a computable function $F$ such that, for every sentence $\varphi$ in the ...

24
votes

Accepted

### "Natural" undecidable problems not reducible to the halting problem

The problems reducible to the halting problem are exactly the problems of complexity $\Delta^0_2$ in the arithmetic hierarchy, and there are indeed many natural problems outside of this class. In this ...

23
votes

Accepted

### Hard-to-compute real numbers

EDIT: This was in a comment below, but I now think it should be part of the main answer:
There are two different ways to ask the question in the OP:
Is there a real number $r$ such that no polytime ...

23
votes

### Why is uncomputability of the spectral decomposition not a problem?

The SVD decomposition falls under the family of phenomena where discontinuity implies non-computability. (Intuitively, this is because, at the point of discontinuity, infinite precision is required.)
...

22
votes

Accepted

### (non-)existence of the aperiodic monotile

This recent preprint claims to find such a tile.
David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss, “An aperiodic monotile”, (2023-03-20) arXiv:2303.10798
A longstanding open ...

Community wiki

22
votes

Accepted

### What is the relationship between Turing Machines and Gödel's Incompleteness Theorem?

It's simple. If the halting problem is undecidable, then PA is not complete, since otherwise, you could solve the halting problem by searching for proofs in PA. And the same argument works for any ...

22
votes

Accepted

### Is there a known Turing machine which halts if and only if the Collatz conjecture has a counterexample?

Let's note that this is not a question of whether Collatz is undecidable.
The statement $\neg\mathrm{Con}(PA)$ is undecidable (by $PA$, assuming $PA$ is consistent) but nevertheless $\neg\mathrm{Con}(...

22
votes

### Why is uncomputability of the spectral decomposition not a problem?

This is primarily an issue of backwards vs. forwards stability. Good SVD algorithms are backwards stable in the sense that the computed singular values and singular vectors are the true singular ...

22
votes

Accepted

### Are the vertical sections of the Ackermann function primitive recursive?

No, already $A(n,3)$ is not primitive recursive. Let me use the essentially equivalent up-arrow notation: $A(n,m)=2\uparrow^{n-1}m$, and argue why $f(n)=2\uparrow^n 3$ is not PR. I claim $f(2n-2)\geq ...

21
votes

### Is it decidable to check if an element has finite order or not?

The decidability of the word problem does not imply the decidability of the order problem, and in fact the following more general result holds.
Theorem. Let $\mathbf{a}, \, \mathbf{b}, \, \mathbf{c}...

21
votes

Accepted

### Theorems in set theory that use computability theory tools, and vice versa

Here are several examples.
There is a natural affinity between forcing and many constructions in the Turing degrees. Specifically, many constructions of degrees by meeting requirements in succession ...

20
votes

### The Halting Problem and Church's Thesis

The invocation of Church's thesis is not a religuous move but rather a warning to the reader that the author is describing informally an effective procedure which could be translated into a ...

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