# 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.

**216**

**31**answers

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

**175**

**15**answers

### Why worry about the axiom of choice?

**164**

**1**answer

### Ultrafilters and automorphisms of the complex field

**145**

**13**answers

### Knuth's intuition that Goldbach might be unprovable

**117**

**11**answers

### Solutions to the Continuum Hypothesis

**109**

**3**answers

### Does there exist a bijection of $\mathbb{R}^n$ to itself such that the forward map is connected but the inverse is not?

**96**

**2**answers

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

**89**

**9**answers

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

**81**

**4**answers

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

**75**

**20**answers

### Proofs of the uncountability of the reals.

**71**

**2**answers

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

**70**

**19**answers

### Injectivity implies surjectivity

**69**

**16**answers

### Most 'unintuitive' application of the Axiom of Choice?

**68**

**5**answers

### Does pointwise convergence imply uniform convergence on a large subset?

**67**

**8**answers

### Succinctly naming big numbers: ZFC versus Busy-Beaver

**67**

**6**answers

### Which graphs are Cayley graphs?

**66**

**5**answers

### How do the compact Hausdorff topologies sit in the lattice of all topologies on a set?

**63**

**13**answers

### Logic in mathematics and philosophy

**63**

**9**answers

### What's wrong with the surreals?

**61**

**8**answers

### Set theories without “junk” theorems?

**61**

**12**answers

### What practical applications does set theory have?

**59**

**11**answers

### Why hasn't mereology succeeded as an alternative to set theory?

**57**

**5**answers

### Inaccessible cardinals and Andrew Wiles's proof

**57**

**4**answers

### Non-Borel sets without axiom of choice

**57**

**3**answers

### Forcing as a new chapter of Galois Theory?

**55**

**9**answers

### Arguments against large cardinals

**51**

**5**answers

### Can the symmetric groups on sets of different cardinalities be isomorphic?

**50**

**10**answers

### How should a “working mathematician” think about sets? (ZFC, category theory, urelements)

**50**

**3**answers

### Does every real function have this weak continuity property?

**50**

**5**answers

### The Logic of Buddha: A Formal Approach

**49**

**6**answers

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

**49**

**2**answers

### How to add essentially new knots to the universe?

**47**

**7**answers

### Is the ultraproduct concept fundamentally category-theoretic?

**47**

**1**answer

### Producing finite objects by forcing!

**46**

**8**answers

### Why should we believe in the axiom of regularity?

**45**

**3**answers

### What the heck is the Continuum Hypothesis doing in Weibel's Homological Algebra?

**45**

**4**answers

### The origin of sets?

**45**

**1**answer

### When does $A^A=2^A$ without the axiom of choice?

**44**

**0**answers

### Set-theoretic reformulation of the invariant subspace problem

**43**

**2**answers

### A question about ordinal definable real numbers

**43**

**1**answer

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

**41**

**7**answers

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

**41**

**4**answers

### Do set-theorists use informal set theory as their meta-theory when talking about models of ZFC?

**41**

**1**answer

### Hilbert's alleged proof of the Continuum Hypothesis in “On the Infinite”

**40**

**13**answers

### Cardinalities larger than the continuum in areas besides set theory

**40**

**2**answers

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

**40**

**4**answers

### A principle of mathematical induction for partially ordered sets with infima?

**40**

**0**answers

### How many algebraic closures can a field have?

**39**

**5**answers

### How many rearrangements must fail to alter the value of a sum before you conclude that none do?

**39**

**3**answers