# Tagged Questions

**12**

votes

**0**answers

267 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 ...

**7**

votes

**1**answer

170 views

### Can Sacks forcing add a Cohen generic real over $L$?

Motivated by this question Forcing the negation of CH without adding Cohen reals over L and Todd Eisworth's comment, the question is the following:
1) Suppose $V$ has no Cohen generic reals over $L$. ...

**28**

votes

**1**answer

950 views

### Should axiomatic set theory be translated into graph theory?

Recently I saw the abstract of a paper by Nash-Williams: ``Should axiomatic set theory be translated into graph theory?''. The abstract, taken from Mathscinet says the following:
The author ...

**8**

votes

**0**answers

132 views

### Are there always large discrete families of normal measures?

Let $\kappa$ be a measurable cardinal. We give the Stone space of all ultrafilters on $\kappa$ the usual topology, where each $x\subseteq\kappa$ determines a basic open $[x]=\{U;x\in U\}$. The ...

**-2**

votes

**1**answer

249 views

### A set theory and a model without the empty set [closed]

Im just asking out of curiosity.
Is there a set theory $\mathcal T$ and a model $\langle M,E\rangle \vDash \mathcal T$ such that no set in M is empty:
$$\left(\forall m\in M \right)\left(\exists n\in ...

**6**

votes

**1**answer

293 views

### Forcing the negation of CH without adding Cohen reals over L

Suppose CH + "there are no Cohen reals over L". Can we force the negation of CH without adding any Cohen real over L?

**2**

votes

**2**answers

520 views

### Can we compute every definable number with knowledge of the halting problem?

Suppose we knew the answer to the halting problem, and the halting problem for this new system with the old halting problem solved. And so on. Would this allow us to compute every definable number?

**4**

votes

**0**answers

143 views

### Two questions about the behavior of the continuum function

The first question asks about the global behavior of the power function in the case of finite gaps.
Question 1. Fix a natural number $m>1.$ For each limit ordinal $\alpha,$ including $0$, let ...

**13**

votes

**0**answers

492 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, ...

**10**

votes

**1**answer

385 views

### What sort of large cardinal can continuum be?

I have stumbled across a related question asking which large cardinal properties can hold for $\aleph_1$. This question is probably also related, asking in what ways $\aleph_0$ is a "large" cardinal.
...

**4**

votes

**1**answer

254 views

### Is tree property inconsistent with Berkeley cardinals in the absence of Axiom of Choice?

On one hand due to Kunen's inconsistency theorem it is known that within $\sf ZF$, large cardinal axioms beyond Reinhardt cardinal are inconsistent with $\sf AC$.
Also some recent results of Bagaria, ...

**5**

votes

**1**answer

319 views

### Ill-founded models of set theory with well-founded ordinals

Let $(\mathcal{M},E)$ be an internally non-well-founded model of set theory i.e of $ZFC^{\neg f}=ZFC\setminus \mathrm{foundation}+\neg \mathrm{foundation}$, then $\mathcal{M}$ includes an infinite ...

**5**

votes

**0**answers

128 views

### Failure of GCH at a strongly compact cardinal

Does Con(ZFC+ there exists a strongly compact cardinal) imply Con(ZFC+ there exists a strongly compact cardinal $\kappa+ 2^\kappa > \kappa^+$)?

**24**

votes

**1**answer

713 views

### Do the analogies between metamathematics of set theory and arithmetic have some deeper meaning?

By "formal analogies" between the metamathematics of $\mathsf{ZFC}$/set theory and $\mathsf{PA}$(=Peano Arithmetic)/first order arithmetic, I mean facts such as the following:
We are considering a ...

**4**

votes

**2**answers

354 views

### Maximal chains and antichains of statements weaker than AC

First, I would like to say that I asked this question (a more general one actually) on math.stackexchange.com.
Consider the set $\varPhi$ of statements in the language of ZF that are weaker than AC ...

**7**

votes

**1**answer

360 views

### Can an ultrapower be undone by forcing?

I am not 100% certain this question is appropriate for MO; I may just be missing something obvious. Also, I vaguely recall a similar question being asked here a while ago, but I can't find it; if it ...

**12**

votes

**1**answer

251 views

### When can Power Sets be Limit Cardinals?

My original question (posted in http://math.stackexchange.com/questions/1584430/can-all-power-sets-be-limit-cardinals) was:
Is it possible to create a model of ZFC, so that the cardinality of each ...

**2**

votes

**1**answer

128 views

### Cardinalities of maximal towers in ${\cal P}(\omega)$

For $A,B\subseteq \omega$ we write $A \subseteq^* B$ if $A\setminus B$ is finite. We call ${\cal T}\subseteq {\cal P}(\omega)$ a tower if it is linearly quasiordered with respect to $\subseteq^*$.
...

