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.
5,254
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"Local" compactness properties beyond $\mathcal{L}_{\omega_1,\omega}$?
Below, all languages are finite for simplicity.
This question is about generalizations of Barwise compactness for logics more complicated than $\mathcal{L}_{\omega_1,\omega}$: properties of the form "...
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Is each $G_\delta$-measurable map $\sigma$-continuous?
Definition. A function $f:X\to Y$ between topological spaces is called
$\bullet$ $G_\delta$-measurable if for each open set $U\subset Y$ the preimage $f^{-1}(U)$ is of type $G_\delta$ in $X$;
$\...
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300
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On the cardinal arithmetic of accessible categories
If $\lambda, \mu$ are regular cardinals, say that $\lambda \trianglelefteq \mu$ if $\lambda \leq \mu$ and
$$\forall X, \, |X| < \mu \implies \mathrm{cf} (P_\lambda(X)) < \mu$$
Here $P_\lambda(X)...
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Why does an $\varepsilon$-incomplete $Q$ remain so in the forcing extension of any $\varepsilon$-complete $P$?
This is Shelah's simpler notion of 'completeness' for not adding reals via forcing. Here $P$ and $Q$ are forcing notions.
Let $S_{\aleph_0}(\kappa)$ be the set of all countable subsets of a cardinal $\...
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683
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$\Delta^0_{\alpha}$ universal sets does not exist
I am taking a course in descriptive set theory, and the exam is approaching on Sunday. In the framework of proving that for an uncountable Polish space $X$ the following holds: $\Delta^0_\alpha(X)\...
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Reference request: choiceless cardinality quantifiers
There is a substantial literature on the logic of cardinality quantifiers. (E.g., the quantifier $Q_\alpha$ where $M \vDash Q_\alpha x \, \varphi (x)$ iff $\vert \{a \in M : M \vDash \varphi[a] \} \...
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203
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The Hausdorff dimension of the set of reals of inner models
Suppose that both $M$ and $N$ are models of $ZFC$ with $M\subseteq N$ so that $M$ is definable in $N$.
Question Can $(\mathbb{R})^M$ have Hausdorff dimension strictly between $0$ and $1$ in $N$? How ...
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422
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What is the nicest bijection $\textbf{R}^p \to \textbf{R}^q$ that you know?
It is well-known that bijection between $\textbf{R}^p$ and $\textbf{R}$ exist (e.g. here, though many other examples exist).
The problem with all these examples of bijections is that typically the ...
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Can every Borel set be partitioned into $\leq\!\aleph_1$ $F_{\sigma \delta}$ sets?
Consider the following two facts, a modified version of which appear in this paper of Arnie Miller from the early 1980's:
$\bullet$ If $\mathbb R$ can be partitioned into $\aleph_1$ closed sets, then ...
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What large cardinals are needed to imply projective sets have the perfect set property?
If there are infinitely many Woodins, then every projective set is determined, whence every projective set has the perfect set property (PSP). Since determinacy is such a stronger property than the ...
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162
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Upper-bounding determinacy
While the converse of Borel determinacy ("If a set of reals is determined, then it is Borel") is boringly disprovable, I'm curious if there is a sense in which something like it is ...
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Does determinacy imply unravellability for the Borel sets (over a weak base theory)?
As far as I know, the only way we currently know how to prove Borel determinacy in $\mathsf{ZFC}$ is to go through unravelability (a rather technical property whose definition can be found in Martin's ...
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219
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Absoluteness of the core model under a proper class of completely Jónsson cardinals
Example 2.4.2 of Larson's The stationary tower (based on Woodin's lecture) describes how we can absorb a generic filter over a set forcing in a definable inner model into a generic extension by $\...
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Intuition for branch uniqueness in inner model theory
In inner model theory, what is the intuition behind the expectation that under appropriate conditions, we should have a single preferred branch to continue an iteration at a limit stage?
At the level ...
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494
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Ring of global sections of a functor $\mathbf{CRing} \to \mathbf{Set}$
Let $U : \mathbf{CRing} \to \mathbf{Set}$ be the forgetful functor. For any functor $F : \mathbf{CRing} \to \mathbf{Set}$ consider the class of natural transformations
$$\mathcal{O}(F) := \mathrm{Hom}(...
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Inner models from highly saturated ideals
Solovay proved that a measurable cardinal is equiconsistent with the existence of an extension of Lebesgue measure to a full measure on $P(\mathbb R)$. The inner model direction is relatively simple. ...
