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0
votes
0answers
149 views

If C is a closed, unbounded subset of Ord, are the sets generated by the new class of ordinals Ord restricted to C, a Universe?

Given any universe, and some closed unbounded subset of Ord, is it possible toy generate a universe in the following way: Restrict Ord to a target club. Then generate all look the sets necessary to ...
8
votes
0answers
262 views

preserving saturated ideals

A reliable source made the following claim: Suppose CH there is an $\omega_2$-saturated ideal on $\omega_1$. Then this is preserved by $\mathrm{Add}(\omega_1,\omega_2)$. Question 1: How do you ...
4
votes
1answer
158 views

Implications between different covering properties of spaces

Let $X$ be a set. A set ${\cal C}\subseteq {\cal P}(X)$ is said to be a cover of $X$ if $\bigcup {\cal C} = X$ and $X\notin {\cal C}$. If ${\frak U}$ and $\frak{W}$ are collections of covers of a ...
5
votes
0answers
147 views

Sierpinski sets and extensions of Lebesgue measure

I am duplicating an old problem from stackexchange: Suppose that every countably generated sigma algebra extending the Borel sigma algebra on $[0, 1]$ admits a measure extending the Lebesgue measure ...
1
vote
1answer
142 views

$\mathbb{P}_{\kappa}$ forces non$(\mathcal{M})$=cov$(\mathcal{M})=\kappa$

Let $\kappa$ be an uncountable regular cardibnal. Consider the finite support iteration $(\langle \mathbb{P}_{\alpha}\rangle _{\alpha \leq \kappa},\langle \mathbb{\dot{Q}}_{\alpha}\rangle _{\alpha ...
9
votes
2answers
502 views

Preserving $\omega_1$ is Inaccessible to the reals

$\omega_1$ is inaccessible to the reals if and only if for all $x \in {}^\omega\omega$, $\omega_1^{L[x]} < \omega_1$. The question is if $\omega_1$ is inaccessible to the reals in $V$ and ...
13
votes
1answer
442 views

Avoiding countable subgroups of a group homeomorphic to the Cantor space

The following question is motivated by the paper [Brian, Mislove, Every compact group can have a non-measurable subgroup]. A positive solution to a variation of the following problem implies a ...
10
votes
0answers
182 views

Generic $\mathbf{\Sigma}_3^1$-absoluteness for class forcings

In the paper "Generic Absoluteness" by Bagaria and Friedman (http://www.logic.univie.ac.at/~sdf/papers/bagfried.pdf) it is shown that in ZFC generic $\mathbf{\Sigma_3^1}$-absoluteness is false for ...
7
votes
2answers
300 views

Topological tameness beyond the Gandy-Harrington topology

The Gandy-Harrington topology on $\omega^\omega$ is the topology generated by all lightface $\Sigma^1_1$ sets; that is, all sets which are continuous-in-the-usual-sense images of $\omega^\omega$. ...
11
votes
1answer
310 views

Non meager rectangle

Suppose $G \subseteq \mathbb{R}^2$ is dense $G_\delta$. Must there (in ZFC) exist non meager sets of reals $A, B$ such that $A \times B \subseteq G$?
4
votes
1answer
210 views

Measure algebra of a total extension of Lebesgue measure

Solovay shows that the existence of a measurable cardinal is equiconsistent with the existence of a countably additive extension of Lebesgue measure that is defined on all sets of real numbers. Given ...
9
votes
1answer
281 views

Saturation of null ideal

In ZFC, can we find more than continuum many non null sets of reals whose pairwise intersections are null?
4
votes
1answer
219 views

Ref. request: Additive probability measure on $\mathcal P({\bf N})$ supplies subset of $\mathbf R$ without Baire property

ZFC proves, among the other things, the existence of a (finitely) additive probability measure $\theta: \mathcal P(\mathbf N) \to \mathbf R$ on the power set of $\mathbf N$ such that $\theta(X) = 0$ ...
9
votes
1answer
384 views

Just a little absoluteness might be cheaper?

Absoluteness is a wonderful thing, but expensive consistency-strength wise. My question is, when can we get large amounts of absoluteness in specific situations for much cheaper? Specifically, fix a ...
8
votes
1answer
192 views

Is the Jordan decomposition of a self-adjoint functional constructive?

Let $A$ be an abstract C*-algebra, and let $\varphi\colon A \rightarrow \mathbb C$ be a bounded linear function. Assuming the axiom of choice, there exist unique positive bounded linear functions ...
2
votes
0answers
55 views

Strongly minimal covering subsets of $\text{Ind}(G)$

Let $G=(V,E)$ be any undirected, simple graph. Let $\text{Ind}(G)$ be the set of independent subsets of $V(G)$. We say that $K\subseteq \text{Ind}(G)$ is a cover (by independent subsets) if $\bigcup K ...
9
votes
0answers
184 views

The Chang model after collapsing an inaccessible limit of Woodins

If $\kappa$ is an inaccessible cardinal and $G \subset \operatorname{Col}(\omega,\mathord{<}\kappa)$ is a $V$-generic filter, then in $V[G]$ the Chang model $L(\text{Ord}^\omega)$ satisfies "every ...
0
votes
0answers
114 views

Finitely additive measure over integers [duplicate]

We know that, with Axiom of Choice (AC), it can be shown that there exists a finitely additive uniform distribution defined for all subsets of the integers (see, e.g., Hrbacek and Jech 1999, Ch. 11). ...
19
votes
3answers
2k views

Who needs Replacement anyway?