**3**

votes

**0**answers

76 views

### Do the classical Laver tables induce periodic systems of jump Laver tables?

Let $A_{n}=(\{1,...,2^{n}\},*_{n})$ where $*_{n}$ is the binary operation on $A_{n}$ such that $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$ for $x,y,z\in\{1,...,2^{n}\}$, $x*1=x+1$ for $x<2^{n}$ and ...

**6**

votes

**1**answer

257 views

### How many closed measure zero sets are needed to cover the real line, really?

This is a refinement of an earlier question.
This question assumes familiarity with combinatorial cardinal characteristics of the continuum.
For the reader's convenience, I reproduce below the ...

**8**

votes

**1**answer

374 views

### How many closed measure zero sets are needed to cover the real line?

This question assumes familiarity with combinatorial cardinal characteristics of the continuum.
Let $\mathcal{E}$ be the $\sigma$-ideal generated by closed measure zero subsets of the real line. It ...

**8**

votes

**0**answers

165 views

### A question about strongly compact cardinals

Is the following equiconsistent with the existence of a strongly compact cardinals:
For every $\lambda > \kappa$ there exists a $\lambda$-strongly compact embedding $j: V \to M$ with the ...

**14**

votes

**2**answers

1k views

### How strong is Cantor-Bernstein-Schröder?

There are several questions here on MO about the Cantor-Bernstein-Schröder ((C)BS) theorem, but I could not find answers to what arose to me recently.
Although I don't think I need to recall it here, ...

**10**

votes

**1**answer

338 views

### “Largish” cardinals

In what follows, $\mathsf{ZCKP}$ refers to the subset of $\mathsf{ZFC}$ consisting of the axioms of Zermelo set theory with choice and foundation ($\mathsf{ZC}$) plus those of Kripke-Platek set theory ...

**13**

votes

**0**answers

250 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 ...

**6**

votes

**0**answers

217 views

### Axiom of choice and a set in the plane that intersects every line in two points

In this question Subset of the plane that intersects every line exactly twice someone ask for a reference of a paper where they proof the result : ''There exist a subset of the plane that intersects ...

**6**

votes

**0**answers

163 views

### “Clubiness” of projective sets of ordinals

I'm sure this is just my google-fu failing me, but: what are sufficient, non-overkill large cardinal axioms which guarantee "Every (boldface) $\Pi^1_n$ set of (real codes for) countable ordinals ...

**0**

votes

**0**answers

372 views

### On the Theory of Infinite Step Processes of Sequential Decision Making

On the border of Set Theory and Game Theory there is a broad class of Infinite Step Processes of Sequential Decision Making that can be characterized by the main following property: a Future ...

**6**

votes

**0**answers

188 views

### Adding minimal subsets to $\aleph_\omega$

Given a cardinal $\kappa,$ recall that $X \subset \kappa$ is called fresh (over $V$), if $X \notin V,$ but $X \cap \alpha \in V$ for all
$\alpha < \kappa.$
Question. Is it consistent that there ...

**6**

votes

**0**answers

250 views

### Finding non convex functions satisfying a weak form of convexity, without the axiom of choice

If a real-valued function $f$ over reals satisfies $$ (1) \; \; \; f({x+y\over2})\le {f(x)+f(y)\over2}, $$and it is continuous, then it is not hard to see that $f$ is indeed convex. On the other ...

**8**

votes

**1**answer

176 views

### Boolean-Valued Models: Why is $\| x=y \| \cdot \| \phi(x) \| \leq \| \phi(y) \|$?

Let $B$ be a complete Boolean algebra. Jech defines a Boolean-valued model $\mathfrak{A}$ of the language of set theory to consist of a Boolean universe $A$ and functions of two variables with values ...

**4**

votes

**1**answer

144 views

### An interpretation for filters of subspaces in Banach spaces

Let $X$ be a separable infinite-dimensional (real or complex) Banach space.
Call a collection $\mathcal{F}$ of closed subspaces of $X$ a filter if it is nonempty, does not contain $\{0\}$, is closed ...

**3**

votes

**0**answers

301 views

### How close to being well-orderable does this make my powerset?

Let's work in a set theory without assuming AC (for instance, but not necessarily, ZF). Fix a set $k$ satisfying $k\times k \simeq k$, and consider its powerset $X = 2^k$. I have a technical condition ...

**3**

votes

**1**answer

210 views

### When can you canonically extend an ultrafilter after forcing?

Suppose that $V$ is a model of $\sf ZFC$, and fix some regular $\kappa$, say $\omega_1$ for practical purposes.
Let $\cal U$ be an ultrafilter on $\omega_1$ in $V$ which is non-principal and even ...

