Questions tagged [set-theory]
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, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
5,829 questions
6 votes
1 answer
149 views
Are there higher amorphous sets in the determinacy world?
Assume $\mathsf{AD}^++V=L(\mathcal{P}(\mathbb{R}))$ as usual for this kind of problem. My question is motivated by the observation that there is no $\omega_1$-amorphous set, i.e., an uncountable set ...
- 1,399
5 votes
3 answers
212 views
Cardinal arithmetic inequalities according to ZF
Suppose $\kappa$, $\lambda$, $\mu$, and $\nu$ are cardinals which may or may not be ordinals. Can we prove without resorting to the axiom of choice either of the following: $\kappa + \lambda \...
- 1,010
2 votes
1 answer
77 views
Cardinality of fibers in a $T_2$-hypergraph with large edges
We say that a hypergraph $H=(V,E)$ is $T_2$ if for all $v, w\in V$ with $v\neq w$, there are disjoint sets $e_1, e_2\in E$ with $v\in e_1, w\in e_2$. For $v\in V$, let $e_v= \{e\in E: v\in e\}$. Given ...
- 53.3k
3 votes
1 answer
350 views
About Cohen forcing
Put $P=\{p\subseteq \omega\times 2: p \mbox{ is a finite function from dom(p) to }2\}$, define $\leq$ on $P$ as: $$f\leq g\iff g\subseteq f$$ for $f,g\in P$. We call $P$ the Cohen forcing. Let $V$ be ...
- 101
1 vote
1 answer
293 views
Question on Cohen forcing
Let $P$ be such that $$P=\{f⊆ω\times 2:f \mbox{ is a finite function}\},$$ define $f≤g\iff g⊆ f$. Then the forcing $P$ adds a cohen real to ground model. Question: Suppose $Q⊆ P$ is a suborder of $P$....
user573840
12 votes
1 answer
689 views
Is every set of cardinals bounded?
Let ZFU$_\text{R}$ be ZF (formulated with Replacement) modified to allow a proper class of urelements. A cardinal $\mathfrak{b}$ is an upper bound of a set $X$ of cardinals if $\mathfrak{a} \leq \...
- 336
4 votes
1 answer
141 views
When is a $< \kappa$-support iteration of $< \kappa$-closed, $\kappa^+$-cc forcings also so?
Consider a $< \kappa$-support iteration (meaning, inverse limits are taken at stages of cofinality of $< \kappa$ and direct limits elsewhere) of length $\alpha$, $\langle \mathbb{P}_\beta, \dot{\...
- 2,033
1 vote
0 answers
221 views
Equality of multisets
I have tested this statement with several examples and it seems to hold true in all cases. Is there an elegant way to prove it, assuming it is indeed correct? A proof that avoids case-by-case analysis ...
6 votes
0 answers
173 views
The cofinality of $\Theta^{L(\mathbb R)}$ under strong forcing axioms
I am looking for literature concerning the cofinality of $\Theta^{L(\mathbb R)}$ under one of the strong forcing axioms $\textsf{PFA},\textsf{MM}$, and $\textsf{MM}^{++}$. Basically, I am looking for ...
- 2,665
7 votes
1 answer
287 views
Partitions of R into meager/measure zero sets
For a $\sigma$-ideal $\mathcal{J}$ on $\mathbb{R}$, consider the following statement. There is a partition $\bigsqcup_{i < \kappa} A_i = \mathbb{R}$ such that $\mathcal{I} = \{X \subseteq \kappa: \...
- 397
4 votes
1 answer
145 views
When is $\Theta$ a (weakly) Lowenheim-Skolem cardinal?
The base theory is $\textsf{ZF}$. The following definitions are due to T. Usuba. Definition 1: An uncountable cardinal $\kappa$ is a weakly Lowenheim-Skolem cardinal if for every pair of ordinals $\...
- 2,665
2 votes
1 answer
150 views
Supernormal sequences
We call a binary sequence $s:\mathbb{N}\to\{0,1\}$ supernormal if for every injective, increasing and computable function $\iota:\mathbb{N}\to\mathbb{N}$, the binary sequence $s\circ\iota:\mathbb{N}\...
