All Questions
1,142 questions
256
votes
16
answers
71k
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 ...
32
votes
2
answers
4k
views
Similarities between Post's Problem and Cohen's Forcing
Remark: I have since learned that G.H. Moore addresses this question in the third reference listed at the end of this post, beginning on p. 157 in which he cites a letter from Kreisel to Gödel dated 4/...
- 7,831
298
votes
34
answers
53k
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 ...
113
votes
2
answers
16k
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 ...
- 3,511
9
votes
1
answer
889
views
Combinatorics for the action of Virasoro / Kac–Schwarz operators: partition polynomials of free probability theory
In the background sections below, I establish the relations among characterizations of the action of Virasoro / Kac–Schwarz operators of 2D gravity models presented in terms of Laurent series by ...
- 10.5k
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 $h(P)\leq ...
- 937
23
votes
1
answer
3k
views
Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)
Gödel's constructible universe ($L$) is defined using definable power set operator in first order logic ($\mathcal{L}_{\omega ,\omega}$). One can produce such a universe in infinitary logics in ...
user46667
32
votes
5
answers
4k
views
How many of the true sentences are provable?
Is there a natural measure on the set of statements which are true in the usual model (i.e. $\mathbb{N}$) of Peano arithmetic which enables one to enquire if 'most' true sentences are provable or ...
- 5,339
23
votes
5
answers
4k
views
Morse-Kelley set theory consistency strength
I've come across several references to MK (Morse-Kelley set theory), which includes the idea of a proper class, a limitation of size, includes the axiom schema of comprehension across class variables (...
- 1,979
18
votes
0
answers
895
views
Is the universality of the surreal number line a weak global choice principle?
I'd like to consider the principle asserting that the surreal
number line is universal for all class linear orders, or in other
words, that every linear order (including proper-class-sized)
linear ...
- 236k
17
votes
2
answers
3k
views
Does ZFC prove the universe is linearly orderable?
It is consistent with ZFC that the universe is well-ordered, e.g. in $V=L$ where global choice holds. I also know that it is consistent that global choice fails (although I have no immediate example ...
- 39.8k
110
votes
10
answers
26k
views
Set theories without "junk" theorems?
Clearly I first need to formally define what I mean by "junk" theorem. In the usual construction of natural numbers in set theory, a side-effect of that construction is that we get such theorems as $...
- 11.8k
107
votes
36
answers
21k
views
Interesting examples of vacuous / void entities
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 ...
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
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
40
votes
3
answers
5k
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 ...
143
votes
12
answers
30k
views
Solutions to the Continuum Hypothesis
Related MO questions: What is the general opinion on the Generalized Continuum Hypothesis? ; Completion of ZFC ; Complete resolutions of GCH How far wrong could the Continuum Hypothesis be? When was ...
- 24.7k
122
votes
4
answers
39k
views
Is the analysis as taught in universities in fact the analysis of definable numbers?
Ten years ago, when I studied in 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 ...
- 10.1k
106
votes
19
answers
12k
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 ...
- 1,069
67
votes
5
answers
10k
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 ...
- 50.8k
67
votes
10
answers
14k
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 ...
- 825
61
votes
5
answers
12k
views
Is the Riemann Hypothesis equivalent to a $\Pi_1$ sentence?
1) Can the Riemann Hypothesis (RH) be expressed as a $\Pi_1$ sentence?
More formally,
2) Is there a $\Pi_1$ sentence which is provably equivalent to RH in PA?
Update (July 2010):
So we have two ...
- 5,502
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
96
votes
16
answers
34k
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 ...
86
votes
7
answers
21k
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 ...
- 114k
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
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
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
39
votes
6
answers
7k
views
A remark of Connes on non-standard analysis
In an interview (at http://www.alainconnes.org/docs/Inteng.pdf) Connes remarks that
I had been working on non-standard analysis, but after a while I had found a catch in the theory.... The point is ...