**6**

votes

**1**answer

329 views

### all subsets borel

Assume Martin's axiom plus $\neg CH$. It is well known, via almost disjoint forcing, that every set of reals of size less than continuum is an example of a metric space whose subsets are all ...

**7**

votes

**0**answers

149 views

### Improving Baumgartner's result?

Q1: Is it consistent with the failure of CH to have an $\aleph_1$-dense subset $A \subseteq \mathbb{R}$ such that for every $X \subseteq \mathbb{R}$ of size $\aleph_1$, there is a $C^{\infty}$ map $F: ...

**6**

votes

**1**answer

250 views

### What is known about the large cardinal strength of Shelah's categoricity conjecture?

Shelah's categoricity conjecture states that for every Abstract Elementary Class $\mathcal{K}$ there is a cardinal $\mu$ depending only on $\operatorname{LS}(K)$ (i.e. the Löwenheim–Skolem number of ...

**1**

vote

**0**answers

73 views

### Does $\mathfrak{P}(A)\cong\mathfrak{P}(B)$ imply $A\cong B$? [duplicate]

Does $\mathfrak{P}(A)\cong\mathfrak{P}(B)\implies A\cong B$ hold for arbitrary sets $A, B$?
Notation:
We write $\mathfrak{P}(M)$ to denote the power set of a set $M$.
$M_1\cong M_2$ is just an ...

**4**

votes

**2**answers

294 views

### A property of uncountable almost disjoint families

Let $\mathcal{A}$ be an uncountable almost disjoint family (not necessarily maximal) of infinite subsets of $\mathbb{N}$. Denote by $\mathcal{A}_{\subseteq}=\{ B\subseteq\mathbb{N}:|B|=\omega \wedge ...

**4**

votes

**0**answers

174 views

### Theories introduced by a class of forcing notions

The following notion is introduced by Mohammad Golshani. Let $V$ be a model of set theory and let $\mathcal{P}$ be a class consisting of non-trivial forcing notions in $V$. Let
$$Th(V, ...

**6**

votes

**1**answer

223 views

### Mahlo cardinal and hyper k-inaccessible cardinal

It is known that every Mahlo cardinal $\kappa$ is hyper $\kappa$-inaccessible. It the converse true, namely: every cadinal $\kappa$ which is hyper $\kappa$-inaccessible is a Mahlo cardinal ?

**3**

votes

**1**answer

237 views

### How can the critical point of an elementary embedding be omega_1?

I've seen an example of an elementary embedding such that $\omega_1$ is the critical point.
I was wondering what's wrong with the following proof that this cannot be:
Let $\phi(x_1,x_2)$ be the ...

**1**

vote

**1**answer

202 views

### What is the cofinality of the positive measure sets of reals?

What is the minimal cardinality of a family of sets of real numbers, each of positive Lebesgue measure, such that every set of real numbers of positive Lebesgue measure contains some member of the ...

**6**

votes

**1**answer

235 views

### Club-guessing at $\omega_2$

Let $S=\{\delta<\omega_2:\text{cf}(\delta)=\omega\}$. A well-known theorem of Shelah tells us that we can find $\langle C_\delta:\delta\in S\rangle$ such that
for every club $C\subseteq\omega_2$ ...

**6**

votes

**0**answers

334 views

### Is there any elementary embedding characterization for $\Pi_{1}^{1}$ - reflecting cardinals?

Similar to one of the characterizations of weakly compact cardinals, a $\Pi_{1}^{1}$ - reflecting cardinal is defined as follows:
A cardinal $\kappa$ is $\Pi_{1}^{1}$ - reflecting if $\kappa$ is ...

**12**

votes

**0**answers

166 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 ...

**18**

votes

**2**answers

540 views

### Removing large cardinals from an uncountable transitive model

The usual way of removing large cardinals from a given model of set theory is to cut off the model below the least large cardinal of interest. But this method may have dramatic effects on the external ...

**6**

votes

**2**answers

250 views

### The role of the index set in the product of uncountably many topological spaces

Let $\langle X_i,\mathcal{T}_i \rangle_{i\in I}$ be a family of topological spaces. Consider $X=\prod_{i\in I} X_i$ with product topology.
Question. Is there a topological property that holds in ...

**6**

votes

**1**answer

586 views

### Replacing Axiom of Choice with Axiom of Countable Choice

Many people find ACC more intuitive than AC ("Pick something from the first set, then something from the second set, then...) and it also doesn't lead to "controversial consequences" (See for eg: ...

**8**

votes

**3**answers

396 views

### Is there a modification of Martin's Axiom which admits non-measurable sets of size less than continuum?

Background/Motivation. We know that some of the usefulness of Martin's Axiom lies in giving certain "smallness" properties to sets of size less than continuum, e.g. we have
that for all infinite ...