- 53.3k
7 votes
0 answers
171 views
Prikry forcing and collapsing
Suppose $\kappa$ is a measurable cardinal, $U$ is a normal measure on $\kappa$, $2^\kappa = \kappa^+$, and after forcing with $\mathrm{Col}(\kappa,\kappa^+)$, $U$ can be extended to a normal measure $...
- 20.7k
7 votes
0 answers
171 views
Almost disjoint sets at higher cardinals
Let $\alpha$ be the least limit ordinal such that there exists a sequence $(X_i:i<(2^{\aleph_0})^+)$ of cofinal subsets of $\alpha$ such that for all $i<j<(2^{\aleph_0})^+,$ the intersection $...
- 33.4k
14 votes
3 answers
746 views
Does ZF alone prove that every complete, atomless Boolean algebra has an infinite antichain?
Here by "antichain" I mean a set of elements that have pairwise-trivial meets, not merely ones that are pairwise-incomparable. Clearly, every atomless Boolean algebra has antichains in this ...
- 143
8 votes
0 answers
276 views
Possible cardinalities of a connected Hausdorff topology
Is there an infinite cardinal $\kappa$ such that whenever $\lambda$ is a cardinal with $\kappa \leq \lambda \leq 2^\kappa$, then there is a topology $\tau$ on $\kappa$ with The space $(\kappa,\tau)$ ...
- 53.3k
21 votes
2 answers
635 views
Theories yielding $\mathit{Con}(\mathsf{ZF+\neg AC})$ without forcing
My question is: What are some examples of consistent (relative to large cardinals) extensions of $\mathsf{ZFC}$ within which there is a forcing-free proof of the consistency of $\mathsf{ZF+\neg AC}$? ...
- 19.5k
3 votes
0 answers
44 views
About the consistency strength of a ubiquitous Perfect Set Property with a singular $\omega_1$ [duplicate]
Kanamori writes in the Higher Infinite on page 135 that "Specker had already made the conceptual move to inner models; through a sequence of implications he had in effect established in ZF that ...
- 1,010
8 votes
1 answer
523 views
Which "specific cases" of order types outside of $M$ could Laver mean? What are examples of undecidable statements in order theory?
Richard Laver finishes his seminal paper "On Fraïssé's order type conjecture", with: Finally, the question arises as to how the order types outside of $M$ behave under embeddability. For ...
- 2,086
3 votes
2 answers
132 views
Size of cutsets in ${\cal P}(\omega)$ having infinite and co-infinite members only
A chain ${\cal C}\subseteq {\cal P}(\omega)$ is a set such that for all $A, B\in {\cal C}$ we have $A\subseteq B$ or $B\subseteq A$. Using Zorn's Lemma one can show that every chain is contained in a ...
- 53.3k
8 votes
1 answer
456 views
Boolean ultrapower - set-theoretic vs algebraic/model-theoretic
I've been looking through the Hamkins/Seabold paper "Well-founded Boolean ultrapowers as large cardinal embeddings". The Boolean ultrapowers are defined there in two different ways: in ...
- 481
13 votes
1 answer
284 views
How far does Cantor-Bendixson rank counting let us build computable isomorphisms between ordinals?
This is tangentially related to this old question of mine. Say that a clean well-ordering is a computable well-ordering $\triangleleft$ of $\mathbb{N}$ such that the following additional data is ...
- 19.5k
3 votes
1 answer
118 views
Cardinality of the collection of maximal antichains in ${\cal P}(\omega)$
An antichain in $\mathcal P(\omega)$ is a set $\mathcal A\subseteq \mathcal P(\omega)$ such that for all $A, B\in \mathcal A$ with $A\neq B$ we have $(A\setminus B)\neq \emptyset$ and $(B \setminus A)\...
- 53.3k
6 votes
1 answer
309 views
Set size comparison via non-existence of surjections
If $X, Y$ are sets, let us say that $X$ is strictly smaller than $Y$, in symbols $X \prec Y$, if $Y$ is non-empty and for every map $f:X\to Y$ we have $Y\setminus\text{im}(f) \neq \varnothing$. Our ...
