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0
votes
0answers
21 views

Lists of sets as objects of ZF axiomatics [migrated]

I have a naive question about foundations of mathematics. A common opinion of most mathematicians is that the essential part of mathematics can be reduced to ZF(C) axioms. I do not quite understand ...
-5
votes
0answers
65 views

what are the practical applications of sets in our daily life? [on hold]

I don`t know the answer to this question?I know I sound stupid writing something in my own question but the computer was forcing me to write something.
8
votes
1answer
116 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 ...
6
votes
3answers
201 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
161 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 $$E:=\big\{\{f,g\}: f, g \in V \land f\neq g\land \exists k\in \mathbb{N} \text{ }\forall n\in\mathbb{N}\setminus\{k\} (f(n) = ...
10
votes
1answer
234 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
307 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
492 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 ...
4
votes
1answer
139 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 $n\geq N$. ...
3
votes
2answers
118 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
127 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
154 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 ...
28
votes
0answers
658 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
108 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
258 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
80 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 ...
2
votes
1answer
314 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
262 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
166 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
101 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
191 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
184 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
425 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), ...
0
votes
0answers
39 views

Uncountability of the set of subsets of $\mathbb N$ [migrated]

Denote by $\mathbb{N}$ the set of natural numbers and by $2^{\mathbb N}$ the set of all subsets of $\mathbb N$. Let $E$ be some subset of $2^{\mathbb N}$ such that for every pair of elements in $E$, ...
9
votes
1answer
260 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
184 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
165 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
162 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
122 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
222 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
142 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 ...
-2
votes
0answers
10 views

Partition on a Closed Set A= [2,3] [migrated]

Is it possible to define a partition on a closed set,such that the union of the partitions will give [2,3] and their intersection to be empty?
9
votes
1answer
164 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
174 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 ...
0
votes
0answers
33 views

How can nontrivial elementary embeddings of the universe to some inner model be surjective? [migrated]

Consider $\kappa$ to the least measurable cardinal, or equivalently $\kappa$ is the critical point for an elementary embedding $j:V \rightarrow M$ from the universe $V$ to an inner model $M$ (critical ...
3
votes
0answers
160 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
244 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
875 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
350 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
173 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 ...
3
votes
0answers
120 views

A Result of Anders Bjorner: Matchings in countably infinite geometric lattices of finite height

Let $L$ be a countably infinite geometric lattice of finite height $r\ge3$. (A geometric lattice of height $r$ is an atomistic semimodular lattice such that every maximal chain has $r+1$ elements.) ...
1
vote
0answers
74 views

Outer measure preserving bijection

Suppose X is a Sierpinski set (So X is uncountable and every null subset of X is countable). Let f be a bijection on X. Must/Does there exist a non null subset Y of X such that for every subset W of ...
1
vote
0answers
170 views

The patterns of possibility for nontrivial automorphisms and nontrivial elementary embeddings of the universe

In their paper "The Role of the Foundation Axiom in the Kunen Inconsistency" (arXiv:1311.0814 [Math.LO]), Daghighi, Golshani, Hamkins, and Jerabek show that the patterns of possibility for the ...
10
votes
3answers
482 views

Axiom of choice for sets of finite sets

The question I am going to ask is really to satisfy my curiosity, as I am not at all an expert of the subject and do not plan to really work on it. Hence, if you think the question is not suitable for ...
7
votes
1answer
171 views

Characterizing L(R) Cardinals in HOD

We're working in L(R) under AD. We know that $\omega_1$ is the least measurable in HOD, $\Theta$ is the least woodin, $\delta^2_1$ is the least strong to the woodin, etc. My question is about ...
7
votes
1answer
304 views

Constructing unnatural transformations

In a nutshell, the question is: is it true that any explicit (not involving axiom of choice) pointwise transformation between sufficiently complicated functors is natural almost everywhere? Let $C$ ...
23
votes
2answers
426 views

Mid point free sets

Given a subset X of unit interval, can we find a subset Y of X of same outer measure as X such that Y does not contain three points of the form x, y and (x+y)/2? I can do this assuming CH but can we ...
5
votes
1answer
250 views

Logical strength of “choice functions exist for well-ordered families”?

A colleague of mine suggested the following weakening of the axiom of choice: If $\mathscr{F} := \{F_\alpha\}$ is a well-ordered family of non-empty sets (i.e., there is a bijection between ...
8
votes
1answer
196 views

Dedekind-finite arithmetic vs natural numbers arithmetic

It is known that the Dedekind-finite cardinals are closed under addition and multiplication, so one may do arithmetic in them, as opposed to only natural numbers. How much can those two arithmetics ...
5
votes
2answers
263 views

The role of the rigid relation principle ($RR$) in the Kunen inconsistency

Consider the rigid relation ($RR$) principle, i.e. "every set admits a rigid binary relation", that is,"that for every set $A$ there is a binary relation $R$ on $A$ such that the structure $(A,R)$ is ...