**3**

votes

**0**answers

95 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

378 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

78 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 ...

**15**

votes

**2**answers

310 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 ...

**0**

votes

**1**answer

82 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 ...

**9**

votes

**0**answers

150 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 ...

**2**

votes

**1**answer

353 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 $\frac{1}{b}$, ...

**8**

votes

**0**answers

150 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

143 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 ...

**9**

votes

**0**answers

144 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 ...

**22**

votes

**5**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 ...

**5**

votes

**1**answer

154 views

### Does $ZF$+$LM$+$A_{\lt2^{\aleph_0}}$ imply $\lnot$$WCH$?

In what follows, I will use the following acronyms:
$AS$:= Freiling's Axiom of Symmetry
$LM$:="Every set of reals is Lebesgue measurable."
$WCH$:="every uncountable subset of $\mathbf R$ can be put ...

**10**

votes

**1**answer

382 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

201 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 ...

**21**

votes

**2**answers

699 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

428 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 ...

**14**

votes

**2**answers

532 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 ...

**8**

votes

**2**answers

357 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

234 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 $V$.
...

**6**

votes

**1**answer

257 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

150 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 ...

**18**

votes

**2**answers

520 views

### History of set-class distinction

I have two questions concerning the history of set theory, both related to the distinction between the notion of a set and the notion of a class:
Who was the first mathematician to make this ...

**2**

votes

**1**answer

181 views

### Wholeness Axiom and Ultimate L

From what I understand:
The Wholeness Axiom(s) is/are the "ultimate axioms of infinity", bordering on inconsistency with ZFC.
Ultimate L (Completion of ZFC) attempts to extend the orderly world of ...

**6**

votes

**1**answer

142 views

### Different ways of making $HOD$ far from $V$

There are different criteria for building a model $V$ of $ZFC$ which is far from its
$HOD$, for example:
$(A)$ Cardinality criteria: For this in a joint work with James Cummings and Sy Friedman, we ...

**9**

votes

**2**answers

745 views

### Non null Turing antichain

This interesting question resulted from a query of Mushfeq: In ZFC, can we find a non null set of pairwise Turing incomparable reals?

**8**

votes

**2**answers

350 views

### Some “axiom of choice” and “dependent choice” issues

I am probably about to ask some fairly basic questions, and yet I have found it quite hard to find the answers to these.
If I understand correctly, mathematicians tend to be quite happy working with ...

**21**

votes

**0**answers

520 views

### Relative null-ness

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

**6**

votes

**0**answers

179 views

### Remote points in $\beta X$

It is known that in general convergence by sequences is not enough to account for all points in $\beta X \setminus X$, where $\beta X$ refers to the Stone-Cech compactification of a topological space ...

**16**

votes

**1**answer

447 views

### Can we find CH in the analytical hierarchy?

Recently I was talking to my friend and I have mentioned to him that it was proven that CH is not provably (over ZFC) equivalent to any statement in second-order arithmetic. However, today I found out ...

**14**

votes

**0**answers

255 views

### A problem of Keisler and Tarski

The following question dates back to Keisler and Tarski: From accessible to inaccessible cardinals, Fund. Math. 53, 1964 and also perhaps Mazur: On continuous mappings of Cartesian products, Fund. ...

**7**

votes

**1**answer

169 views

### Forcing Extension of Countable Linearly Iterable Structures

Let $V$ satisfy there exists a measurable cardinal. Let $\kappa$ be a measurable cardinal and $U$ be the normal measure on $\kappa$ witnessing this. Let $\mathbb{P}$ be a forcing of size less than ...

**5**

votes

**0**answers

59 views

### Is there an analytic $\mathrm{P}$- ideal on $\omega$ which is not $\Sigma^0_2$ and not $\Pi^0_3$-complete?

Soleski proved that for any analytic $\mathrm{P}$-ideal on $\omega$ is $\Pi^0_3$.
The usually example , such as the density zero ideal $Z_0$ is $\Pi^0_3$-complete, $I_{\frac{1}{n}}$ is ...

**4**

votes

**1**answer

159 views

### Sets not containing the vertices of unit triangles (Question posed by Erdős)

Following this post, I have been thinking about the problem posed by Erdős,
Does there exist a constant $c > 0$ such that every subset $A$ of the plane of area more than $c$ contains the ...

**7**

votes

**2**answers

309 views

### Large cardinal consistency strength and size

My understanding is that large cardinals are ordered by "consistency strength", but how does this correlate with their size (cardinality)?
More specifically, are there any systematic results on the ...

