forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations,...

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6
votes
1answer
302 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
2answers
524 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
0answers
145 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
0answers
498 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
1answer
407 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
1answer
299 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
1answer
329 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
0answers
131 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
1answer
755 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
2answers
366 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
1answer
371 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
1answer
263 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
1answer
129 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
0answers
78 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
1answer
270 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
1answer
386 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
0answers
168 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
2answers
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
1answer
359 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
0answers
255 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
0answers
223 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 ...
8
votes
2answers
303 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
0answers
386 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
0answers
190 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
1answer
309 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 hand,...
8
votes
1answer
180 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
1answer
145 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
0answers
309 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
1answer
225 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
1answer
330 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
0answers
152 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
1answer
252 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 $\...
0
votes
0answers
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
2answers
297 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
0answers
176 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, \mathcal{P})...
6
votes
1answer
238 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
1answer
247 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
1answer
204 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
1answer
238 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
0answers
336 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
0answers
169 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
2answers
550 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
2answers
255 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 $...
5
votes
1answer
600 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
3answers
400 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 ...
8
votes
1answer
264 views

Can one force there to be an elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$ for some inaccessible $\lambda$?

Is it consistent that there exists an inaccessible cardinal $\lambda$ and a forcing extension $V[G]$ so that $$V[G]\models\text{There is some non-trivial elementary embedding $j:V_{\lambda}^{V[G]}\...
10
votes
1answer
206 views

Is there a (first-order) sentence which admits $(\aleph_2,\aleph_0)$ iff a Kurepa tree exists?

In Chang and Keisler's Model Theory I came across the following theorem (Theorem 7.2.13): Theorem There exists a (first-order) sentence $\sigma$ such that for all infinite cardinals $\alpha$, $\sigma$...
8
votes
1answer
270 views

Transitive models and CH

The following was asked on stackexchange but I think it also belongs here: http://math.stackexchange.com/questions/1513446/transitive-models-and-ch Suppose $M, N$ are two countable transitive models ...
7
votes
1answer
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
0answers
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 $\...