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Choice function for elementary embedding $j: V_\lambda\prec V_\lambda$
Work in ZF. Let $\lambda$ be strongly inaccessible. Let $\kappa$ be such that for every $\alpha\lt\lambda$, there is some $j: V_\lambda\prec V_\lambda$, with critical point $\kappa$, such that $j(\...
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Are there analogues of real-valued measurability for larger powersets?
Apologies if the question is somewhat naïve - I'm not a set theorist, just an enthusiast.
One possible intuition we could have about the generalized continuum hypothesis is that power sets are so ...
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307
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A $\mathsf{ZF}$ example of a nonreflexive group which is isomorphic to its double dual?
Given a group $G$ denote by $G^\ast=\mathrm{Hom}(G,\Bbb Z)$ its dual and by $j\colon G\to G^{\ast\ast}$ the canonical homomorphism $g\mapsto (f\mapsto f(g))$. A group is reflexive iff $j$ is an ...
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Can the set of endomorphisms of $(\mathbb R,+)$ have cardinality strictly between $\frak c$ and $\frak{c^c}$?
Let $\frak c$ be the cardinality of the reals. I know that in ZF the set of endomorphisms of $(\mathbb R,+)$ can have at least two different cardinalites:
If we allow the axiom of choice, you can ...
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203
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ladder system uniformization at successors of singulars
Shelah proved (paper 667) that if GCH holds and $\lambda$ is singular, then for every stationary $S \subseteq \{ \alpha < \lambda^+ : \text{cf}(\alpha) = \text{cf}(\lambda) \}$, there is a ladder ...
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538
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Does second order ZFC conservatively extend first order ZFC?
If I replace the axiom schema of specification in ZFC by a single axiom in second order logic, and similarly do same thing for the axiom schema of replacement, is this version of "second order ZFC" ...
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Can every set be measurable?
The Solovay model shows that ZF (plus an inaccessible) is consistent with every subset of $\mathbb{R}$ being measurable. How far can we go in that direction? We can always have non-measurable sets ...
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Would Ultimate L really "[reduce] all questions of set theory to axioms of strong infinity"?
According to these slides, the axiom $V = \mathrm{Ultimate} \,L$ has the following consequences (p. 55):
It implies the Continuum Hypothesis.
It reduces all questions of set theory to axioms of ...
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224
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Metrically Ramsey ultrafilters
On Thuesday I was in Kyiv and discussed with Igor Protasov the system of MathOverflow and its power in answering mathematical problems. After this discussion Igor Protasov suggested to ask on MO the ...
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650
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Is there any theorem achieving Conway's "Mathematician's Liberation Movement"
John Conway on in the appendix to part zero of ONAG describes a "Mathematician's Liberation Movement". The goal would be to give mathematicians the freedom to create mathematical theories with the ...
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346
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Model for "$\kappa$ limit cardinal iff $2^\kappa$ limit cardinal"
Is there a model of ${\sf (ZFC)}$ such that in the model we have that $\kappa$ is a limit cardinal if and only if $2^\kappa$ is a limit cardinal?
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239
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Topological applications of $\mathfrak{p}=\mathfrak{t}$
I'm working on the Malliaris-Shelah's recent result of $\mathfrak{p}=\mathfrak{t}$, but I'm more interested in what possible topological applications can derivate from this equality.
Searching in ...
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273
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Proper classes in Bounded Zermelo set theory
I want to know if there is a standard terminology for this among set theorists working with element-based set theories like ZFC. I will follow the convention that a class in any given set theory is a ...
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235
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Universally meager spaces and large cardinals
Definition: (Todorcevic) A subset $A$ of a topological space $X$ is called universally meager if for every Baire space $Y$ and every continuous $f : Y \to X$ which is nowhere constant (not constant on ...
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189
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Specializing fat trees
The discussion is about trees of height $\omega_1$ that are not necessarily thin, namely, no cardinality constraints on the size of each level. A classcial theorem of Baumgartner states that it is ...
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226
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Is there a 'local' version of Near Coherence of Filters?
The axiom Near Coherence of Filters (NCF) is known to be independent of ZFC.
Axiom (NCF I): For any two free ultrafilters $\mathcal D$ and $\mathcal E$ on $\mathbb N$, there exist finite-to-one ...
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219
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When can we force two frames to be homeomorphic?
Recall that if $M,N$ are two structures of the same type, then
$M$ is $\mathcal{L}_{\infty,\omega}$ elementarily equivalent to $N$ precisely when $M$ and $N$ are isomorphic in some forcing extension. ...