**18**

votes

**1**answer

2k views

### How far wrong could the Continuum Hypothesis be?

I hear it's consistent with ZFC to have
$$ 2^{\aleph_0} = \aleph_n $$
for any $n = 1, 2, 3, \dots $. How much worse can it get?
More precisely: are there models of ZFC with $2^{\aleph_0} \gt ...

**0**

votes

**1**answer

186 views

### A question regarding models of $ZF+I_0$ [Revised]

In his answer to user42090's mathoverflow question"Minimal Generalized Contnuum Hypothesis & Axiom of Choice", Prof. Hamkins writes:
"...one can build the analogue of the symmetric models for ...

**3**

votes

**1**answer

308 views

### “Lebesgue-measurable” cardinals and real-closed fields

I understand the motivation behind measurable cardinals is to ask the question: "is there any set large enough to admit a non-trivial measure on all of its subsets?"
Hence, it's also worthwhile to ...

**9**

votes

**1**answer

270 views

### tree properties on $\omega_1$ and $\omega_2$

Are the following mutually consistent (relative to large cardinals)?
(1) There are no $\omega_2$-Aronszajn trees.
(2) There is an $\omega_1$-Kurepa tree.
In the models I know of the tree property ...

**1**

vote

**1**answer

163 views

### tree property at $\aleph_2$ and $\aleph_4$

It is claimed that if there are two weakly compact cardinals, then there is a generic extension in which $\aleph_2$ and $\aleph_4$ have the tree property. Assuming one knows Mitchell's forcing, what ...

**6**

votes

**0**answers

236 views

### The least admissible above a dominating real

Let $\mathbb{P}$ be the usual forcing which adds a dominating real: conditions in $\mathbb{P}$ are pairs $(p, f)$ with $p:\omega\rightarrow\omega$ finite partial and $f:\omega\rightarrow\omega$ total, ...

**11**

votes

**1**answer

231 views

### Partition relation at successor cardinal

Is it consistent that there are regular cardinals $\kappa < \lambda$, such that $\lambda$ is a successor cardinal and for every coloring $d\colon[\lambda]^2\to\kappa$ there is some ...

**3**

votes

**1**answer

122 views

### Does totally proper forcing imply countable distributivity?

For a suitable model $M$ for $Q$ and a condition $q \in Q$ we say that $q$ is $(M,Q)$-generic if whenever $r \leqslant q$, $D \in M$ dense, $D \subset Q$, $r$ is compatible with an element of $D \cap ...

**0**

votes

**2**answers

291 views

### A question regarding the relation between Freiling's Axiom of Symmetry and real-valued measurable cardnals

A major argument against Freiling's Axiom of Symmetry is the following (this from the wikipedia article of the same name):
"The naive probabalistic notion used by Freiling tacitly assumes that there ...

**8**

votes

**0**answers

210 views

### On an unpublished result of Magidor

In 1970th, Magidor proved the following important results:
(1) Assuming the existence of a supercompact cardinal, it is consistent that $\aleph_\omega$
is strong limit and ...

**26**

votes

**2**answers

1k views

### What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simply connected?

This question concerns a set-theoretic aspect that I found interesting in the recent question asked by user Nick R., namely, Is
$\mathbb{R}^3\setminus\mathbb{Q}^3$ simply connected? He had asked ...

**2**

votes

**1**answer

175 views

### Notation: $Sigma$ and $Pi$ of intersections

In Jech - Set Theory, the proof of Theorem 31.7, I came along some notations I wish to understand correctly.
For a countable elementary substructure $M \prec H_\lambda$ and $A \in M$ and a generic ...

**28**

votes

**5**answers

2k views

### Why should we care about “higher infinities” outside of set theory?

Let's say you are a prospective mathematician with some addled ideas about cardinality.
If you assumed that the natural numbers were finite, you'd quickly vanish in a puff of logic. :)
If you ...

**4**

votes

**2**answers

207 views

### Natural examples of $\bf\Sigma^0_3$ equivalence relations

I have been reading about Borel equivalence relations and I have noticed that while $\bf\Sigma^0_3$ equivalence relations are mentioned, there is a conspicuous absence of natural examples (other than ...

**1**

vote

**1**answer

147 views

### Injective subset function

Let $X$ be a non-empty set and let $F: X \to {\cal P}(X)$ be a function with the following property:
for $A \subseteq X$ we have $|A| \leq |\bigcup F(A)|$.
Does this imply that there is an ...

**3**

votes

**1**answer

244 views

### On generic forcing conditions

Let $P$ be a forcing poset, and $Q \in V^P$ a forcing poset in $V^P$. Let $M \prec H(\lambda)$ ($\lambda$ sufficiently large) countable with $P,Q \in M$.
What I want to know is if then the following ...