**133**

votes

**0**answers

11k views

### Ultrafilters and automorphisms of the complex field

It is well-known that it is consistent with $ZF$ that the only automorphisms of the complex field $\mathbb{C}$ are the identity map and complex conjugation. For example, we have that ...

**36**

votes

**0**answers

1k views

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

The invariant subspace problem (ISP) for Hilbert spaces asks whether every bounded linear operator $A$ on $l^2$ (with complex scalars) must have a closed invariant subspace other than $\{0\}$ and ...

**29**

votes

**0**answers

1k views

### 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?

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 ...

**24**

votes

**0**answers

523 views

### Where do uncountable models collapse to?

Suppose $T$ is a complete first-order theory (in an finite, or at worst countable, language). Given any model $\mathcal{M}\models T$ of cardinality $\kappa$, we can ask whether $\mathcal{M}$ can be ...

**21**

votes

**0**answers

533 views

### Relative null-ness

Here, "measure" always means Lebesgue measure on $\mathbb{R}$. This question is partly motivated by my answer ...

**21**

votes

**0**answers

714 views

### categorical characterization of large cardinals

Question 1. Is there a categorical representation of Kunen's inconsistency result?
Question 2. Is there a categorical characterization of very large cardinals (in particular for strong and ...

**21**

votes

**0**answers

857 views

### Supercompact and Reinhardt cardinals without choice

A friend of mine and I ran into the following question while reading about proper forcing, and have been unable to resolve it:
Definition. A cardinal $\kappa$ is supercompact if for all ordinals ...

**20**

votes

**0**answers

565 views

### A question about small sets of reals

In ZFC, does there exist an uncountable set of reals $A$ such that for every closed measure zero set of reals $B$, we have that $ A + B = \{a+b : a \in A, b \in B\} \neq \mathbb{R}$?
This question is ...

**20**

votes

**0**answers

1k views

### Subfields of $\mathbb{C}$ isomorphic to $\mathbb{R}$ that have Baire property, without Choice

While sitting through my complex analysis class, beginning with a very low level introduction, the teacher mentioned the obvious subfield of $\mathbb{C}$ isomorphic to $\mathbb{R}$, and I then ...

**17**

votes

**0**answers

400 views

### The cofinality of $(\mathbb{N}^\kappa,\le)$ for uncountable $\kappa$?

For a partially ordered set $P$, a set $A\subseteq P$ is cofinal if for each element of $P$ there is a larger element in $A$. The cofinality of $P$, ${\rm cof}(P)$, is the minimal cardinality of a ...

**17**

votes

**0**answers

435 views

### Souslin trees on the first inaccessible cardinal

This may be well-known or simply deducible from the existing theorems, but I didn't find an answer in my set theory books:
Is there a model of $ZFC$ in which there are no $\kappa$-Souslin trees where ...

**17**

votes

**0**answers

388 views

### Are there lightweight foundations for arbitrarily extendable objects?

My experience with foundations is rather scant, but I've run into some types of objects that seem to resist the sort of set-theoretic encoding schemes via Kurowski tuples that are rather common for ...

**17**

votes

**0**answers

578 views

### Antichains of Cardinals in ZF Without Choice…

With the Axiom of Choice, the cardinals form a nice linearly ordered "set". In the absence of the Axiom of Choice, the cardinals form a partially ordered "set". Broadly, I am wondering what ...

**15**

votes

**0**answers

628 views

### Souslin trees and weakly compact cardinals

In Souslin trees on the first inaccessible cardinal it is asked if it is consistent that there are no $\kappa-$Souslin trees at the least inaccessible cardinal $\kappa$.
In this question I would like ...

**14**

votes

**0**answers

613 views

### Lebesgue Measurability and Weak CH

Let $LM$ denote "all subsets of $\Bbb{R}$ are Lebesgue measurable", and
$WCH$ (weak continuum hypothesis) denote "every uncountable subset of $\Bbb{R}$ can be be put into 1-1 correspondence with ...

**13**

votes

**0**answers

239 views

### Non meager union of lines

Suppose every subset of real line of size $\aleph_1$ is meager. Can we conclude that any union of $\aleph_1$ lines in plane is also meager? If we replace meager by null, this was negatively answered ...

**13**

votes

**0**answers

369 views

### Is $\mathbb{Z}^{\omega}$ ever the union of a chain of proper subgroups each isomorphic to $\mathbb{Z}^{\omega}$?

Recall that the covering number $cov(B)$ is the least cardinal $\kappa$ such that $\kappa$ meagre sets cover the real line. Andreas Blass and John Irwin http://www.math.lsa.umich.edu/~ablass/bb.pdf ...

**13**

votes

**0**answers

423 views

### Ideas behind Gitik's solution of PCF conjecture

Recently Moti Gitik has refuted Shelah's PCF conjecture (see Short extenders forcings II ) by proving the following theorem:
Theorem. Assuming the consistency of infinitely many strong cardinals, one ...

