All Questions
6,026 questions
60
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15
answers
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Abstract thought vs calculation
Jeremy Avigad and Erich Reck claim that one factor leading to abstract mathematics in the late 19th century (as opposed to concrete mathematics or hard analysis) was the use of more abstract notions ...
60
votes
7
answers
9k
views
Does anyone still seriously doubt the consistency of $ZFC$?
As someone self-taught in set theory beginning with Donald Monk’s excellent book on MK set theory, $ZFC$ has always seemed like a weak set theory.
Despite this, the majority of professional ...
60
votes
6
answers
7k
views
Has decidability got something to do with primes?
Note: I have modified the question to make it clearer and more relevant. That makes some of references to the old version no longer hold. I hope the victims won't be furious over this.
Motivation:
...
- 2,824
60
votes
7
answers
9k
views
In what respect are univalent foundations "better" than set theory?
It was an ambitious project of Vladimir Voevodsky's to provide new foundations for mathematics with univalent foundations (UF) to eventually replace set theory (ST).
Part of what makes ST so appealing ...
60
votes
8
answers
10k
views
Why should we believe in the axiom of regularity?
Today I started reading Maddy's Believing the axioms. As I knew beforehand, it includes some discussion of ZFC axioms. However, I really hoped for a more extensive discussion of axiom of foundation/...
- 28.2k
60
votes
8
answers
6k
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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.
My ...
- 236k
59
votes
8
answers
8k
views
Cauchy reals and Dedekind reals satisfy "the same mathematical theorems"
The succinct question
The conjecture of Birch and Swinnerton-Dyer (to take a random example) mentions L-functions and hence the complex numbers and hence the real numbers (because the complexes are ...
- 41.4k
59
votes
9
answers
7k
views
Has anyone thought about creating a formal proof wiki with verifier?
Mathematics has undergone some rather nice developments recently with the adoption of new techologies, things like on-line journals, the arXiv, this website, etc. I imagine there must be many further ...
58
votes
9
answers
8k
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 ...
- 32.4k
58
votes
5
answers
8k
views
How to make Ext and Tor constructive?
EDIT: This post was substantially modified with the help of the comments and answers. Thank you!
Judging by their definitions, the $\mathrm{Ext}$ and $\mathrm{Tor}$ functors are among the most non-...
- 33.8k
58
votes
3
answers
4k
views
What is the geometry of an undecidable diophantine equation?
As an arithmetic algebraic geometer of the highest moral fiber, I am trained to look at Diophantine equations in terms of the geometry of the corresponding scheme. For instance, if the Diophantine ...
- 148k
57
votes
6
answers
6k
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 ...
- 3,511
56
votes
7
answers
8k
views
What is the smallest unsolved Diophantine equation?
If $P=\sum_{\bf i} a_{\bf i}x^{\bf i}\in {\mathbb Z}[x_1,\dots,x_d]$, let $|P|=\sum_{\bf i}|a_{\bf i}|x^{\bf i}$ and $h(P)=|P|(2,\dots,2)$, so that there is only a finite number of $P$ with $h(P)\leq ...
- 937
55
votes
2
answers
5k
views
Automatically solving olympiad geometry problems
Warning: I am only an amateur in the foundations of mathematics.
My understanding of this Wikipedia page about Tarski's axiomatization of plane geometry (and especially the discussion about ...
- 41.4k
54
votes
5
answers
15k
views
The unification of Mathematics via Topos Theory
In her paper The unification of Mathematics via Topos Theory, Olivia Caramello says "one can generate a huge number of new results in any mathematical field without any creative effort". Is ...
- 1,190
54
votes
1
answer
3k
views
In the two-person Killing the Hydra game, what is the winning strategy?
My question is which player has a winning strategy in the
two-player version of the Killing the Hydra game?
In their amazing paper,
Kirby, Laurie; Paris, Jeff, Accessible independence results for ...
- 236k
53
votes
7
answers
8k
views
Zorn's lemma: old friend or historical relic?
It is often said that instead of proving a great theorem a mathematician's fondest dream is to prove a great lemma. Something like Kőnig's tree lemma, or Yoneda's lemma, or really anything from this ...
- 18.7k
53
votes
7
answers
7k
views
Are there any undecidability results that are not known to have a diagonal argument proof?
Is there a problem which is known to be undecidable (in the algorithmic sense), but for which the only known proofs of undecidability do not use some form of the Cantor diagonal argument in any ...
- 114k
53
votes
5
answers
20k
views
Categorical foundations without set theory
Can there be a foundations of mathematics using only category theory, i.e. no set theory? More precisely, the definition of a category is a class/set of objects and a class/set of arrows, satisfying ...
user2529
53
votes
2
answers
3k
views
Silver's approach to the inconsistency of $\mathrm{ZFC}$
As all probably know, Jack Silver passed away about one month ago. The announcement released, with delay, by European Set Theory Society includes a quote by Solovay about his belief on inconsistency ...
- 2,381
53
votes
1
answer
4k
views
When does $A^A=2^A$ without the axiom of choice?
Assuming the axiom of choice the following argument is simple, for infinite $A$ it holds: $$2\lt A\leq2^A\implies 2^A\leq A^A\leq 2^{A\times A}=2^A.$$
However without the axiom of choice this doesn't ...