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334
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What arithmetic is interpretable in Mayberry's Euclidean set theory?
John Mayberry published what he calls a Euclidean set theory in his book The Foundations of Mathematics in the Theory of Sets. It is ZF with the axiom of infinity replaced by an axiom saying "the ...
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Is it true that $\kappa\to[\kappa]^2_2$ iff $\kappa\to[\kappa]^2_\kappa$ for inaccessible $\kappa$?
Recall that the square partition relation $\kappa\to[\lambda]^k_\eta$ holds iff for every $f:[\kappa]^k\to\eta$ there exists $H\in[\kappa]^\lambda$ such that $f"[H]^k\neq\eta$. I.e. said in words, ...
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Using van de Wiele's characterization as a definition?
This fall I'm teaching a class on generalized computability theory (broadly construed). One thing I want to talk about briefly is E-recursion.
Now, E-recursion is generally defined in terms of the ...
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227
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Second-order separation schema in Zermelo and Zermelo--Fraenkel
It is a nice theorem of Zermelo that if we replace the Replacement schema with its second-order counterpart, "The image of a set under any function is again a set", then we necessarily get a model ...
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434
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The minimum cardinality of an almost disjoint reaping family
The following cardinal arose in the discussion surrounding the question posed here by Dominic van der Zypen:
Define $\kappa$ to be the minimum cardinality of a family $\mathcal{A}$ of infinite ...
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282
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Higher dimensional $\Delta$-system lemma
Consider the following statement (called $\Delta^d(\kappa^+, \lambda)$ where $d\in \omega$ and $\kappa<\lambda$ are cardinals): For every $A': [\lambda]^d\to [\lambda]^{\leq \kappa}$, there exist $...
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170
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Non-wellorderable ultrafilters with wellorderable bases
There are some models in which $2^\omega$ is not wellorderable but there is a free ultrafilter over $\omega$. What about the consistency of: $2^\omega$ is not wellorderable + AC for countable sets of ...
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369
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Baire category of tall ideals
Problem. Is it consistent with ZFC that $\mathfrak t=\omega_1$ and each $\omega_1$-generated tall $P$-ideal is of the second Baire category?
(Asked 01.10.2016 by David Chodounsky at page 20 of Volume ...
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398
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The cone property in the enumeration degrees
A Borel partial order is the partial order corresponding to a Borel preorder of some Polish space. For example, the Turing and enumeration degrees, $\mathcal{D}$ and $\mathcal{E}$ respectively, are ...
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169
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Sharply less regular cardinals in set theory
If $\lambda<\kappa$ are two regular cardinals, then every locally $\lambda$-presentable category is locally $\kappa$-presentable. The analogous statement does not hold for $\lambda$-accessible ...
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383
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The reals in $L$
Assume "$0^\#$ exists".
We know that $0^\#$ is a $\Pi^1_2$-singleton. That means, there is a Shoenfield tree $S$ on $\omega \times (\omega \times \omega_1)$ so that $$x = 0^\# \leftrightarrow S_x \...
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349
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Cohen's model yet again
It has been discussed already whether a countable OD set necessarily contains an OD element. See e.g.
A question about ordinal definable real numbers
.
A negative answer was obtained in Archive for ...
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188
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Examples of analytic $\mathcal{I}$-mad families
If $\mathcal{I}$ is an ideal (proper and containing the finite sets) on $\omega$, call a family of subsets $\mathcal{A}\subseteq[\omega]^\omega$ $\mathcal{I}$-almost disjoint if for all distinct $A,B\...
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What's Reeb's take on naive integers?
Georges Reeb's "claim Q" is the statement that "naive integers don't fill up $\mathbb{N}$". To anyone familiar with model theory this could easily be interpreted as the existence of nonstandard models ...
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168
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Are there always large discrete families of normal measures?
Let $\kappa$ be a measurable cardinal. We give the Stone space of all ultrafilters on $\kappa$ the usual topology, where each $x\subseteq\kappa$ determines a basic open $[x]=\{U;x\in U\}$. The ...
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229
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A question about strongly compact cardinals
Is the following equiconsistent with the existence of a strongly compact cardinals:
For every $\lambda > \kappa$ there exists a $\lambda$-strongly compact embedding $j: V \to M$ with the ...
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371
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PCF conjecture and fixed points of the $\aleph$-function
Recently Moti Gitik refuted Shelah's PCF conjecture, by producing a countable set $a$ of regular cardinals with $|\operatorname{pcf}(a)| \geq \aleph_1.$ See his papers
Short extenders forcings I and ...