# 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.

**30**

**2**answers

### Similarities between Post's Problem and Cohen's Forcing

**27**

**1**answer

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

**175**

**15**answers

### Why worry about the axiom of choice?

**40**

**2**answers

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

**17**

**1**answer

### Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions?

**96**

**2**answers

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

**49**

**6**answers

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

**46**

**8**answers

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

**36**

**0**answers

### Concerning the various proofs from the axiom of choice that R^3 admits of surprising geometrical decompositions into circles, skew lines and so on: can we prove in any instance that there are no Borel such decompositions? Or that AC is required?

**15**

**2**answers

### Does ZFC prove the universe is linearly orderable?

**16**

**3**answers

### Pi1-sentence independent of ZF, ZF+Con(ZF), ZF+Con(ZF)+Con(ZF+Con(ZF)), etc.?

**5**

**1**answer

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

**216**

**31**answers

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

**117**

**11**answers

### Solutions to the Continuum Hypothesis

**81**

**4**answers

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

**55**

**9**answers

### Arguments against large cardinals

**17**

**8**answers

### Forcing as a tool to prove theorems

**24**

**4**answers

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

**14**

**0**answers

### Is the universality of the surreal number line a weak global choice principle?

**89**

**9**answers

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

**57**

**4**answers

### Non-Borel sets without axiom of choice

**61**

**8**answers

### Set theories without “junk” theorems?

**66**

**5**answers

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

**34**

**5**answers

### Completion of ZFC

**24**

**2**answers

### Large cardinal axioms and Grothendieck universes

**26**

**11**answers

### Does the Axiom of Choice (or any other “optional” set theory axiom) have real-world consequences? [closed]

**24**

**3**answers

### Who needs Replacement anyway?

**26**

**2**answers

### Can there be an embedding j:V to L, from the set-theoretic universe V to the constructible universe L, when V not= L?

**29**

**4**answers

### Is it possible to define higher cardinal arithmetics

**22**

**6**answers

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

**21**

**1**answer

### Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)

**16**

**3**answers

### Singularizing forcing of “small” cardinality?

**11**

**3**answers

### How elementary can we go?

**10**

**2**answers

### How to construct a continuous finite additive measure on the natural numbers

**5**

**7**answers

### Uncountable family of infinite subsets with pairwise finite intersections

**19**

**4**answers

### Why is there no Borel function mapping every countable set of reals outside itself?

**9**

**1**answer

### Resembling the Levy Collapse

**10**

**1**answer

### totally ordered chain in the powerset with big cardinality

**13**

**1**answer

### $T_2$-spaces where all non-empty open sets are homeomorphic

**10**

**1**answer

### Selective ultrafilter and bijective mapping

**69**

**16**answers

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

**75**

**20**answers

### Proofs of the uncountability of the reals.

**71**

**2**answers

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

**67**

**8**answers

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

**43**

**2**answers

### A question about ordinal definable real numbers

**40**

**13**answers

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

**30**

**3**answers

### Is there a computable model of ZFC?

**17**

**4**answers

### Cardinality of the permutations of an infinite set

**12**

**5**answers

### Subset of the plane that intersects every line exactly twice

**37**

**3**answers