first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, ...

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112
votes
0answers
9k views

Ultrafilters and automorphisms of the complex field

It is well-known that it is consistent with $ZF$ that the only automorphisms of the complex field $\mathbb{C}$ are the identity map and complex conjugation. For example, we have that ...
111
votes
26answers
12k views

What are some reasonable-sounding statements that are independent of ZFC?

Every now and then, somebody will tell me about a question. When I start thinking about it, they say, "actually, it's undecidable in ZFC." For example, suppose A is an abelian group such that every ...
82
votes
15answers
14k views

Why worry about the axiom of choice?

As I understand it, it has been proven that the axiom of choice is independent of the other axioms of set theory. Yet I still see people fuss about whether or not theorem X depends on it, and I don't ...
73
votes
11answers
14k views

Knuth's intuition that Goldbach might be unprovable

Knuth's intuition that Goldbach's conjecture (every even number greater than 2 can be written as a sum of two primes) might be one of the statements that can neither be proved nor disproved really ...
70
votes
11answers
7k views

Checkmate in $\omega$ moves?

Is there a chess position with a finite number of pieces on the infinite chess board $\mathbb{Z}^2$ such that White to move has a forced win, but Black can stave off mate for at least $n$ moves for ...
68
votes
16answers
15k views

What if Current Foundations of Mathematics are Inconsistent? [closed]

The title of the question is also the title of a talk by Vladimir Voevodsky, available here. Had this kind of opinion been expressed before? EDIT. Thanks to all answerers, commentators, voters, ...
66
votes
41answers
15k 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 ...
65
votes
9answers
10k views

Solutions to the Continuum Hypothesis

(A related MO question: What is the general opinion on the Generalized Continuum Hypothesis? ) Background The Continuum Hypothesis (CH) posed by Cantor in 1890 asserts that $ \aleph_1=2^{\aleph_0}$. ...
52
votes
3answers
5k views

Does every non-empty set admit a group structure (in ZF)?

It is easy to see that in ZFC, any non-empty set $S$ admits a group structure: for finite $S$ identify $S$ with a cyclic group, and for infinite $S$, the set of finite subsets of $S$ with the binary ...
51
votes
28answers
5k views

Can infinity shorten proofs a lot?

I've just been asked for a good example of a situation in maths where using infinity can greatly shorten an argument. The person who wants the example wants it as part of a presentation to the general ...
50
votes
14answers
4k views

When are two proofs of the same theorem really different proofs

Many well-known theorems have lots of "different" proofs. Often new proofs of a theorem arise surprisingly from other branches of mathematics than the theorem itself. When are two proofs really the ...
48
votes
6answers
10k views

How many orders of infinity are there?

Define a growth function to be a monotone increasing function $F: {\bf N} \to {\bf N}$, thus for instance $n \mapsto n^2$, $n \mapsto 2^n$, $n \mapsto 2^{2^n}$ are examples of growth functions. Let's ...
48
votes
8answers
5k views

Is there any formal foundation to ultrafinitism?

Ultrafinitism is (I believe) a philosophy of mathematics that is not only constructive, but does not admit the existence of arbitrarily large natural numbers. According to wikipedia, it has been ...
48
votes
2answers
5k views

Does Godel's incompleteness theorem admit a converse?

Let me set up a strawman: One might entertain the following criticism of Godel's incompleteness theorem: why did we ever expect completeness for the theory of PA or ZF in the first place? Sure, one ...
47
votes
8answers
3k views

Can we unify addition and multiplication into one binary operation? To what extent can we find universal binary operations?

The question is the extent to which we can unify addition and multiplication, realizing them as terms in a single underlying binary operation. I have a number of questions. Is there a binary ...
47
votes
3answers
6k views

Nelson's program to show inconsistency of ZF

At the end of the paper Division by three by Peter G. Doyle and John H. Conway, the authors say: Not that we believe there really are any such things as infinite sets, or that the Zermelo-Fraenkel ...
46
votes
12answers
8k views

Logic in mathematics and philosophy

What are the relations between logic as an area of (modern) philosophy and mathematical logic. The world "modern" refers to 20th century and later, and I am curious mainly about the second half of ...
43
votes
26answers
6k views

Sexy vacuity …

I included this footnote in a paper in which I mentioned that the number of partitions of the empty set is 1 (every member of any partition is a non-empty set, and of course every member of the empty ...
43
votes
5answers
3k views

Does anyone know a polynomial whose lack of roots can't be proved?

