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7
votes
1answer
247 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
508 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
387 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
167 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
196 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
193 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
290 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
200 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
301 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
281 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
191 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
282 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
897 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
139 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
181 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
195 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
270 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
253 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
317 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
317 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
534 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
152 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
130 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
146 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
170 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
748 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 ...
4
votes
0answers
120 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 ...
7
votes
1answer
300 views

Under $\neg CH$, have countable unions of rationally independent numbers inner measure zero?

In their 1943 paper On non-denumerable graphs, Erdos and Kakutani suggest as likely the following proposition. (EK*) Suppose CH fails and $\lbrace M_n : n \in \omega \rbrace$ is a countable family of ...
3
votes
0answers
87 views

Partition refinement of a clopen covering in $\Box (\omega+1)^\omega$

Consider $\omega+1$ with the interval topology, that is $U\subseteq (\omega+1)$ is open if and only if $U\subseteq\omega$ or $(\omega+1)\setminus U$ is finite. We write $(\omega+1)^\omega$ for the ...
3
votes
1answer
373 views

Formulating Kunen's inconsistency and Reinhardt cardinals in term of category theory

It is known that one can formulate certain large cardinal axioms (e.g. Vopenka's principle--see Mike Shulman's answer to Harry Gindi's mathoverflow question "Reasons to believe Vopenka's ...
1
vote
1answer
295 views

Is second-order ZFC categorical with regard to its proper class models

Second-order ZFC offers partial categoricity in the sense that, given any two models, one of them must be isomorphic to an initial segment of the other [1]. However, this leaves questions regarding ...
2
votes
1answer
185 views

Is there a 'largest' second-order categorical axiomatization of set theory, extended from ZFC2

While it's possible to obtain categorical second-order axiomatizations of set theory by extending ZFC2 with additional axioms (see [1]), these axioms tend to be somewhat arbitrary (e.g. adding an ...
4
votes
0answers
116 views

Is the lowenheim-skolem number of nth order logic larger than the corresponding number for 2nd order logic

According to this paper, by Vaananen, the $LS$ number for $2^{nd}$ order logic is given by "the supremum of $Π_{2}$-definable ordinals", where "The Lowenheim-Skolem number $LS(L)$ of $L$ is the ...
4
votes
1answer
222 views

What consistency results follow the assumption: $\forall\alpha(\aleph_{\alpha+1}\nleq2^{\aleph_\alpha})$?

In a recent question on Math.SE it was asked whether or not For every infinite cardinal $\mathfrak m$ there is no $\aleph$ number, $\kappa$, such that $\mathfrak m<\kappa<2^{\mathfrak m}$. By ...
5
votes
1answer
204 views

$\text{cov}(\mathcal{M})$ vs. $\mathfrak{b}$ vs. $\mathfrak{s}$

Let me first recall some pretty standard notations: $\text{cov}(\mathcal{M})$ is the covering number of the ideal $\mathcal{M}$ of all meager subsets of $\mathbb{R}$; $\mathfrak{b}$ is the bounding ...
15
votes
1answer
440 views

Whatever happened to $L(j)$?

So this question probably shows my inner model theoretic ignorance, but: In "Two remarks on elementary embeddings of the universe" (http://projecteuclid.org/download/pdf_1/euclid.pjm/1102969567), ...
9
votes
1answer
284 views

Assuming AD, is every infinite cardinal closed under power set in a choice model?

Assume AD+DC. Assume $\kappa$ is an infinite cardinal and $N$ is a (set or class) transitive model of ZFC containing $\kappa$. Is it true that for all $\alpha<\kappa$, $N$ thinks that the power ...
7
votes
0answers
224 views

A question about finitely additive extensions of Lebesgue measure

Suppose $m:P([0, 1]) \to [0, 1]$ is a finitely additive measure extending the Lebesgue measure. Must there exist some $X \subseteq [0, 1]$ such that $m(X \cap I) = |I|/2$ for every sub interval $I ...
5
votes
1answer
176 views

Universally Baire Tree Representation of Projective Sets

In Feng, Magidor, and Woodin "Universally Baire Sets of Reals", they show that if $A$ is a $\mathbf{\Pi}_2^1$ set and $U$ and $V$ are any pair of trees witnessing the universal baireness of $A$, then ...
5
votes
2answers
178 views

Extending hyperconnected spaces

A hyperconnected space is a topological space such that every two non-empty open sets have non-empty intersection. Let's call a space $(X,\cal{T})$ maximally hyperconnected if it is hyperconnected and ...
1
vote
0answers
124 views

What can be said about a Boolean-valued structure from what the Boolean-valued forcing extension thinks about it?

Suppose that $\phi$ is a formula in the language of set theory such that there are some $n_{1},...,n_{k}$ such that if $V\models\phi(x)$, then $x=(X,R_{1},...,R_{k})$ and ...
3
votes
1answer
258 views

A question about Cantor's Power Set theorem without the Axiom of Choice

Assume that we are working in ZF set theory without the Axiom of Choice. If S is an infinite set, let $S(f)$ denote the set of all finite subsets of $S$, let $S(I)$ denote the set of all infinite ...
5
votes
0answers
149 views

Reference to forcing with a sigma ideal $\cong$ Cohen forcing

This is a historical question: Who was the first person to notice the following? If $V \models \kappa$ is measurable and $P$ adds $\kappa$ Cohen reals, then in $V^P$, letting $\hat{I}$ to be the ...
9
votes
1answer
178 views

Are the failure of SCH and “$cf([\mu]^{cf (\mu)},\subset)>\mu^+$ for some singular” equiconsistent?

Is it true that the following two statements are equiconsistent? (1) $2^\mu>\mu^+$ for some strong limit singular cardinal $\mu$ (2) $cf([\mu]^{cf (\mu)},\subset)>\mu^+$ for some singular ...
4
votes
1answer
186 views

Meager set of full measure

Let X be a compact Hausdorff topological group and let m be the Haar measure on X. Can we find a meager set in X whose complement is m-null? I can do it when X is separable but I don't know if there ...
3
votes
0answers
167 views

name for an intermediate notion between huge and 2-huge

I am employing a large cardinal notion that has been used explicitly before, and I am wondering if someone has given it a good succinct name. A cardinal $\kappa$ is huge if there is an elementary $j ...
6
votes
0answers
255 views

Reference for “if the set $A$ is Suslin, then every $\Sigma^1_1(A)$ set is Suslin”

Does anyone know of a reference for one or both of the following facts (in $\mathsf{ZF}$)? If the set of reals $A$ is Suslin, then every $\Sigma^1_1(A)$ set of reals is Suslin. If $T$ is a tree on ...
13
votes
2answers
897 views

When does Vopěnka's principle hold?

Vopěnka's principle (VP) is the statement that, given any proper class $\{\mathcal{A}_\eta: \eta\in ON\}$ of first-order structures in the same language, there are some $\alpha\not=\beta$ with ...
7
votes
1answer
364 views

Explicit counter example to Vopěnka's principle in the constructible universe?

Vopěnka's principle is a large cardinal axiom which has many equivalent formulations. One of them, which I find especially appealing, is the following: if the universe is satisfies Vopěnka's principle ...
2
votes
1answer
181 views

Uncountable cardinals and Prufer $p$-groups

Let $A$ be an elementary Abelian uncountable $p$-group. Is it known if there is an action of a Prufer $q$-group (here $q$ is a prime not necessarily distinct from $p$) $C_{q^{\infty}}$ onto $A$ such ...