- 53.3k
24 votes
1 answer
1k views
Is Zorn's Lemma equivalent to the Axiom of Choice for individual sets?
It is well-known that in $\mathsf{ZF}$, the Axiom of Choice and Well-ordering Theorem are equivalent. What is perhaps less well-known is that there is a "local" version of this equivalence. ...
- 1,538
15 votes
1 answer
1k views
What was the definition of strongly inaccessible in 1958?
I'm reading Erdős and Hajnal's paper "On the structure of set-mappings" from 1958 and also a companion paper "Some remarks concerning our paper…". In it they define a partition ...
- 321
14 votes
0 answers
361 views
For what cardinality is the cofinite topology on a set symmetrizable?
A symmetric on a set $X$ is any function $d:X\times X\to[0,\infty)$ such that for every $x,y\in X$ the following two conditions are satisfied: $d(x,y)=0$ if and only if $x=y$; $d(x,y)=d(y,x)$. A ...
- 509
12 votes
1 answer
371 views
Comparability of power sets and (AC)
For sets $X, Y$ we write $X \leq Y$ if there is an injective map $f:X\to Y$. Let (S) be the statement: For any sets $X, Y$, either ${\cal P}(X) \leq {\cal P}(Y)$, or ${\cal P}(Y) \leq {\cal P}(X)$, ...
- 53.3k
-2 votes
0 answers
110 views
Axiom of Foundation and Proper Class epsilon chains
In Kunen, it is emphasized that Axiom of Foundation only requires all non-empty subSETS to have an epsilon-least element. But what about proper classes that (might) have infinite-descending epsilon ...
- 127
18 votes
2 answers
976 views
Where is the first repetition in the cumulative hierarchy up to elementary equivalence?
This is a sequel to my MSE question about elementary equivalences between the $V_α$. Given that there are only $ℶ_1$ first-order theories in the language of set theory, by pigeonhole principle, there ...
- 1,383
2 votes
0 answers
241 views
$0 = 1$ and (nigh-)inconsistent LCAs
This question is twofold. For one I would like to know which large cardinal notions which got any (at least minimal) traction have been known to be inconsistent. I know, for example, of Berkeley and ...
- 199
3 votes
1 answer
134 views
Disjoint maximal chain and maximal antichain in ${\cal P}(\omega)$
If $(P,\leq)$ is a partially ordered set, we say that $C\subseteq P$ is a chain if $a\leq b$ or $b\leq a$ for all $a,b\in C$. An antichain is a set $A\subseteq P$ with $a\not \leq b$ and $b\not\leq a$ ...
- 53.3k
10 votes
1 answer
484 views
Pure buttons in the modal logic of forcing
I've been trying to understand what it means for a button to be pure in the context of the modal logic of forcing, and it would help to have an example of a button which is not a pure button. Based on ...
- 345
5 votes
1 answer
217 views
Chromatic number of the antichain hypergraph on $\mathcal P(\omega)$
If $H=(V, E)$ is a hypergraph, the its chromatic number $\chi(H)$ is the smallest non-empty cardinal $\kappa$ such that there is a map $c:V \to \kappa$ such that for every $e\in E$ containing more ...
- 53.3k
7 votes
1 answer
245 views
Ascending chains in $\mathcal{P}(\kappa)/I_{\mathrm{NS}}$
$\newcommand{\NS}{\mathrm{NS}}\mathcal{P}(\kappa)/I_{\NS}$ is the Boolean algebra of subsets of $\kappa$ under nonstationary symmetric difference, with $\mathbf{0} = [\emptyset] = I_{\NS}$, $\mathbf{1}...
- 2,033
6 votes
1 answer
2k views
What is the order type of the hyperwebster? [closed]
I was recently asked by a friend what was the order type of all integers ordered by their English names (e.g. $3 < 2$, because three comes before two in the dictionary) and while I found several ...
- 87
3 votes
1 answer
252 views
Chromatic number of the maximal chain hypergraph on ${\cal P}(\omega)$
If $H=(V, E)$ is a hypergraph, the its chromatic number $\chi(H)$ is the smallest non-empty cardinal $\kappa$ such that there is a map $c:V \to \kappa$ such that for every $e\in E$ containing more ...
