**7**

votes

**1**answer

136 views

### Consistency Strength of “HC is elementary in V[G]”

Let $P$ be the Levy-collapse of the ordinals, so $P$ is a class forcing notion that makes every ordinal countable.
Note that since $P$ is weakly homogeneous, for any formula $\phi(\overline{a})$ ...

**5**

votes

**0**answers

130 views

### Sacks minimality without choice

The usual argument for the minimality result for Sacks forcing uses choice.
Theorem (Sacks): Let $s \subseteq \mathbb S_\kappa$ be generic for the forcing to add a Sacks subset to $\kappa$, where $\...

**11**

votes

**1**answer

414 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 ...

**6**

votes

**1**answer

212 views

### Elementary chains in forcing extensions of $M_1$

Let $M_1$ be the canonical inner model with one Woodin cardinal $\delta$. Now suppose that $\mathbb{P}$ is a forcing notion of size $< \delta$, which preserves $\omega_1$ and that $G$ is a generic ...

**7**

votes

**0**answers

152 views

### $V$ as a $HOD$ of its class generic extension

By an old result of Roguski, The theory of the class $HOD$, any model $V$ of $ZFC$ has a class generic extension $V[G]$ such that $HOD$ of $V[G]$ equals $V$.
This result is also stated and generalized ...

**9**

votes

**0**answers

228 views

### (A little bit) Beyond the E-recursive

The E-recursive functions are a particular generalization of classical recursion theory to the entire set-theoretic universe, $V$. They are defined via a schemes: see http://www.math.harvard.edu/~...

**7**

votes

**0**answers

208 views

### A strengthening of Chang's conjectures

In the Handbook of Set Theory, Foreman has (essentially) the following proposition (3.9):
Suppose $\kappa_n>...>\kappa_0$ and $\lambda_n>...>\lambda_0$ are regular cardinals and $(\...

**2**

votes

**1**answer

196 views

### Is Extensionality needed for the incompleteness of very weak set theories?

$ST$ is the weak set theory built upon identity theory and containing
the axiom for empty set,
the axiom for adjunction and
the axiom for extensionality.
It is known that $ST$ interprets ...

**6**

votes

**0**answers

168 views

### PCF conjecture and fixed points of the $\aleph$-function

Recently Moti Gitik refuted Shelah's PCF conjecture, by producing a countable set $a$ of regular cardinals with $|pcf(a)| \geq \aleph_1.$ See his papers
Short extenders forcings I and Short ...

**5**

votes

**0**answers

159 views

### Can one take roots of rank-into-rank embeddings infinitely many times?

If $\lambda$ is a cardinal, then 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=\bigcup_{\...

**5**

votes

**1**answer

249 views

### A kind of anti-Ramsey result

In contrast to classic results for arithmetic progressions of arbitrary length in one set at least of any finite partition of IN, it is easy to construct a partition in two sets of integers A and B ...

**2**

votes

**0**answers

108 views

### A question regarding an analogue of the Kleene $T$-predicate for Koepke's ordinal computability

Does Koepke's notion of ordinal computability admit an analogue of the Kleene $T$-predicate? If so, is the existence of such a $T$-predicate independent of $ZFC$? Also, if one assumes the existence ...

**6**

votes

**0**answers

91 views

### By replacing the Laver tables with nearly distributive fake Laver tables, can one produce algebras where the period of 1 grows fast?

I am now generalizing the notion of the classical Laver table and the fake Laver table to a larger class of algebras. A sequence of algebras $(\{1,...,2^{n}\},*_{n})_{n\in\omega}$ is said to be a ...

**6**

votes

**0**answers

128 views

### How distributive are the fake Laver tables?

The Laver table $A_{n}$ is the unique algebra $(\{1,...,2^{n}\},*)$ such that $x*1=x+1$ for $1\leq x<2^{n}$, $2^{n}*1=1$, and $x*(y*z)=(x*y)*(x*z)$.
Let's now replace the Laver table $A_{n}$ with ...

**1**

vote

**0**answers

187 views

### How may we define a bijection from $\wp(\mathbb{Q})$ to $\mathbb{R}$ in $ZFC$? [closed]

Some expressed difficulty understanding that there are more members in $\mathbb{R}$ than in $\mathbb{Q}$ according to classical set theories because there between all real numbers is a rational number....

**17**

votes

**0**answers

434 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 ...

