All Questions
1,142 questions
51
votes
3
answers
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Is each squared finite group trivial?
A semigroup $S$ is defined to be squared if there exists a subset $A\subseteq S$ such that the function $A\times A\to S$, $(x,y)\mapsto xy$, is bijective.
Problem: Is each squared finite group ...
- 42k
51
votes
30
answers
8k
views
Taking a theorem as a definition and proving the original definition as a theorem
Gian-Carlo Rota's famous 1991 essay, "The pernicious influence of mathematics upon philosophy" contains the following passage:
Perform the following thought experiment. Suppose that you are ...
50
votes
0
answers
2k
views
How many algebraic closures can a field have?
Assuming the axiom of choice given a field $F$, there is an algebraic extension $\overline F$ of $F$ which is algebraically closed. Moreover, if $K$ is a different algebraic extension of $F$ which is ...
- 39.8k
47
votes
5
answers
9k
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Proof assistants for mathematics
This question is related to (maybe even the same in intent as) Intro to automatic theorem proving / logical foundations?, but none of the answers seem to address what I'm looking for.
There are a lot ...
47
votes
4
answers
5k
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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 ...
- 44.4k
47
votes
4
answers
4k
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 ...
- 236k
45
votes
8
answers
10k
views
What is Realistic Mathematics?
This post is partially about opinions and partially about more precise mathematical questions. Most of this post is not as formal as a precise mathematical question. However, I hope that most readers ...
- 25.5k
43
votes
4
answers
5k
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Lists as a foundation of mathematics
I am wondering if there is a foundation of mathematics where not sets or "set-like objects" (such as objects of a suitable topos as in ETCS) are the primitive notion, but rather lists. These ...
- 63.1k
41
votes
3
answers
2k
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What is the minimal size of a partial order that is universal for all partial orders of size n?
A partial order $\mathbb{B}$ is universal for a class $\cal{P}$ of partial orders if every order in $\cal{P}$ embeds
order-preservingly into $\mathbb{B}$.
For example, every partial order
$\langle\...
- 236k
41
votes
2
answers
2k
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On the difference between two concepts of even cardinalities: Is there a model of ZF set theory in which every infinite set can be split into pairs, but not every infinite set can be cut in half?
An interesting question has arisen over at this
math.stackexchange
question
about two concepts of even in the context of infinite
cardinalities, which are equivalent under the axiom of
choice, but ...
- 236k
41
votes
4
answers
2k
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What is the probability two random maps on n symbols commute?
It is well known that two randomly chosen permutations of $n$ symbols commute with probability $p_n/n!$ where $p_n$ is the number of partitions of $n$. This is a special case of the fact that in a ...
- 38.6k
40
votes
7
answers
8k
views
What is the general opinion on the Generalized Continuum Hypothesis?
I'm community wikiing this, since although I don't want it to be a discussion thread, I don't think that there is really a right answer to this.
From what I've seen, model theorists and logicians ...
39
votes
3
answers
3k
views
Can one show that the real field is not interpretable in the complex field without the axiom of choice?
We all know that the complex field structure $\langle\mathbb{C},+,\cdot,0,1\rangle$ is interpretable in the real field $\langle\mathbb{R},+,\cdot,0,1\rangle$, by encoding $a+bi$ with the real-number ...
- 236k
39
votes
5
answers
8k
views
Is there a constructive proof of Cantor–Bernstein–Schroeder theorem ?
An important feature of the Cantor-Schroeder-Bernstein theorem is that it does not rely on the axiom of choice. However, its various proofs are non-constructive, as they depend on the law of excluded ...
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38
votes
4
answers
2k
views
On sentences true in all finite groups
Let $w$ be a group word with two variables $x$ and $y$.
Is the sentence $(\forall x)(\exists y)w=1$
true in every group if it is true
in every finite group?
The same question about the sentence $(\...
- 893
38
votes
5
answers
5k
views
Does "compact iff projections are closed" require some form of choice?
There are many equivalent ways of defining the notion of compact space, but some require some kind of choice principle to prove their equivalence. For example, a classical result is that for $X$ to be ...
- 53.3k
37
votes
7
answers
8k
views
Model theoretic applications to algebra and number theory(Iwasawa Theory)
One of my favorite results in algebraic geometry is a classical result of AX (see http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/) I'll recall ...
- 2,396
36
votes
3
answers
3k
views
Latest status of core model theory?