- 53.3k
11 votes
1 answer
491 views
Two definitions of regularity for ultrafilters
I am interested in the possible equivalence of two definitions of 'regular ultrafilter'. For context, an ultrafilter $\mathcal{D}$ is called $\lambda$-complete if, given any sequence $\langle X_{\...
- 333
6 votes
1 answer
355 views
Vopenka's principle with class-sized structures
Consider the following class-ized version of Vopenka's Principle: ($\mathsf{CVP}$) For every class $\mathcal{C}$ of graphs (or other structures of fixed set-sized similarity type), there are distinct ...
- 19.5k
5 votes
1 answer
579 views
Does every ultrafilter on real numbers contain a meager set?
I am trying to compare the structure of $z$-ultrafilters and closed ultrafilters on the Mysior plane. We know that any zero-set $Z\subseteq \mathbb{R}\times \{0\}$ of the Mysior plane is either meager ...
- 2,267
7 votes
1 answer
171 views
Admissibility spectrum and recursively large ordinals
Some people already have asked questions concerning Sy Friedman's results: (1) For $x\in\mathbb{R}$ if every $x$-admissible ordinals are stable, then $0^\#\in L[x]$. (2) There can be, by a class-...
- 503
18 votes
1 answer
538 views
Blowing up the power of $\aleph_\omega$ while preserving $\aleph_\omega$
There are nowadays various forcing techniques that give us models of $\qquad \aleph_\omega$ is a strong limit and $2^{\aleph_\omega}>\aleph_{\omega+1}$, assuming enough large cardinals in the ...
- 541
15 votes
0 answers
402 views
Forcing measurability of $\mathbf\Sigma^1_3$ when $\aleph_1$ is inaccessible to the reals
It is known, by Shelah, that Lebesgue measurability of all $\mathbf\Sigma^1_3$ sets of reals implies $\forall x\in\mathbb{R}(\aleph_1^{L[x]}<\aleph_1^V)$. Several decades ago, Yasuo Yoshinobu asked ...
- 503
7 votes
1 answer
366 views
Minimal cutsets containing no maximal antichain
If $(P,\leq)$ is a poset, we say that $C\subseteq P$ is a chain if $a\leq b$ or $b\leq a$ for all $a, b\in C$. Moreover, $A \subseteq P$ is an antichain if $a\not\leq b$ and $b\not\leq a$ whenever $a,...
- 53.3k
9 votes
1 answer
823 views
Applications of inner model theory in algebraic topology?
Is there any prior research or plausible avenue for application of nontrivial aspects of inner model theory to algebraic topology?
- 591
10 votes
0 answers
337 views
CH-preserving powerfully ccc forcing of size $2^{\aleph_1}$
Is it consistent with, or even implied by, CH that there is a CH-preserving, powerfully ccc complete Boolean algebra $\mathbb{B}$ of size $2^{\aleph_1}$? (Powerfully ccc means that ccc holds in every ...
- 8,161
3 votes
0 answers
318 views
A proof by Woodin of the Kunen Theorem
Akihiro Kanamori in his book ''The higher infinity'', in the page 320 when presented the second proof of the Kunen inconsistency Theorem, states that there are a function $S:\kappa\rightarrow P(\...
- 391
8 votes
2 answers
905 views
Is an ultrapower essential for defining the hyperreals?
If AC holds, the hyperreals are typically defined using the ultraproduct construction. Without AC, such as in ...
- 1,391
5 votes
2 answers
251 views
Relationship between tower number ($\mathfrak{t}$) and splitting number ($\mathfrak{s}$)
Recall that a tower is an infinite family $T$ of subsets of $\omega$ such that $T$ is well-ordered by the relation $\supset^*$ of almost inclusion and has no infinite pseudointersections. So, the ...
- 267
18 votes
3 answers
718 views
Possible cardinalities of maximal chains in ${\cal P}(\omega)$
Let ${\cal P}(\omega)$ denote the power-set of $\omega$. We order it by set inclusion $\subseteq$ and say that ${\cal C}\subseteq {\cal P}(\omega)$ is a chain if for all $A, B\in {\cal C}$ we have $A\...
- 53.3k