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Questions tagged [set-theory]

forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.

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256 votes
16 answers
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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 ...
34 votes
2 answers
2k views

Are the Sierpiński cardinal $\acute{\mathfrak n}$ and its measure modification $\acute{\mathfrak m}$ equal to some known small uncountable cardinals?

This question was motivated by an answer to this question of Dominic van der Zypen. It relates to the following classical theorem of Sierpiński. Theorem (Sierpiński, 1921). For any countable partition ...
Taras Banakh's user avatar
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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/...
Benjamin Dickman's user avatar
297 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 ...
Konrad Swanepoel's user avatar
49 votes
2 answers
6k views

What interesting/nontrivial results in Algebraic geometry require the existence of universes?

Brian Conrad indicated a while ago that many of the results proven in AG using universes can be proven without them by being very careful (link). I'm wondering if there are any results in AG that ...
Harry Gindi's user avatar
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41 votes
5 answers
11k views

Do sets with positive Lebesgue measure have same cardinality as R?

I have been thinking about which kind of wild non-measurable functions you can define. This led me to the question: Is it possible to prove in ZFC, that if a (Edit: measurabel) set $A\subset \mathbb{...
Sune Jakobsen's user avatar
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 ...
user avatar
97 votes
10 answers
15k views

Reflection principle vs universes

In category-theoretic discussions, there is often the temptation to look at the category of all abelian groups, or of all categories, etc., which quickly leads to the usual set-theoretic problems. ...
Peter Scholze's user avatar
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 ...
pinaki's user avatar
  • 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 (...
Richard Rast's user avatar
  • 1,979
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 ...
Asaf Karagila's user avatar
  • 39.7k
8 votes
2 answers
698 views

Order type of $\alpha$-computable well-orderings

One of the nice features of the first admissible ordinal after $\omega$, i.e. $\omega_1^{CK}$, is that it is the collection of ordinals whose order type is that of a computable well-ordering on $\...
Nate Ackerman's user avatar
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 $...
Jacques Carette's user avatar
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 ...
Konrad Swanepoel's user avatar
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 ...
Joel David Hamkins's user avatar
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 ...
skeptical scientist's user avatar
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 ...
Joel David Hamkins's user avatar
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 ...
Gil Kalai's user avatar
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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 ...
Anixx's user avatar
  • 10.1k
107 votes
9 answers
36k views

solving $f(f(x))=g(x)$

This question is of course inspired by the question How to solve f(f(x))=cosx and Joel David Hamkins' answer, which somehow gives a formal trick for solving equations of the form $f(f(x))=g(x)$ on a ...
Kevin Buzzard's user avatar
82 votes
5 answers
19k views

Inaccessible cardinals and Andrew Wiles's proof

In a recent issue of New Scientist (16 Aug 2010), I was surprised to read that a part of Wiles' proof of Taniyama-Shimura conjecture relies on inaccessible cardinals. Here's the article Richard Elwes,...
Cosmonut's user avatar
  • 1,101
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 ...
user8996's user avatar
  • 825
63 votes
4 answers
7k views

When size matters in category theory for the working mathematician

I think a related question might be this (Set-Theoretic Issues/Categories). There are many ways in which you can avoid set theoretical paradoxes in dealing with category theory (see for instance ...
jg1896's user avatar
  • 3,318
22 votes
1 answer
4k views

Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions?

Let the proof theoretic ordinal $\alpha$ of a theory $T$ be the least recursive ordinal such that $T$ does not prove that $\alpha$ is well-founded. This ordinal is intended to quantify in some sense ...
user40919's user avatar
  • 711
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 ...
92 votes
3 answers
14k views

Is every sigma-algebra the Borel algebra of a topology?