What is the "strongest" core model to this day? In particular, how far are we from a core model for supercompact cardinals? There are rumors of some notes from a workshop in 2004:
https://...
- 1,312
36
votes
2
answers
3k
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What can be expressed in and proved with the internal logic of a topos?
The title of this post expresses what I really want, which is to learn how to wield the internal logic of a topos more effectively. However, to bring it down to earth, I'll ask a few basic questions ...
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35
votes
3
answers
5k
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Counterintuitive consequences of the Axiom of Determinacy?
I just read Dr Strangechoice's explanation that if all subsets of the real numbers are Lebesgue measurable, then you can partition $2^\omega$ into more than $2^\omega$ many pairwise disjoint nonempty ...
- 7,633
34
votes
1
answer
3k
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Does "every" first-order theory have a finitely axiomatizable conservative extension?
I originally asked this question on math.stackexchange.com here.
There's a famous theorem (due to Montague) that states that if $\sf ZFC$ is consistent then it cannot be finitely axiomatized. ...
- 3,137
34
votes
5
answers
2k
views
Forcing as a replacement of induction and diagonal arguments
Let me give some examples motivating the question.
The use of forcing instead of induction: For this consider Cantor's theorem:
Theorem 1. Any two countable dense linear orders $I, J$ without end ...
- 32.2k
34
votes
3
answers
4k
views
Alternatives to the law of the excluded middle
As a sentential logic, intuitionistic logic plus the law of the excluded middle gives classical logic.
Is there a logical law that is consistent with intuitionistic logic but inconsistent with ...
- 331
33
votes
3
answers
8k
views
Category of categories as a foundation of mathematics
In
Lawvere, F. W., 1966, “The Category of
Categories as a Foundation for
Mathematics”, Proceedings of the
Conference on Categorical Algebra, La
Jolla, New York: Springer-Verlag,
1–21.
...
- 2,399
33
votes
5
answers
7k
views
Why do people say Gödel's sentence is true when it is true in some models but false in others?
I am a beginner, so this question may be naive.
Suppose we have a (sufficiently strong) consistent first order logic system. Gödel's first incompleteness theorem says there exists a Gödel sentence $g$ ...
33
votes
1
answer
1k
views
Is this conjecture strictly weaker than P=NP?
My three computability questions are related to the following group theory question (first asked by Bridson in 1996):
For which real $\alpha\ge 2$ the function $n^\alpha$ is equivalent to the Dehn ...
user6976
33
votes
3
answers
2k
views
Wiki for consequences of axiom of choice?
I raised the following question as part of another MO question, but I am following the suggestion of Nate Eldredge to make it a question in its own right.
For many years, there has a been a valuable ...
- 82.7k
33
votes
0
answers
2k
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Is there a (discrete) monoid M injecting into its group completion G for which BM is not homotopy equivalent to BG?
For a (discrete) monoid $M$, the classifying space $BM$ is the
geometric realization of the nerve of the one object category whose
hom-set is $M$. (This definition gives the usual classfiying space
...
- 5,309
32
votes
4
answers
4k
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Is a random subset of the real numbers non-measurable? Is the set of measurable sets measurable?
One might say, "a random subset of $\mathbb{R}$ is not Lebesgue measurable" without really thinking about it. But if we unpack the standard definitions of all those terms (and work in ZFC), it's not ...
- 2,210
32
votes
5
answers
9k
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How many binary operations are associative?
Let $X$ be a finite set of $n$ elements, and consider a binary operation $\odot: X \times X \rightarrow X$. There are $n^{n^2}$ such binary operations, as the $n \times n$ table entries can each
be ...
- 151k
31
votes
6
answers
4k
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A Model where Dedekind Reals and Cauchy Reals are Different
Is there a model where Dedekind reals and Cauchy reals are different? I'd appreciate if someone can refer me to any related work in case such a model exists.
- 365
31
votes
3
answers
3k
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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 $e\mapsto p_e$, ...
- 236k
31
votes
2
answers
1k
views
Define $\mathbb{N}$ in the ring $\mathbb{Z}$ without Lagrange's theorem
It is well-known that the set of nonnegative integers $\mathbb{N}$ is definable in the ring of integers $\mathbb{Z}$. Indeed, by Lagrange's four squares theorem we have $\mathbb{N} = \{n \in \mathbb{Z}...
user40023
31
votes
3
answers
5k
views
Should there be a true model of set theory?
