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.
5,270
questions
7
votes
1
answer
291
views
What is the height (or depth) of $[\mathbb{N}]^\infty$?
(This question assumes familiarity with combinatorial cardinal characteristics of the continnum.)
Let $[\mathbb{N}]^\infty$ be the family of infinite subsets of $\mathbb{N}$,
partially ordered by $\...
4
votes
1
answer
271
views
Is it consistent that the gaps between cardinals $\kappa$ and $2^\kappa$ "get larger and larger"?
Is the following statement consistent in $\mathsf{ZFC}$?
For every ordinal $\beta$ there is an ordinal $\lambda_0$ such that for all ordinals $\lambda\geq\lambda_0$ we have $2^{\aleph_{\lambda}...
23
votes
2
answers
2k
views
What is the largest Laver table which has been computed?
Richard Laver proved that there is a unique binary operation $*$ on $\{1,\ldots,2^n\}$ which satisfies $$a*1 \equiv a+1 \mod 2^n$$
$$a* (b* c) = (a* b) * (a * c).$$
This is the $n$th Laver table $(A_n,...
14
votes
1
answer
725
views
Changing cofinalities above supercompact cardinals
Question. Suppose $\kappa$ is a supercompact cardinal and $\lambda > \kappa$
is measurable (or even larger large cardinal if necessary).
Is there a set generic extension of the universe in which $\...
0
votes
1
answer
298
views
Can epsilon-induction be derived from the transitive closure operator?
I was wondering (and could not seem to prove or disprove) if epsilon-induction could be derived from the transitive closure operator for binary relations, if we do not have the Foundation Axiom.
The ...
8
votes
2
answers
381
views
"Clubiness" of projective sets of ordinals
I'm sure this is just my google-fu failing me, but: what are sufficient, non-overkill large cardinal axioms which guarantee "Every (boldface) $\Pi^1_n$ set of (real codes for) countable ordinals ...
9
votes
1
answer
485
views
Two questions about higher Souslin trees
Assume $V=L$ and let $\kappa$ be a Mahlo cardinal. Let $L[G]$ be the generic extension obatined by Mitchell forcing to make $2^{\aleph_0}=\aleph_2=\kappa.$
It is known that in the extension there are ...
4
votes
1
answer
273
views
Completing class-sized Fields
Let's say that an ordered Field is a class (proper or not) which satisfies the axioms of ordered fields. We work in NBG set theory with global choice.
Let's say that an ordered Field is real closed ...
18
votes
0
answers
899
views
Souslin trees and weakly compact cardinals
In Souslin trees on the first inaccessible cardinal it is asked if it is consistent that there are no $\kappa-$Souslin trees at the least inaccessible cardinal $\kappa$.
In this question I would like ...
3
votes
1
answer
735
views
Is tree property inconsistent with Berkeley cardinals in the absence of Axiom of Choice?
On one hand due to Kunen's inconsistency theorem it is known that within $\sf ZF$, large cardinal axioms beyond Reinhardt cardinal are inconsistent with $\sf AC$.
Also some recent results of Bagaria, ...
8
votes
1
answer
349
views
How long does the slow inefficient algorithm for computing the product in classical Laver tables take?
Let $(A_{n},*)$ denote the $n$-th classical Laver table. Let
$X_{n}$ be the set of all finite sequences of elements from $A_{n}$.
Define a function $E_{n}:X_{n}\rightarrow X_{n}$ by letting
$E_{n}((...
-1
votes
1
answer
155
views
Subgraphs of $\mathbb{R}^2$ in the Hadwiger-Nelson problem
In the setting of the Hadwiger-Nelson problem, two points of $\mathbb{R}^2$ form an edge if and only if their distance is $1$. The resulting graph $G$ has chromatic number $\chi(G)\in \{4,5,6,7\}$ and ...
4
votes
2
answers
263
views
Can one satisfaction class code another?
