**5**

votes

**0**answers

61 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

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

**7**

votes

**0**answers

119 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 ...

**4**

votes

**0**answers

87 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 ...

**8**

votes

**0**answers

125 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 ...

**18**

votes

**5**answers

1k 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 ...

**4**

votes

**1**answer

124 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 ...

**8**

votes

**1**answer

325 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

173 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 ...

**18**

votes

**2**answers

604 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 ...

**14**

votes

**1**answer

346 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 ...

**-4**

votes

**0**answers

72 views

### Accepted Universal Set and Proof Verification [closed]

I noticed in my discrete math textbook that the Rosen makes reference to the universal set, denoted $U$, and establishes various results, like the complement of a set $A$ with respect to $U$, or the ...

**13**

votes

**2**answers

496 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

337 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 ...

**9**

votes

**0**answers

194 views

+50

### 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

226 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

137 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

493 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

159 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 ...

**5**

votes

**1**answer

132 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

709 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?

**7**

votes

**2**answers

325 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

504 views

### Relative null-ness

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

**-3**

votes

**0**answers

58 views

### How can we be sure that unnameable numbers have corresponding Dedekind cuts (without invoking Uncountable AC)? [migrated]

Most of the real numbers are unnameable, so there cannot be any definitions of those numbers, and Dedekind cuts depend on a definition. Is there a proof there exists a Dedekind cut even for unnameable ...

**5**

votes

**0**answers

164 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

428 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 ...

**13**

votes

**0**answers

237 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

159 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

55 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

154 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

286 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

179 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 ...

**-2**

votes

**0**answers

9 views

### Can someone explain the reasoning behind Cantor's diagonal argument? [migrated]

I'm taking a class that's covering cardinalities, and I was introduced to Cantor's diagonal argument today, and I'm having trouble following the logic. The theorem states, "If s[1], s[2], … , s[n], … ...

**3**

votes

**1**answer

188 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

**0**answers

200 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

154 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 ...

**5**

votes

**0**answers

223 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, ...

**10**

votes

**1**answer

222 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 ...

**2**

votes

**1**answer

118 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

**1**answer

223 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

201 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

170 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

196 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

138 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

239 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 ...

**6**

votes

**1**answer

179 views

### Consistency of the nonrigidity of $P(\omega_1)/NS$

Is it consistent with ZFC that there exists an automorphism of $P(\omega_1)/\mathrm{NS}_{\omega_1}$ which is not the identity?

**1**

vote

**0**answers

109 views

### Defining Global Choice in terms of strong limit cardinals over $ZF$

In his answer to user33038's mathoverflow question "What axioms are stronger than the Axiom of choice?", Prof. Hamkins writes:
"What's more, the axiom of choice is equivalent over $ZF$ to the ...