As I understand it, there is a program in set theory to produce an ultimate, canonical model of set theory which, among other things, positively answers the Continuum Hypothesis and various questions ...
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29
votes
2
answers
3k
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Does Taranovsky's system of ordinal notations make sense?
Dmytro Taranovsky has a Web page on which he claims to define a system of ordinal notations strong enough to provide an ordinal analysis of full second-order arithmetic. I think (perhaps unjustly) ...
- 32.5k
29
votes
2
answers
2k
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What do coherent topoi have to do with completeness?
There is a theorem of Deligne in SGA4 that a "coherent" topos (e.g. one on a site where all objects are quasi-compact and quasi-separated) has enough points (i.e. isomorphisms can be detected via ...
- 25.6k
29
votes
3
answers
3k
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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 ...
29
votes
0
answers
2k
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Did Grothendieck overestimate topoi?
I was reading the Russian translation of Recoltes et Semailles and in the footnote where Grothendieck lists his 12 contributions (including schemes) we find the following lines:
Из этих тем ...
28
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0
answers
2k
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Is Feferman's unlimited category theory dead?
In 2013 Solomon Feferman in Foundations of unlimited category theory: what remains to be done (The Review of Symbolic Logic, 6 (2013) pp 6-15, link) laid out three desirable axioms for "...
- 1,446
28
votes
2
answers
2k
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The true reason for the incompleteness of formal systems
A 3/4 year ago, I read Gödel's beautiful paper "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme 1". There is one thing, I never understood.
In a footnote, Gödel ...
- 281
28
votes
0
answers
827
views
Can one divide by the cardinal of an amorphous set?
This question arose in a discussion with Peter Doyle.
It is provable in ZF that one can divide by any positive finite cardinal $k$: if $X \times \{1,\ldots,k\} \simeq Y \times \{1,\ldots,k\}$ then $X \...
- 44.4k
28
votes
13
answers
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Are there any good nonconstructive "existential metatheorems"?
Are there any good examples of theorems in reasonably expressive theories (like Peano arithmetic) for which it is substantially easier to prove (in a metatheory) that a proof exists than it is ...
27
votes
5
answers
4k
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What is induction up to $\varepsilon_0$?
This is a question asked out of curiosity, and because I can't understand the Wikipedia page.
I have often been told that PA cannot prove the validity of induction up to $\varepsilon_0$, which has ...
- 156k
27
votes
1
answer
932
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A cardinal inequality for finiteness
Nearly ten years ago, I explained in a blog post that, assuming only ZF, a cardinal number $\mathfrak{n}$ is finite if and only if it satisfies this monstrous inequality:
$$2^{2^{2^{2^{\mathfrak{n}}}}}...
- 44.4k
27
votes
4
answers
7k
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Did Pogorzelski claim to have a proof of Goldbach's Conjecture?
In 1977, Henry Pogorzelski published what some believed was a claimed proof of Goldbach's Conjecture in Crelle's Journal (292, 1977, 1-12). His argument has not been accepted as a proof of Goldbach's ...
27
votes
2
answers
2k
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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 ...
- 12.4k
27
votes
5
answers
3k
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Algebraic description of compact smooth manifolds?
Given a compact smooth manifold $M$, it's relatively well known that $C^\infty(M)$ determines $M$ up to diffeomorphism. That is, if $M$ and $N$ are two smooth manifolds and there is an $\mathbb{R}$-...
- 6,546
27
votes
4
answers
3k
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What "metatheory" did early set theory/logic researchers use to prove semantic results?
Things like the first-order completeness theorem and the Löwenheim-Skolem theorem are considered foundational in mathematical logic.
The modern approach seems to be, usually, to interpret a "model" ...
- 4,907
27
votes
2
answers
2k
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Is there a name for a family of finite sequences that block all infinite sequences?
Let ${\bf N}^\omega = \bigcup_{m=1}^\infty {\bf N}^m$ denote the space of all finite sequences $(N_1,\ldots,N_m)$ of natural numbers. For want of a better name, let me call a family ${\mathcal T} \...
- 114k
27
votes
4
answers
10k
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Finite axiom of choice: how do you prove it from just ZF?
The axiom of choice asserts the existence of a choice function for any family of sets F. Suppose, however, that F is finite, or even that F just has one set. Then how do we prove the existence of a ...
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