Let $M$ be a model of ${\sf ZFC}$. A satisfaction class $S$ for $M$ is subset of $M$'s ordered pairs which satisfies in $M$ the standard Tarskian compositional axioms. E.g.:
$M\vDash \forall \phi, \...
4
votes
1
answer
655
views
Large cardinals without choice?
For any given extension $T$ of ZFC (or perhaps NBGC or something), we can ask whether there is an extension $T'$ of ZF which does not prove AC such that
$Con(T) \leftrightarrow Con(T')$
$Con(T) \to ...
4
votes
2
answers
325
views
Is injectivity of $2^{(\ldots)}$ weaker than $\mathsf{GCH}$? [duplicate]
The following statement cannot be proven in $\mathsf{ZFC}$:
(S) : If $A, B$ are sets with $|A| < |B|$, then $2^{|A|} = |{\cal P}(A)| < |{\cal P}(B)| = 2^{|B|}$.
Obviously, $\mathsf{...
10
votes
2
answers
417
views
Slim Kurepa tree at a singular strong limit cardinal of uncountable cofinality
For a strong limit cardinal $\kappa$ the notion of $\kappa$-Kurepa tree is trivial: the full binary tree is a $\kappa$-Kurepa tree. Accordingly, we consider the following strengthening:
A slim $\...
4
votes
1
answer
260
views
Does "Every infinite set is splittable" imply $\mathsf{AC}$? [duplicate]
We say an infinite set $X$ is splittable if there are $X_1, X_2\subseteq X$ with $X_1\cap X_2 = \emptyset$, $X_1\cup X_2 = X$ and there are bijections $\varphi:X_1\to X_2$ and $\psi:X_1\to X$.
Does ...
7
votes
1
answer
1k
views
Does the consistency strength hierarchy coincide with the "arithmetic consequence" hierarchy at ZF + Reinhardt?
In these slides (see especially slide 26), Steel emphasizes the phenomenon that for all known "natural" extensions of ZFC, the ordering by consistency strength agrees with the ordering by containment ...
6
votes
3
answers
644
views
When does the generalized Cantor space embed in a $\kappa$-compact space
The generalized Cantor space is the space $2^\kappa$, with basic open sets
$$
[\sigma] := \{f\in 2^\kappa : \sigma\subseteq f\},
$$
for $\sigma\in 2^{<\kappa}$.
A space is $\kappa$-compact if ...
5
votes
1
answer
385
views
Cardinality of connected Hausdorff topologies
Let $X$ be an infinite set and let $C(X)$ denote the collection of connected Hausdorff topologies on $X$. Suppose $N\subseteq C(X)$ has the property that whenever $\tau\neq\sigma \in N$ then $(X,\tau)$...
7
votes
1
answer
644
views
When are unions of isomorphic groups isomorphic?
I was thinking about how to prove $\operatorname{Br}(K)\cong H^2(\operatorname{Gal}(\bar{K}/K),\bar{K}^*)$ without having to introduce inductive limits and all the profinite stuff. So, I started ...
4
votes
2
answers
859
views
Group & modules of arbitrary cardinality [closed]
How do I see that there is a group of an arbitrary cardinality? Is this also true for abelian groups? Also, given a commutative ring $R\neq 0$ how do I see that there is an $R$-module of arbitrary ...
8
votes
1
answer
289
views
$\mathsf{AD}_\mathbb{R}$ and Elementary Embeddings
Suppose $\mathsf{AD}_\mathbb{R} + V = L(\mathscr{P}(\mathbb{R})) + \mathsf{DC}$ holds. (We can use more if it is helpful.)
I believe under $\mathsf{AD}_\mathbb{R}$, every $A \subseteq \mathbb{R}$ is ...
6
votes
0
answers
146
views
Forcing in GBC, the ctm approach
There is a nice, detailed survey about forcing in GBC in the appendix of the dissertation of Jonas Reitz. At page 115 the author wrote: " If $ \Gamma $
is a finite collection of sentences forced by $ \...