The set theory ETCS famously comes without the Replacement axiom schema (or an equivalent) that is part of ZFC. One (to me, not apparently useful) set that one cannot build in ETCS is $\coprod_{n\in ...
11
votes
0answers
255 views

Adding a saturated ideal

Is it consistent that there is no $\omega_2$-saturated ideal on $\omega_1$, but one is introduced by an $\omega_2$-closed forcing? Some motivation: If $\delta$ is a Woodin cardinal, then it remains ...
10
votes
0answers
234 views

c.c.c forcing notions and adding minimal generic reals

Is the following statement consistent: ``There is no non-trivial c.c.c forcing notion adding a minimal generic real''? The question is related to Prikry's question: Is it consistent that any ...
5
votes
0answers
116 views

Homogeneous $\omega$-monolithic compact space

Under CH, is the cardinality of every homogeneous $\omega$-monolithic compact space $X$ not greater than $2^{\omega}$?
0
votes
1answer
211 views

Provability of unprovability

I have three questions (without any real background, this is just something I've been wondering about recently) Can PA prove that PA can't prove nor disprove, say, Goodstein theorem (or any natural ...
7
votes
1answer
248 views

For a ring $k$ and a set $X$, what are the $k$-algebra homomorphisms $k^X \to k$?

Let $k$ be a commutative ring. Feel free to assume it's a field. Let $X$ be a set. This question is only interesting when $X$ is infinite. Write $k^X$ for the $k$-algebra of functions $X \to k$, ...
11
votes
3answers
516 views

The size of Lindelof space

Question. Suppose that $X$ is a Lindelof space such that every point of $X$ is a $G_{\delta}$-point. Then is it true that $|X| ≤ 2^{\omega}$?
9
votes
1answer
391 views

Is $\clubsuit_{\omega_1}$ enough to get Suslin tree?

This is problem 15.3 in Arnie Miller's problem list: (Juhasz) Suppose there exists $\langle A_{\alpha} : \alpha \in L \rangle$, where $L$ is the set of limit ordinals below $\omega_1$ and for each ...
4
votes
1answer
173 views

Cohen algebra and $\mathcal P(\omega)/ \mathrm{fin}$

Let $C$ be the Cohen algebra, the boolean completion of the partial order of finite partial functions from $\omega$ to 2, ordered by reverse inclusion. Does there exist an ideal $I$ on $C$ such that ...
4
votes
1answer
200 views

Does the minor graph of graphs on $\mathbb{N}$ have an uncountable independent set?

By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\}: a\neq b \in V\}$. We write $V(G) := V$. If $S, T$ are disjoint subsets of $V(G)$ we say that ...
4
votes
2answers
199 views

Minimum cardinality of lower-bounding subset of $[\omega]^\omega$

Let $[\omega]^\omega$ denote the collection of all infinite subsets of $\omega$. Let us call $S\subseteq [\omega]^\omega$ lower-bounding if for all $a\in [\omega]^\omega$ there is $s\in S$ such that ...
5
votes
1answer
295 views

Uncountably many countable graphs with no homomorphism between them

By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\}: a\neq b \in V\}$. A graph homomorphism between graphs $G, H$ is a map $f:V(G)\to V(H)$ such ...
2
votes
0answers
202 views

Is there a $\Sigma^0_3$-complete ideal on $\omega$?

In Kechris's book "Classical Descriputive set theory" Chapter 23 (Exercise 23.4), it claim that there is a $\Sigma^0_\xi$-complete ideal on $\omega$ for each $\xi\geq 3$. There is a candidate ...
5
votes
1answer
309 views

Details for Woodin's forcing argument for a saturated ideal from the Levy collapse

Theorem 2.65 in Woodin's book shows that a saturated ideal on $\omega_1$ exists after Levy-collapsing a Woodin cardinal $\delta$ to $\omega_2$. I am confused about the part of the argument where he ...
9
votes
1answer
288 views

Can there be a tree of height $\omega_2$ having all levels countable, with no cofinal branch?