- 39.8k
53
votes
1
answer
6k
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 ...
- 39.8k
52
votes
5
answers
7k
views
Metamathematics of buts
Something I learned (probably in middle school) that always bothered me is that the truth value of "and" and "but" are basically the same. If you were going to assign a truth-...
- 6,870
52
votes
3
answers
7k
views
Function extensionality: does it make a difference? why would one keep it out of the axioms?
Yesterday I was shocked to discover that function extensionality (the statement that if two functions $f$ and $g$ on the same domain satisfy $f\left(x\right) = g\left(x\right)$ for all $x$ in the ...
- 33.8k
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 ...
51
votes
3
answers
3k
views
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 ...
- 41.9k
51
votes
4
answers
7k
views
How undecidable is the spectral gap?
Nature just published a paper by Cubitt, Perez-Garcia and Wolf titled Undecidability of the Spectral Gap, there is an extended version on arxiv which is 146 pages long. Here is from the abstract:"Many ...
- 1,731
50
votes
4
answers
6k
views
Do set-theorists use informal set theory as their meta-theory when talking about models of ZFC?
Here, Noah Schweber writes the following:
Most mathematics is not done in ZFC. Most mathematics, in fact, isn't done axiomatically at all: rather, we simply use propositions which seem "intuitively ...
- 509
50
votes
1
answer
6k
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 ...
- 17.6k
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
49
votes
5
answers
5k
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". ...
- 18.7k
49
votes
1
answer
2k
views
Producing finite objects by forcing!
It is a trivial fact that forcing can not produce finite sets of ground model objects. However there are situations,
where we can use forcing to prove the existence of finite objects with some ...
- 32.2k
49
votes
0
answers
3k
views
Concerning proofs from the axiom of choice that ℝ³ admits surprising geometrical decompositions: Can we prove there is no Borel decomposition?
This question follows up on a comment I made on Joseph O'Rourke's
recent question, one of several questions here on mathoverflow
concerning surprising geometric partitions of space using the axiom
of ...
- 236k
48
votes
5
answers
7k
views
What axioms are used to prove Gödel's Incompleteness Theorems?
I understand Gödel's Incompleteness Theorems to be statements about effectively generated formal systems, which basically makes them theorems about algorithms. This is cool, because despite being ...
- 11.3k
47
votes
5
answers
10k
views
Set theory and Model Theory
This question probably doesn't make any sense, but I don't see why, so I ask it here hoping someone will illuminate the matter:
There is this whole area of study in Set Theory about the consistency, ...
- 1,219
47
votes
3
answers
7k
views
Clearing misconceptions: Defining "is a model of ZFC" in ZFC
There is often a lot of confusion surrounding the differences between relativizing individual formulas to models and the expression of "is a model of" through coding the satisfaction relation with ...
- 2,762
47
votes
7
answers
7k
views
What is an explicit bijection in combinatorics?
A standard way of demonstrating that two collections of combinatorial objects have the same cardinality is to exhibit a bijection between them. Browsing through some examples (here, there, yonder) ...
- 48.8k
47
votes
4
answers
5k
views
How to rewrite mathematics constructively?
Many mathematical subfields often use the axiom of choice and proofs by contradiction. I heard from people supporting constructive mathematics that often one can rewrite the definitions and theorems ...
- 581
47
votes
4
answers
5k
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 ...
- 44.4k
47
votes
4
answers
4k
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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
47
votes
5
answers
9k
views
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 ...
46
votes
15
answers
11k
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 ...
46
votes
4
answers
5k
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 ...
- 29k
46
votes
8
answers
12k
views
What are some proofs of Godel's Theorem which are *essentially different* from the original proof?
I am looking for examples of proofs of Godel's (First) Incompleteness Theorem which are essentially different from (Rosser's improvement of) Godel's original proof.
This is partly inspired by ...
46
votes
2
answers
2k
views
Applications of Zorn’s lemma that aren’t chain-complete/directed-complete?
Zorn’s Lemma applies to posets in which every chain has an upper bound. However, in all applications I know, the poset is also evidently chain-complete — chains have least upper bounds. A few ...
- 21.3k
45
votes
8
answers
10k
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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
45
votes
6
answers
8k
views
Situation with Artemov's paper?
Artemov's paper on Goedel's theorem has been on the arxiv since 2019. There was a (less than fully friendly) discussion of this on FoM. At stackexchange, I found only a brief mention at this MSE ...
- 16.6k
45
votes
5
answers
6k
views
Nonstandard analysis in probability theory
I am quite new at nonstandard analysis, and recently I became aware of its use in probability theory mainly through the following two books:
Nelson (1987). Radically Elementary Probability Theory
...
- 1,302
45
votes
5
answers
64k
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How large is TREE(3)?
Friedman, in _Lectures notes on enormous integers shows that TREE(3) is much larger than n(4), itself bounded below by $A^{A(187195)}(3)$ (where $A$ is the Ackerman function and exponentiation ...
- 3,630
44
votes
5
answers
3k
views
History of (proposal of) set-theoretic foundations
It is often said that set theory is the de facto foundation of mathematics. Regardless of the truth of this claim, this seems to be the story told to students (and mathematicians) who poke their ...
- 4,265