4
votes
1
answer
401
views
"Lexicographic" ordering on ${\cal P}(\omega)$
For $A\neq B\in {\cal P}(\omega)$ we set $$\mu(A,B) = \min\big((A\setminus B)\cup (B\setminus A)\big).$$ We define $A < B$ if and only if $A \neq B$, and
$A = B\cap \mu(A,B)$ (that is $A$ is an ...
8
votes
1
answer
292
views
Theorem of Bukovsky characterizing ground models
It was mentioned in a talk that Bukovsky proved the following are equivalent for inner models $M \subseteq V$:
(1) There is a partial order $\mathbb P \in M$ and a $\mathbb P$-generic filter $G \in V$...
6
votes
1
answer
208
views
Can There be Rudin-Keisler Immediate Sucessors?
There are several well-studied orderings on the set $\omega^*$ of ultrafilters on the natural numbers. Three popular ones are $\le_i$ for $i = 1,2,3$. We define $\mathcal U \le_i \mathcal V$ to mean ...
11
votes
1
answer
778
views
Is the dual of the product of infinite cyclic groups a free abelian group ?
By a theorem of Specker, the group $\mathrm{Hom}(\prod_{\aleph_0} \mathbb{Z},\mathbb{Z})$ is isomorphic to $\bigoplus_{\aleph_0}\mathbb{Z}$ and is in particular a free abelian group. I wonder, if this ...
9
votes
1
answer
555
views
Interpreting Robinson arithmetic in a very weak set theory
It is known that adjunctive set theory interprets Robinson arithmetic, and that extensionality is not needed for that. (Montagna and Mancini, "A minimal predicative set theory", Notre Dame Journal of ...
5
votes
2
answers
592
views
Least inner model of ZF without power set axiom
I'm interested in the existence and properties of an analogue version of $L$ for models of ZF$^-$ (ZF without the power set axiom), which for simplicity I'll call $L^-$. By "analogue" I mean the least ...
16
votes
2
answers
946
views
$\mathfrak{ufo}$: An unidentified combinatorial cardinal characteristic of the continuum?
An ultrafilter ornament is a chain of free filters on $\mathbb{N}$ that are not ultrafilters, whose union is an ultrafilter.
Let $\mathfrak{ufo}$ be the minimal cardinality of
an ultrafilter ...
10
votes
2
answers
307
views
Limits of rearranged sequences along ultrafilters
Suppose that a bounded sequence of real numbers $s_i$ ($i\in\omega$) has a limit $\alpha$ along some ultrafilter $\mu_1\in \beta{\Bbb N}\setminus{\Bbb N}$. Then given another ultrafilter $\mu_2\in \...
5
votes
1
answer
433
views
Cofinality of countable ordinals in ZF, and in toposes
A countable limit ordinal $\kappa$ has cofinality $\omega$. One proves this in ZF, say, using the usual trick for representing $\kappa$ as a countable set of reals having closed convex span $[0,1]$ (...
2
votes
4
answers
1k
views
The paradox with the first uncountable ordinal
Suppose we have a set $M = (0,1) \subset R$ of reals well-ordered as the first uncountable ordinal.
Let $M(a) = \lbrace x \in M : x < a \rbrace$. For every $a \in M$ set $M(a)$ is countable. That'...
7
votes
1
answer
363
views
Iteration of random reals
Consider two random reals $x, y$ over a transitive model $V$ of ZFC. More specifically, if $\mathcal C^V={}^\omega2$ is the Cantor space, composing the canonical homeomorphism with the projections $\...
19
votes
3
answers
1k
views
Set-theoretic forcing over sites?
All texts I have read on set-theoretic independence proofs consider several different sorts of constructions separately, such as Boolean-valued models (equivalently, forcing over posets), permutation ...
2
votes
0
answers
144
views
When do wide initial segments ruin admissibility?
Suppose $\alpha$ is admissible and $\beta<\alpha$. We know that $L_\alpha$ is an admissible set (by definition), but adding subsets of $\beta$ to $L_\alpha$ might break admissibility: while set ...