- 1,131
36
votes
3
answers
2k
views
Internal logic of the topos of simplicial sets
I am looking for a closed statement (i.e. not depending on any parameter objects) which is true in the internal logic of the topos of simplicial sets, but is not an intuitionistic tautology. Ideally, ...
- 66.8k
29
votes
1
answer
1k
views
Is the Golomb countable connected space topologically rigid?
The Golomb space $\mathbb G$ is the set of positive integers endowed with the topology generated by the base consisting of the arithmetic progressions $a+b\mathbb N_0$ with relatively prime $a,b$ and $...
- 41.9k
23
votes
2
answers
3k
views
Is there any research on set theory without extensionality axiom?
In practice (say, in computer science), collections with many "labels" ("identities"), or collections which occur in many copies, are more frequently used than sets. Such collections do not satisfy ...
- 889
23
votes
1
answer
2k
views
Can we axiomatize Omnific Integers without the Surreal Number system?
Omnific integers are the counterpart in the Surreal numbers of the integers. The surreal numbers are usually defined using set theory, and then the omnific integers are defined as a particular subset (...
- 4,597
20
votes
7
answers
2k
views
Does every set admit a rigid binary relation? (and how is this related to the Axiom of Choice?)
Let us say that a set B admits a rigid binary relation, if there is a binary relation R such that the structure (B,R) has no nontrivial automorphisms.
Under the Axiom of Choice, every set is well-...
- 236k
20
votes
3
answers
5k
views
Pi1-sentence independent of ZF, ZF+Con(ZF), ZF+Con(ZF)+Con(ZF+Con(ZF)), etc.?
Let
ZF1 = ZF,
ZFk+1 = ZF + the assumption that ZF1,...,ZFk are consistent,
ZFω = ZF + the assumption that ZFk is consistent for every positive integer k,
... and similarly define ZFα ...
- 9,833
10
votes
2
answers
803
views
Reference for Wang tile
I am working on projects in solving ground state of generalized Ising models. One recent work involves tiling with basic tiles that filled the whole lattice. For example, we could obtain results:
...
- 867
185
votes
11
answers
52k
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 ...
- 2,745
135
votes
43
answers
38k
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 ...
128
votes
13
answers
24k
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 ...
- 5,492
77
votes
8
answers
12k
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 ...
- 9,833
68
votes
4
answers
12k
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 ...
- 25.5k
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
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
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
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 ...
37
votes
6
answers
6k
views
Who needs Replacement anyway?
The set theory ETCS famously comes without the Replacement axiom schema (or an equivalent) that is part of ZFC. One (to me, not apparently useful) set that one cannot build in ETCS is $\coprod_{n\in \...
- 35.5k
34
votes
8
answers
8k
views
Arithmetic fixed point theorem
I want to understand the idea of the proof of the artihmetic fixed point theorem. The theorem is crucial in the proof of Gödel's first Incompletness theorem.
First some notation: We work in $NT$, the ...
- 63.1k
32
votes
3
answers
7k
views
Is the statement that every field has an algebraic closure known to be equivalent to the ultrafilter lemma?
The existence and uniqueness of algebraic closures is generally proven using Zorn's lemma. A quick Google search leads to a 1992 paper of Banaschewski, which I don't have access to, asserting that ...
- 118k
30
votes
3
answers
3k
views
Can there be an embedding j:V → L, from the set-theoretic universe V to the constructible universe L, when V ≠ L?
Main Question. Can there be an embedding $j:V\to L$ of the
set-theoretic universe $V$ to the constructible universe $L$, if
$V\neq L$?
By embedding here, I mean merely a proper class isomorphism from
$...
- 236k
28
votes
2
answers
7k
views
Large cardinal axioms and Grothendieck universes
A cardinal $\lambda$ is weakly inaccessible, iff a. it is regular (i.e. a set of cardinality $\lambda$ can't be represented as a union of sets of cardinality $<\lambda$ indexed by a set of ...
- 23.5k