In Ebbinghaus-Flum-Thomas's Introduction to Mathematical Logic, the following assertion is made: If ZFC is consistent, then one can obtain a polynomial $P(x_1, ..., x_n)$ which has no roots in the ...
43
votes
5answers
4k views

Decidability of chess on an infinite board

The recent question Do there exist chess positions that require exponentially many moves to reach? of Tim Chow reminds me of a problem I have been interested in. Is chess with finitely many men on an ...
42
votes
8answers
4k views

Is all ordinary mathematics contained in high school mathematics?

By high school mathematics I mean Elementary Function Arithmetic (EFA), where one is allowed +,×, xy, and a weak form of induction for formulas with bounded quantifiers. This is much weaker than ...
42
votes
2answers
3k views

How would you solve this tantalizing Halmos problem?

1-ab invertible => 1-ba invertible has a slick power series "proof" as below, where Halmos asks for an explanation of why this tantalizing derivation succeeds. Do you know one? Geometric series. In ...
41
votes
3answers
2k views

Forcing as a new chapter of Galois Theory

There is a (very) long essay by Grothendieck with the ominous title La Longue Marche à travers la théorie de Galois (The Long March through Galois Theory). As usual, Grothendieck knew what he was ...
40
votes
14answers
4k views

What is the high-concept explanation on why real numbers are useful in number theory?

The utopian situation in mathematics would be that the statement and the proof of every result would live "in the same world", at the same level of mathematical complexity (in a broad sense), unless ...
40
votes
18answers
4k views

What are some results in mathematics that have snappy proofs using model theory?

I am preparing to teach a short course on "applied model theory" at UGA this summer. To draw people in, I am looking to create a BIG LIST of results in mathematics that have nice proofs using model ...
39
votes
9answers
4k views

How do they verify a verifier of formalized proofs?

In an unrelated thread Sam Nead intrigued me by mentioning a formalized proof of the Jordan curve theorem. I then found that there are at least two, made on two different systems. This is quite an ...
39
votes
4answers
4k views

What was Gödel's real achievement?

When I first heard of the existence of Gödel's theorem, I was amazed not just at the theorem but at the fact that the question could be made precise enough to answer: how on earth, even in ...
38
votes
15answers
9k views

Most 'unintuitive' application of the Axiom of Choice?

It is well-known that the axiom of choice is equivalent to many other assumptions, such as the well-ordering principle, Tychonoff's theorem, and the fact that every vector space has a basis. Even ...
38
votes
5answers
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 ...
37
votes
30answers
8k views

nontrivial theorems with trivial proofs

A while back I saw posted on someone's office door a statement attributed to some famous person, saying that it is an instance of the callousness of youth to think that a theorem is trivial because ...
35
votes
7answers
2k views

How would one even begin to try to prove that a simple number-theoretic statement is undecidable?

This question is closely related to this one: Knuth's intuition that Goldbach might be unprovable. It stems from my ignorance about non-standard models of arithmetic. In a comment on the other ...
35
votes
5answers
2k views

What's wrong with the surreals?

Of all the constructions of the reals, the construction of the surreals seems the most elegant to me. It seems to immediately capture the total ordering and precision of Dedekind cuts at a ...
35
votes
0answers
505 views

Reasons to prefer one large prime over another to approximate characteristic zero

Background: In running algebraic geometry computations using software such as Macaulay2, it is often easier and faster to work over $\mathbb F_p = \mathbb Z / p\mathbb Z$ for a large prime $p$, rather ...
34
votes
1answer
4k views

A question about ordinal definable real numbers

If ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) is consistent, does it remain consistent when the following statement is added to it as a new axiom? "There exists a denumerably ...
33
votes
7answers
6k views

Reductio ad absurdum or the contrapositive?

