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
331 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
1answer
155 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
255 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
298 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
178 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
242 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
251 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
205 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
239 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
170 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
556 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
607 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
403 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
267 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
272 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
137 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 $\...
11
votes
1answer
416 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
1answer
216 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
0answers
153 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
0answers
229 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
0answers
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
1answer
198 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
0answers
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
0answers
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
1answer
250 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
0answers
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
0answers
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
0answers
129 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
0answers
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
0answers
436 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
2answers
266 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
2answers
860 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
0answers
178 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
0answers
121 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
1answer
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
0answers
588 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
2answers
404 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
1answer
476 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
0answers
396 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
1answer
345 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
0answers
320 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
0answers
160 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
1answer
253 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
0answers
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
3answers
419 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 ...