For many years I had the idea that if a well-founded tree is both very tall and very narrow, then it must have a cofinal branch. For example, it is a fun exercise to show that any $\omega_1$-tree ...
7
votes
1answer
193 views

Introducing meets while preserving directed closure

A poset $\mathbb{P}$ is called well-met iff every pair of compatible conditions in $\mathbb{P}$ has a greatest lower bound. Question: Suppose $\mathbb{P}$ is a separative partial order which is ...
11
votes
1answer
294 views

minimal collapsing without GCH

Suppose $\kappa$ is a regular cardinal. Does there necessarily exist a poset $\mathbb P$ that collapses $\kappa^+$ while preserving all other cardinals?
22
votes
1answer
939 views

How hard is it to destroy a diamond? (with a real)

If we start with $V\models\lozenge$, it is not hard to force the failure of diamond. You can blow up the continuum, or destroy all the Suslin trees. You can blow up the continuum of $\aleph_1$, and ...
5
votes
0answers
141 views

Analytic equivalence relations whose classes are sometimes Borel

There are analytic equivalence relations for which the statement "All classes are Borel" is independent of $ZFC$. In all the examples I know about, the classes are non Borel in $L$ or $L[z]$ for some ...
2
votes
1answer
183 views

Can the epsilon induction condition be presented with successor case and limit case, similar to transfinite induction?

The epsilon induction looks like this: $\forall x \Big(\forall y (y \in x \rightarrow P(y)) \rightarrow P(x)\Big) \rightarrow \forall x P(x)$ Here, the quantifiers "run over" any sets and not only ...
9
votes
1answer
198 views

Does $Add(\kappa,1)^L$ ever collapse cardinals?

In general, we know that adding a subset to a regular cardinal $\kappa$ can collapse cardinals. If, for example, there is $\gamma < \kappa$ with $2^\gamma >\kappa$, then $Add(\kappa,1)$ will ...
7
votes
3answers
273 views

Borel coloring of a graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$

The following question was asked in a comment by Joel David Hamkins in Graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$. Let $V$ be the set of all functions $f:\mathbb{N}\to\mathbb{N}$. ...
2
votes
1answer
255 views

Graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$

Let $V$ be the set of all functions $f:\mathbb{N}\to\mathbb{N}$. Let two distinct functions $f,g:\mathbb{N}\to\mathbb{N}$ form an edge if and only if they differ in exactly one input $n\in\mathbb{N}$. ...
11
votes
2answers
321 views

Graph $G$ with $\omega(G) = 2$ but $\chi(G) \geq \aleph_0$

Given an infinite cardinal $\kappa$, is there a graph $G$ that has no clique consisting of more than 2 points, but $\chi(G) = \kappa$?
3
votes
1answer
321 views

Is this version of van der Waerden's Theorem consistent with ZFC?

One way to phrase van der Waerden's Theorem is: For every finite coloring of $\mathbb N$ and every finite $F \subseteq \mathbb N$, there exist $a,b \in \mathbb N$ such that $a + b \cdot F$ is ...
10
votes
2answers
547 views

Does van der Waerden's Theorem hold for $\omega_1$?

One way to phrase van der Waerden's Theorem is: For every finite coloring of $\mathbb N$ and every finite $F \subseteq \mathbb N$, there exist $a,b \in \mathbb N$ such that $a + b \cdot F$ is ...
5
votes
1answer
158 views

Order dimension of $\omega^\omega/(fin)$

Let $\omega^\omega$ denote the collection of all functions $f:\omega\to\omega$. For $f,g\in\omega$ we say $f\simeq g$ if and only if $\exists N \in \omega$ such that $f(n) = g(n)$ for all $n\geq N$. ...
4
votes
2answers
135 views

${\frak b}$ and ${\frak d}$ defined with $\leq$ instead of $\leq^*$

Let $\omega^\omega$ denote the collection of all functions $f:\omega\to\omega$. For $f,g\in\omega$ we define $f\leq g$ if $f(n)\leq g(n)$ for all $n\in\omega$; $f\leq^* g$ if there is $N\in\omega$ ...
6
votes
1answer
150 views

completions of regular suborders

Suppose $\mathbb{P}$ is a regular suborder of the separative partial order $\mathbb{Q}$ (see below for definitions). Must there always exist some complete boolean algebra $\mathbb{B}$ such that: ...
6
votes
0answers
172 views

Where does this strengthening of I1 stand?

Let's call a cardinal $\delta$ an $\text{I1}$-tower cardinal if for each $A\subseteq V_{\delta}$, there exists a $\kappa<\delta$ such that whenever $\kappa<\alpha<\delta$ there is some ...
29
votes
0answers
760 views

Bidi: A new cardinal characteristic of the continuum?

This question assumes familiarity with combinatorial cardinal characteristics of the continuum. Identify an infinite set $a\subseteq\mathbb{N}$ with its increasing enumeration. Thus, for each natural ...
5
votes
0answers
123 views

The dominating number $\mathfrak{d}$ and convergent sequences

All spaces considered below are compact Hausdorff. If $K$ is a space, then $w(K)$ is its weight. For a Boolean algebra $\mathcal{A}$, $K_\mathcal{A}$ denotes its Stone space. I am interested in ...