14
votes
1
answer
730
views
Is there a class of mathematical structures with non-isomorphic natural representations as a standard Borel space?
Background. The field of Borel equivalence relation theory
provides a robust, unifying theory that organizes most of the
classification problems of classical mathematics into a hierarchy,
allowing us ...
1
vote
1
answer
300
views
Ordinal of injectivity for a smooth regular curve with a finite arc-length
Let $\gamma: [a,b]\to\mathbb{R}^d$ defined by $$\gamma(t)=(\gamma_1(t),\dots,\gamma_d(t)) $$ be a smooth (i.e., $\gamma\in C^\infty (\mathbb{R}))$ and regular ($\gamma^\prime(t)\neq \vec 0$) curve ...
3
votes
1
answer
117
views
Generic sections of non-null sets are non-null
Consider the Cantor space $\mathcal C={}^\omega2$ with the usual product measure, and let $r$ be a random real (over a transitive model $V$ of ZFC). Let $B\subset \mathcal C^V\times\mathcal C^V$ a ...
3
votes
1
answer
140
views
Levels of L resembling each other, take 2
(Everything below is assuming $V=L$.)
Fix an uncountable regular cardinal $\kappa$, and let $$E_\kappa=\{\mu<\kappa: \mbox{there is an elementary substructure of $L_\kappa$ isomorphic to $L_\mu$}\}...
6
votes
1
answer
254
views
Fine structure question: when do levels of $L$ look "a lot" like each other?
(Everything is assuming $V=L$.)
Fix an uncountable regular cardinal $\kappa$, and let $$E_\kappa=\{\mu<\kappa: \mbox{there is an elementary substructure of $L_\kappa$ isomorphic to $L_\mu$}\},$$ ...
15
votes
1
answer
399
views
Consistency strength of $\aleph_2$-Souslin hypothesis
Question 1. What is known about the consistency strength of $\aleph_2$-Souslin hypothesis?
Question 2. What is known about the consistency strength of having both $\aleph_2$-Souslin hypotheis and $\...
1
vote
0
answers
599
views
First few research papers [closed]
I was planning on posting this on academia.stackexchange, but I want an answer from mathematicians who've dealt with a similar issue when they were beginning graduate students. If this site doesn't ...
5
votes
1
answer
249
views
A button for individual reals
Hamkins introduced the notion of a "button" in forcing. This is a set-theoretic statement that can be forced, and can never be made false by further forcing. An example is $V \not= L$. Another ...
7
votes
1
answer
378
views
References for higher descriptive set theory surveys
A student of Adi Jarden and mine attempts at generalizing results on selection principles from the Baire space $\omega^\omega$ to the higher Baire space $\kappa^\kappa$ ($\kappa$ uncountable), and ...
5
votes
1
answer
567
views
When is the generalized Cantor space $\kappa$-compact?
My M.Sc. student has the following question, that I assume has an answer in the literature, and we are looking for references.
The generalized Cantor space is the space $2^\kappa$, with basic open ...
8
votes
2
answers
621
views
Does the class category of ZF-algebras satisfy the Multiverse axioms?
I read with interest both Hamkins Multiverse Axioms and Joyal and Moerdijk's algebraic set theory. Both of these perspectives takes a set-theory universe as an object, and consider collections of set-...
15
votes
2
answers
2k
views
Is platonism regarding arithmetic consistent with the multiverse view in set theory?
A "truth" platonist for arithmetic believes, given a statement in the language of arithmetic, that the problem whether the statement is true has a definite answer.
Prof. Hamkins has argued for a ...
9
votes
2
answers
1k
views
Using the multiverse approach to decide the law of the exluded middle?
Recently, in response to deciding the Continuum Hypothesis $CH$, Hamkins and Gitman have proposed consider a multiverse of set-theoretic universes, some in which $CH$ is true, some in which $\neg CH$ ...