This question arises from the excellent question posed on math.SE by Salvo Tringali, namely, Correspondence between Borel algebras and topology. Since the question was not answered there after some ...
Joel David Hamkins's user avatar
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 ...
Asaf Karagila's user avatar
  • 39.7k
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 ...
Andrew Critch's user avatar
29 votes
4 answers
3k views

When $2^\alpha = 2^\beta$ implies $\alpha=\beta$ ($\alpha,\beta$ cardinals)

Sorry if this is a silly question. I was wondering, under what axioms of set theory is it true that if $\alpha$,$\beta$ are cardinals, and $2^\alpha=2^\beta$, then $\alpha=\beta$? Do people use these ...
Steve D's user avatar
  • 4,425
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 ...
Ioachim Drugus's user avatar
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 (...
Keshav Srinivasan's user avatar
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-...
Joel David Hamkins's user avatar
19 votes
9 answers
5k views

Forcing as a tool to prove theorems

It is often mentioned the main use of forcing is to prove independence facts, but it also seems a way to prove theorems. For instance how would one try to prove Erdös-Rado, $\beth_n^{+} \to (\aleph_1)...
Rachid Atmai's user avatar
  • 3,804
19 votes
2 answers
2k views

Woodin's unpublished proof of the global failure of GCH

An unpublished result of Woodin says the following: Theorem. Assuming the existence of large cardinals, it is consistent that $\forall \lambda, 2^{\lambda}=\lambda^{++}.$ In the paper "The ...
Mohammad Golshani's user avatar
19 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 ...
Scott Aaronson's user avatar
16 votes
1 answer
2k views

totally ordered chain in the powerset with big cardinality

Let $B$ be some set. The problem is to find a set $A\subset\mathcal{P}(B)$ of subsets of $B$ which is totally ordered by inclusion and such that there exists a bijection $A\leftrightarrow \mathcal{P}(...
Toink's user avatar
  • 642
15 votes
1 answer
3k views

V=L and a Well-Ordering of the Reals

A fairly simple question: I've read in multiple sources that Godel proved that if we accept the axiom of constructibility in ZFC, then we can create an explicit formula that well-orders the real ...
Jimmy Miller's user avatar
6 votes
1 answer
298 views

Does the critical sequence for subalgebras of elementary embeddings with finitely many generators have order type $\omega$?

Suppose that $\lambda$ is a cardinal. Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$. If $j,k\in\mathcal{E}_{\lambda}$, then define $j[k]=\...
Joseph Van Name's user avatar
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 ...
AgCl's user avatar
  • 2,745
157 votes
5 answers
28k views

What makes dependent type theory more suitable than set theory for proof assistants?

In his talk, The Future of Mathematics, Dr. Kevin Buzzard states that Lean is the only existing proof assistant suitable for formalizing all of math. In the Q&A part of the talk (at 1:00:00) he ...
MWB's user avatar
  • 1,667
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 ...
Scott Aaronson's user avatar
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/...
Wojowu's user avatar
  • 28.2k
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 ...
Jason's user avatar
  • 2,762
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 \...
David Roberts's user avatar
  • 35.5k
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 $...
Joel David Hamkins's user avatar
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 ...
algori's user avatar
  • 23.5k
21 votes
6 answers
3k views

Where in ordinary math do we need unbounded separation and replacement?

[I have updated the question after initial comments in the hope of clarifying it.] I do quite a bit of reasoning, typically about topology and metric spaces, in "non-standard" foundations, such as ...
14 votes
4 answers
2k views

What can be preserved in mathematics if all constructions are carried out in ZF?

This is inspired by this discussion. I see that the debates about the necessity of the axiom of choice in this or that statement are still ongoing. In this regard, I became interested in whether there ...
Sergei Akbarov's user avatar
12 votes
1 answer
517 views

Selective ultrafilter and bijective mapping

For arbitrary (selective) ultrafilter $\mathcal{F}$ does there exist bijection $\phi:[\omega]^2\to\omega$ with the property: $\forall B\in\mathcal{F} : \phi([B]^2)\in\mathcal{F}$ ?
ar.grig's user avatar
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