From time to time, when I write proofs, I'll begin with a claim and then prove the contradiction. However, when I look over the proof afterwards, it appears that my proof was essentially a proof of ...
33
votes
4answers
2k views

The origin of sets?

The history of set theory from Cantor to modern times is well documented. However, the origin of the idea of sets is not so clear. A few years ago, I taught a set theory course and I did some digging ...
32
votes
15answers
6k views

Strong induction without a base case

Strong induction proves a sequence of statements $P(0)$, $P(1)$, $\ldots$ by proving the implication "If $P(m)$ is true for all nonnegative integers $m$ less than $n$, then $P(n)$ is true." for ...
32
votes
8answers
4k 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 ...
32
votes
4answers
2k views

Which topological spaces admit a nonstandard metric?

My question is about the concept of nonstandard metric space that would arise from a use of the nonstandard reals R* in place of the usual R-valued metric. That is, let us define that a topological ...
32
votes
3answers
2k views

Inconsistent theory with long contradiction

What can one say about an inconsistent theory $T$ which has no contradictions (i.e. deductions of $P \wedge \neg P$) of length shorter than $n$, where $n$ is some huge number? There have been some ...
30
votes
2answers
5k views

Is the analysis as taught in universities in fact the analysis of definable numbers?

Ten years ago when I studied in the university I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows: All numbers are divided into two classes: those ...
30
votes
10answers
3k views

Believing the Conjectures

In Believing the axioms (I and II), Penelope Maddy proposes five "rules of thumb" that she then uses to justify large cardinal axioms in set theory. These extrinsic rules are modeled after the ...
30
votes
7answers
2k views

Is the ultraproduct concept fundamentally category-theoretic?

Once again, I would like to take advantage of the large number of knowledgable category theorists on this site for a question I have about category-theoretic aspects of a fundamental logic concept. ...
30
votes
1answer
879 views

Is $π$ definable in $(\Bbb R,0,1,+,×,<,\exp)$?

(This question is originally from Math.SE, where it didn't receive any answers.) Is there a first-order formula $\phi(x) $ with exactly one free variable $ x $ in the language of ordered fields ...
29
votes
6answers
2k views

Is the non-triviality of the algebraic dual of an infinite-dimensional vector space equivalent to the axiom of choice?

If $V$ is given to be a vector space that is not finite-dimensional, it doesn't seem to be possible to exhibit an explicit non-zero linear functional on $V$ without further information about $V$. The ...
29
votes
3answers
2k views

The set-theoretic multiverse as a (bi)category

Joel Hamkin's The set-theoretic multiverse has featured in MO questions before, e.g., here and here. But I was wondering about the best category theoretic angle to take on it. In the paper Joel ...
29
votes
1answer
2k views

Does $2^X=2^Y\Rightarrow |X|=|Y|$ imply the axiom of choice?

The Generalized Continuum Hypothesis can be stated as $2^{\aleph_\alpha}=\aleph_{\alpha+1}$. We know that GCH implies AC (Jech, The Axiom of Choice, Theorem 9.1 p.133). In fact, a relatively weak ...
28
votes
7answers
5k views

Arguments against large cardinals

I started to learn about large cardinals a while ago, and I read that the existence, and even the consistency of the existence of an inaccessible cardinal, i.e. a limit cardinal which is additionally ...
28
votes
3answers
2k views

For which Millennium Problems does undecidable -> true?

Three good answers were received — by Alex Gavrilov, Bjørn Kjos-Hanssen, and Terry Tao — and the bounty has been awarded (somewhat arbitrarily) to Alex Gavrilov. The answers ...
28
votes
3answers
2k views

Why do stacked quantifiers in PA correspond to ordinals up to $\epsilon_0$?

I am trying to understand why induction up to exactly $\epsilon_0$ is necessary to prove the cut-elimination theorem for first-order Peano Arithmetic; or, as I understand, equivalently, why the length ...