**13**

votes

**0**answers

260 views

### Are there additive subgroups of reals of dimension 1 with no subgroups of dimension strictly between 0 and 1?

I will use $dimA$ to denote the Hausdorff dimension of a set $A \subseteq \mathbb{R}$. Being a null set means having Lebesgue measure zero.
In the 1966 paper "Additive gruppen mit vorgegebener ...

**13**

votes

**0**answers

399 views

### How to prove projective determinacy (PD) from I0?

Martin and Steel (in 1987?) showed that if there are infinite many Woodin cardinals then every projective set of reals is determined (PD).
However, it is mentioned in many texts that in 1983/1984 ...

**13**

votes

**0**answers

432 views

### To what extent does (co)homology of groups made discrete depend on set theory?

There's a well-known paper by Milnor, "On the homology of Lie groups made discrete," that discusses the relation between the homology of a Lie group $G$ and the underlying discrete group $G^\delta$. ...

**12**

votes

**0**answers

159 views

### Do the Laver tables converge to the Sierpinski triangle with a line segment sticking out in the hyperspace topology?

Let $(\{1,...,2^{n}\},*_{n})$ denote the $n$-th Laver table.
Let
$$C_{n}=\{(\frac{x}{2^{n}},\frac{x*_{n}y}{2^{n}})|x,y\in\{1,2,3,...,2^{n}\}\}$$
for all $n\in\mathbb{N}$.
Then since $C_{n}$ is a ...

**12**

votes

**0**answers

217 views

### A meager subgroup of the real line, which cannot be covered by countably many closed subsets of measure zero?

Is there a ZFC-example of a subgroup $H$ of the real line $\mathbb R$ such $H$ is meager, has zero Lebesgue measure, but cannot be covered by countably many closed subsets of measure zero in $\mathbb ...

**12**

votes

**0**answers

249 views

### Analytic contraction of the Stone-Cech compactification of $\mathbb C$

Let $S$ be the Stone-Cech compactification of $\mathbb C$. Then any meromorphic function $f:\mathbb C \to \mathbb{CP}^1$ extends to $S$.
Do the meromorphic functions separate the points of $S$?
...

**12**

votes

**0**answers

370 views

### How much choice is required to prove concretizability theorems in category theory?

A concretization of a category is a faithful functor to the category of sets. A category is concretizable if there exists such a functor.
An evident necessary condition for concretizability is ...

**12**

votes

**0**answers

451 views

### Some questions about $0^{\sharp}$ and forcing over $L$

1-Let $P\in L$ be a nontrivial set forcing or even a tame class forcing (tameness in the sense of Sy Friedman; see for example his Handbook paper), and let $G$ be $P-$generic over $L$. Then it is ...

**12**

votes

**0**answers

680 views

### Can a model of $V\neq L$ contain a class giving the $L$-ordering on all its sets?

This question is inspired by the excellent question by Douglas Ulrich When is $L$-Rank definable in inner models of $V=L$?
Suppose $M \in L$ is a countable model of $ZFC$, and furthermore suppose $M ...

**12**

votes

**0**answers

297 views

### How to measure the strength of Zermelo over bounded Zermelo?

Bounded Zermelo is Zermelo set theory with only bounded separation. It has the same strength as simple type theory or MacLane set theory or ETCS. It is a finitely axiomatized fragment of Zermelo, so ...

**11**

votes

**0**answers

173 views

### Is there a locally compact, $\omega_1$-compact, not $\sigma$-countably compact space of size $\aleph_1$?

There are old ZFC examples due to Eric van Douwen that satisfy all the properties in the title, except that they are of cardinality $2^{\aleph_0}$, so the answer to the title question is YES if the ...

**11**

votes

**0**answers

464 views

### Arguments against Freiling's argument against Continuum Hypothesis

Freiling's axiom of symmetry ($\sf AS$) is known as a justification for falsity of Continuum Hypothesis. Freiling in his 1986 paper, Axioms of symmetry: throwing darts at the real number line, ...

**11**

votes

**0**answers

299 views

### Large cardinals arising from alternate set theories

My question is whether there are model-theoretic large cardinal axioms associated with $NFU$ - or more generally, with set theories other than $ZFC$.
Large cardinal properties generally come in one ...

**11**

votes

**0**answers

197 views

### Generic $\mathbf{\Sigma}_3^1$-absoluteness for class forcings

In the paper "Generic Absoluteness" by Bagaria and Friedman (http://www.logic.univie.ac.at/~sdf/papers/bagfried.pdf) it is shown that in ZFC generic $\mathbf{\Sigma_3^1}$-absoluteness is false for ...

**11**

votes

**0**answers

273 views

### Adding a saturated ideal

Is it consistent that there is no $\omega_2$-saturated ideal on $\omega_1$, but one is introduced by an $\omega_2$-closed forcing?
Some motivation:
If $\delta$ is a Woodin cardinal, then it remains ...