**9**

votes

**2**answers

264 views

### Two questions about the “grasp” cardinal function

For a topology $\mathcal{T}$ on a set $S$, where $\mathcal{T}$ does not have a finite base, I define the grasp $g(\mathcal{T})$ to be the least infinite cardinal $\kappa$ such that $\mathcal{T}$ has a ...

**11**

votes

**2**answers

857 views

### On Hamkins' answer to a problem by Michael Hardy

Based on a post by Michael Hardy and Hamkins' answer to it Andreas Blass, Will Brian, Joel Hamkins, Michael Hardy and Paul Larson introduced a new cardinal characteristic of the continuum $\mathfrak{...

**4**

votes

**0**answers

176 views

### On the Axiom of Choice for Conglomerates and Skeletons

Say that $\mathcal{X}$ is a conglomerate if $\mathcal{X} = \{X_i: i \in I\}$, where each $X_i$ and $I$ are classes. The Axiom of Choice for Conglomerates is the statement: Whenever $\mathcal{X}$ and $\...

**4**

votes

**0**answers

119 views

### Characterization of $L[T_{2n+1}]$ as a direct limit of mice

I am asking for a reference request/proof sketch for the result of Steel
that characterizes $L[T_{2n+1}]$ as a direct limit of mice.
Given that both $L[T_{2n+1}]$ and $M_{2n}$ have a $\Sigma_{2n+2}$ ...

**2**

votes

**1**answer

172 views

### Set theoretic issue of localization of abelian categories

For a small abelian category in which every object is also a set, consider its localization with respect to a Serre subcategory (thus a quotient category), is it true that under this localization ...

**26**

votes

**0**answers

577 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 ...

**12**

votes

**2**answers

400 views

### Woodin on Posner-Robinson for the hyperjump and sharp

The Posner-Robinson theorem states that, if $X$ is noncomputable, there is some $G$ such that $X\oplus G=G'$; that is, even though genuine jump inversion only works above $0'$, every (nontrivial) $X$ ...

**11**

votes

**1**answer

473 views

### Can you have many independent reals?

Working in $\sf ZFC$, is it provable, or at least consistent (say, over $L$), that you have $\aleph_1$ forcings, $\Bbb P_\alpha$ such that:
$\Bbb P_\alpha$ is c.c.c.
$\Bbb P_\alpha$ adds a real ...

**3**

votes

**0**answers

391 views

### “Nicely” strong measure zero sets

This question is essentially an expanded version of the unanswered half of Two strengthenings of "strong measure zero".
A set $X$ of reals is strong measure zero if, for any $f: \omega\...

**7**

votes

**1**answer

343 views

### Is every ordinal potentially definable?

It is easy to see that, if $V\models\alpha>\omega_1^{CK}$, then $\alpha$ is not recursive in any forcing extension of $V$. The argument goes as follows:
The relation "$\Phi_e=r$" is $\Pi^0_2$.
...

**11**

votes

**0**answers

316 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 ...

**9**

votes

**0**answers

158 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, ...

**12**

votes

**1**answer

252 views

### Forcings that are not equivalent to Levy collapse

Assume GCH and that $\kappa$ is a regular uncountable cardinal. Let $\mathbb{P}$ be a separative, $<\kappa$-directed closed, nowhere trivial, $\kappa^+$-cc poset of size $\kappa^+$. Must $\mathbb{...

**3**

votes

**0**answers

100 views

### A question regarding forcing in $NGBC^{-f}$+$BAFA$

Suppose one has a model $M$$\vDash$$NGBC^{-f}$+BAFA. Does there exist a (class) forcing extension $M[G]$$\vDash$$NGBC^{-f}$+$BAFA$ that has a submodel $N$$\vDash$$NGB^{-f}$+$BAFA$+$\lnot$$AC$? Can ...

**19**

votes

**3**answers

418 views

### Which partitions of $[0,1]$ are collection of level sets of a real continuous function?

Let $f:[0,1]\to[0,1]$ be given. The level sets of $f$ (ie the collection of all sets of the form $\{x\in[0,1]:f(x)=y\}$, for each fixed $y\in[0,1]$) partition the domain of $f$. I am curious for set ...

**0**

votes

**1**answer

79 views

### Prime ideals containing the finite members of ${\cal P}(\omega)$

Let ${\frak P}$ denote the collection of prime ideals containing the finite members of ${\cal P}(\omega)$, and order ${\frak P}$ by set inclusion.
What is the cardinality of ${\frak P}$, and what's ...

**16**

votes

**2**answers

349 views

### Can $L$ be thin?

I have recently been wondering if the following is consistent with ZFC:
For every infinite ordinal $\alpha$: $|V_\alpha\cap L|=|\alpha|$.
Intuitively, this states that for $L$ is very "thin", in ...

**2**

votes

**1**answer

163 views

### Boolean algebras and free filters generated by chains

Suppose $\mathbf{B}$ is a complete Boolean algebra with an infinite domain $B$. Suppose $\mathbf{B}$ is atomic (i.e. every element is the supremum of some set of atoms). This algebra contains the co-...

**9**

votes

**0**answers

157 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 ...

**3**

votes

**1**answer

396 views

### Does Borel's proof for existence of normal numbers make an essential use of axiom of choice?

A normal number is a real number whose infinite sequence of digits in every base $b$ is distributed uniformly in the sense that each of the $b$ digit values has the same natural density $\frac{1}{b}$, ...

**12**

votes

**0**answers

232 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 ...

**5**

votes

**1**answer

163 views

### $\kappa$-support iterations of $<\kappa$-strategically closed forcing

Let $\kappa$ be an uncountable regular cardinal, and suppose that $\langle \mathbb{P}_\alpha,\dot{\mathbb{Q}}_\alpha:\alpha<\delta\rangle$ is a $\kappa$-support iteration of $<\kappa$-...

**10**

votes

**0**answers

155 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 ...

**28**

votes

**7**answers

2k 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/...

**5**

votes

**1**answer

208 views

### A variant of Freiling's Axiom of Symmetry and a weak form of the Continuum Hypothesis in models where all sets of reals are Lebesgue measurable

Consider the following variant of Freiling's Axiom of Symmetry, $\mathsf{AS}$, which will be denoted $A_{< 2^{\aleph_0}}$:
given any function $f$ from $\mathbb{R}$ into the families of of subsets ...

**10**

votes

**1**answer

431 views

### Cardinality of definable sets of reals

Throughout this question we assume ZFC.
If CH holds, then the following is obvious:
(S) Every definable infinite subset of $\mathbb R$ has size either $\aleph_0$ or $2^{\aleph_0}$.
(It's true ...

**4**

votes

**1**answer

257 views

### A question regarding a common critique of Freiling's Axiom of Symmetry

(In what follows, Freiling's Axiom of Symmetry is simply the following:
($A_{\aleph_0}$) :( $\forall$$f$: $\mathbf R$ $\rightarrow$$\mathbf R_{\aleph_0}$)($\exists$$x_1$,$x_2$)($x_2$$\notin$$f$($x_1$...

**20**

votes

**2**answers

759 views

### Antirandom reals

This is a crossposting of http://math.stackexchange.com/questions/1446602/anti-random-reals, which has not gotten any answers; after thinking about the problem, I've become more convinced that it ...

**16**

votes

**1**answer

490 views

### Two strengthenings of “strong measure zero”

A set $X\subseteq\mathbb{R}$ is strong measure zero if, for every sequence $(\epsilon_i)_{i\in\mathbb{N}}$ of positive reals, there is a sequence $(I_i)_{i\in\mathbb{N}}$ of open intervals covering $X$...

**14**

votes

**2**answers

557 views

### Who needs RCS iterations?

According to this paper of Chaz Schlindwein, any countable support iteration of semi-proper forcings is semi-proper. This seems like a breakthrough simplification, and I wonder why it is not more ...

**9**

votes

**2**answers

412 views

### Ordering of large cardinals by cardinality

Let Type A and Type B be two types of large cardinals from, say, Cantor's Attic (http://cantorsattic.info/Upper_attic)
Now assuming that ZFC + Type A + Type B is consistent (ie, both Type A and Type ...

**10**

votes

**0**answers

240 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 $V$.
...

**9**

votes

**1**answer

316 views

### Pontryagin dual of the surreal numbers?

Has any work been done on the Pontryagin dual of the surreal numbers (suitably topologized)? I have not been able to find anything and am not sure if this is still unknown.
Alternatively, has this ...

**5**

votes

**1**answer

159 views

### almost disjoint ladder system on $\omega_2$

Suppose $\langle s_\alpha : \alpha \in \omega_2 \cap \mathrm{cof}(\omega_1) \rangle$ is a sequence such that each $s_\alpha$ is an increasing cofinal map from $\omega_1$ to $\alpha$. Is it possible ...