**11**

votes

**0**answers

254 views

### Absoluteness of “$\kappa$-homogeneously Suslin” for sets of reals

What is known about the absoluteness, or lack thereof, of the notion of "$\kappa$-homogeneously Suslin" for sets of reals?
For example, if $A$ is $\kappa$-homogeneously Suslin and $\lambda > ...

**11**

votes

**0**answers

583 views

### Can one exhibit an explicit Kuratowski infinite set without invoking Replacement?

The customary formulation of the Axiom of Infinity within Zermelo-Fraenkel set theory asserts the existence of an inductive set: a set $ I$ with $\varnothing\in I$ such that $x\in I$ implies ...

**11**

votes

**0**answers

514 views

### Is every $\sigma$-algebra of sets *abstractly* the Borel algebra of a topology on perhps some other set?

Is every sigma-algebra the Borel algebra of a topology?
inspires the present question which asks for less.
Question: Given a $\sigma$-algebra ${\cal A}$ on a set $X$, does there exist a topology ...

**10**

votes

**0**answers

245 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 ...

**10**

votes

**0**answers

311 views

### Is it consistent with ZFC that no nontrivial forcing notion has automatic mutual genericity?

A nontrivial forcing notion $\newcommand\Q{\mathbb{Q}}\Q$ exhibits
automatic mutual genericity, if whenever $G,H\subseteq\Q$ are
distinct $V$-generic filters (existing, say, in some forcing
extension ...

**10**

votes

**0**answers

236 views

### When does $HOD^{V[G]} \subseteq V$?

Assume that $\mathbb{P}\in HOD$ is non-trivial. It is well-known that if $\mathbb{P}$
satisfies some homogeneity properties, then $HOD^{V[G]} \subseteq V$, where $G$ is $\mathbb{P}$-generic over $V$.
...

**10**

votes

**0**answers

258 views

### c.c.c forcing notions and adding minimal generic reals

Is the following statement consistent:
``There is no non-trivial c.c.c forcing notion adding a minimal generic real''?
The question is related to Prikry's question: Is it consistent that any ...

**10**

votes

**0**answers

261 views

### Is there a “hereditary” construction for $L$?

Recall that $L$, Godel's constructible universe is constructed by defining the following hierarchy:
$L_0=\varnothing$, for a limit ordinal $\delta$, $L_\delta=\bigcup_{\alpha<\delta}L_\alpha$, and ...

**10**

votes

**0**answers

238 views

### Namba forcing and semiproperness

This question is the result of leaving "Proper and Improper Forcing" on my nightstand by accident.
Is the statement "Namba forcing is semiproper" known to be equiconsistent with some more standard ...

**10**

votes

**0**answers

349 views

### Intermediate submodels without Boolean algebras

My question is motivated by the following well-known fact regarding intermediate submodels of generic extensions. I would like to know if it can be proven using posets without the need for Boolean ...

**10**

votes

**0**answers

371 views

### Ultrafilter theorem and translation invariant measures

The usual Vitali construction of a non-Lebesgue measurable set generalizes to a proof that there are no (non-trivial) translation invariant measures on $\mathcal P\mathbb R$.
On the other hand, there ...

**10**

votes

**0**answers

516 views

### Existence (or non) of “definable” ultrafilters

This is a question which I suspect has an absurdly easy answer, but I'm not seeing it.
Let $\langle\cdot,\cdot\rangle:\omega^2\rightarrow\omega$ be your favorite pairing map (for me, this is the ...

**10**

votes

**0**answers

250 views

### full support iteration of semiproper forcings

Suppose $\langle P_\alpha,\dot{Q}_\alpha:\alpha\leq\omega_1\rangle$ is a full support iteration of (semi)proper forcings. Is the full limit $P_{\omega_1}$ (semi)proper or at least stationary set ...

**9**

votes

**0**answers

144 views

### Idea behind the proof of consistency of club filter of $\omega_1$ is ultrafilter + ZF + DC

I've been trying to understand Radin Forcing and some of its applications, one of which is the use of it to prove the consistency of ''Club filter of $\omega_1$ is an ultrafilter + ZF + DC''. However, ...

**9**

votes

**0**answers

153 views

### From interpolation to separation

Lusin's separation theorem states that, if $A$ and $B$ are disjoint analytic subsets of a Polish space, then there is a Borel set $X$ separating them ($A\subseteq X$, $B\cap X=\emptyset$). Craig's ...

**9**

votes

**0**answers

148 views

### Homogeneity of a variant of Prikry forcing

Prikry forcing is easily seen to be cone homogeneous (for any $p, q \in \mathbb{P}$, there are $p' \leq p, q' \leq q$ and an isomorphism $\Phi: \mathbb{P}/p' \simeq \mathbb{P}/q'$); in particular for ...

**9**

votes

**0**answers

195 views

### The Chang model after collapsing an inaccessible limit of Woodins

If $\kappa$ is an inaccessible cardinal and $G \subset \operatorname{Col}(\omega,\mathord{<}\kappa)$ is a $V$-generic filter, then in $V[G]$ the Chang model $L(\text{Ord}^\omega)$ satisfies "every ...