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.
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questions with no upvoted or accepted answers
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Infinite time Turing machines, semi-decidable sets and descriptive set theory
Definition A set of reals $A$ is said to be ittm-eventually-semi-decidable if there is an Infinite Time Turing Machine programme $P_e$ so that $x\in A$ iff $P_e(x)$ has converged on “1” on its ...
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130
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Better scales and Failures of SCH
Assume $\mu$ is a singular cardinal of countable cofinality. Recall that a scale for $\mu$ consists of an increasing sequence $\vec{\mu}$ of regular cardinals $\langle \mu_n:n<\omega\rangle$ ...
- 7,345
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223
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How much is known about the consistency strength of toposes and topos-like categories?
It's a well-known fact that the theory of a well-pointed topos with a natural numbers object (NNO) has the same consistency strength as MacLane set theory (also known as bounded Zermelo). There are ...
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"Relative plausibility" of some infinitary theories
We work in $\mathsf{ZFC+V=L}$.
Define a plausible theory to be a theory $T\subseteq\mathcal{L}_{\omega_1,\omega}$ in an $\omega_1$-finite language which is $\omega_1$-c.e. and $\omega_1$-finitely ...
- 19.1k
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152
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A model where the uniformity of the meager ideal is strictly below the almost disjointness number
I'm looking for a model satisfying the inequality described in the title. Recall that the uniformity of the meager ideal, denoted $\operatorname{non}(\mathcal M)$ (or $\operatorname{non}(\mathcal B)$) ...
- 1,832
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203
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Which first-order theories admit a compact-like superstructure?
Positive set theory is an approach to rectifying Russel's paradox by restricting the syntactic form of formulas for which we allow comprehension. It can be motivated by the construction of certain ...
- 10.3k
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220
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Determinacy of symmetric games
Is it consistent that for all ordinals $α$ and $λ$ and infinite regular cardinals $κ$, games on $V_λ$ with game length $κα$ and $\mathrm{OD}(\mathrm{On}^κ)$ payoff that depends only on the set of all ...
- 6,102
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217
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adding one point from the Stone-Cech compactification
Let $X$ be any non-compact Tychonoff space and $\beta X$ be its Stone-Čech compactification.
The following fact is known: any point $p$ from the reminder $\beta X \setminus X$ is not a $G_{\delta}$-...
- 71
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$0^\#$ in weak theories vs large cardinals in $L$
To better understand the transition from large cardinal axioms consistent with the constructible universe $L$ to large cardinal axioms transcending $L$, I am looking for natural equiconsistencies ...
- 6,102
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207
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Turing-independent joins
Define $Y \subseteq 2^{\omega}$ to be Turing independent if for every finite $F \subseteq Y$ and $y \in Y \setminus F$, the Turing join of $F$ does not compute $y$.
Define $X \subseteq 2^{\omega}$ ...
- 71
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172
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Making the precaliber number bigger than all Knaster numbers
Write $\mathfrak m_k$ for the Martin's axiom number for $k$-Knaster, i.e., for the smallest size of a family of dense subsets of some $k$-Knaster poset for which there is no generic filter. (A poset ...
- 662
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172
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Asymptotically discrete ultrafilters
Definition 1. A ultrafilter $\mathcal U$ on $\omega$ is called discrete (resp. nowhere dense) if for any injective map $f:\omega\to \mathbb R$ there is a set $U\in\mathcal U$ whose image $f(U)$ is a ...
- 40.9k
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If $j_{1},...,j_{n}:V_{\lambda+1}\rightarrow V_{\lambda+1}$ are elementary embeddings, then does $j_{1}(A)=...=j_{n}(A)=A$ for some linear order $A$?
Suppose that $j_{1},\dots,j_{n}:V_{\lambda+1}\rightarrow V_{\lambda+1}$ are elementary embeddings. Then does there necessarily exist a linear ordering $A$ of $V_{\lambda}$ such that $j_{1}(A)=\dots=j_{...
- 27.8k
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274
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Co-cones in the Turing degrees
Let the cocone of a Turing degree ${\bf d}$ be the set $cc({\bf d}):\{{\bf c}: {\bf c}\not\ge_T {\bf d}\}$. I'm curious what's known about the various partial orders (isomorphic to ones) of the form $...
- 19.1k
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Equivalent strictly convex norms in spaces of small density
Can one construct in ZFC a Banach space of density character $\omega_1$ that does not have an equivalent strictly convex norm?
Maybe one may apply some kind of a Löwenheim–Skolem-type argument to a ...
- 11.3k
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273
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A strange planar set and the Continuum Hypothesis
Call a number abnormal if its decimal expansion doesn't feature every digit an infinite number of times. Call a triangle in ${\Bbb R}^2$ abnormal if at least one of its angles spans an abnormal ...
- 17.5k
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195
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Query about iterated collapse forcing
I have heard it said that it is possible to use Woodin's iterated collapse forcing to prove that the theory $\textsf{ZF}$+"there exists a non-trivial elementary embedding $j:V_{\lambda+2} \rightarrow ...
- 2,005
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205
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Possible cofinalities of cuts of ultraproducts
Suppose $\kappa$ is a regular cardinal, $\bar{\mu}=(\mu_i: i<\kappa)$ is an increasing sequence of regular cardinals ($>\kappa$) and set
$pcut(\bar \mu)=\{ (\lambda_1, \lambda_2):$ for some ...
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A non-special Aronszajn tree with a stationary set that is non-stationary with respect to the tree
Is there any example of a ($\omega_1$-)Aronszajn tree $T$ that is non-special and there exists a stationary subset $S\subset \omega_1$ such that $S$ is not stationary with respect to $T$?
A tree ...
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201
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Does Solovay's $\Sigma$-construction preserve "niceness"?
Suppose I have two forcing notions $\mathbb{P}$ and $\mathbb{Q}$, and a $\mathbb{Q}$-name $\nu$ for a real. Let $G$ be $\mathbb{P}$-generic, and suppose $r$ is a real in $M[G]$ such that for some $H$ ...
- 19.1k
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Natural theories for the failure of gap-1 transfer principle
The gap-1 transfer principle says that the transfer principle $(\kappa^+, \kappa) \to (\lambda^+, \lambda)$ holds for all infinite cardinals $\kappa, \lambda.$
It is known that for some sentence $\...
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155
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Woodin's theorem about the existence of sharps for the Chang's model
In The sharp for the Chang model is small, Mitchell has stated a result of Woodin about the Chang's model, and he has produced a result using much weaker assumptions.
As it is stated in the paper, ...
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243
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Tree property using side conditions
The following problems were asked during the high and low forcing workshop:
Question 1. Can one force tree property at $\kappa^{++}$ for $\kappa$ singular using side conditions?
Question 2. Can one ...
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240
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Adding minimal subsets to $\aleph_\omega$
Given a cardinal $\kappa,$ recall that $X \subset \kappa$ is called fresh (over $V$), if $X \notin V,$ but $X \cap \alpha \in V$ for all
$\alpha < \kappa.$
Question. Is it consistent that there ...
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168
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Sacks minimality without choice
The usual argument for the minimality result for Sacks forcing uses choice.
Theorem (Sacks): Let $s \subseteq \mathbb S_\kappa$ be generic for the forcing to add a Sacks subset to $\kappa$, where $\...
- 2,087
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245
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$V$ as a $HOD$ of its class generic extension
By an old result of Roguski, The theory of the class $HOD$, any model $V$ of $ZFC$ has a class generic extension $V[G]$ such that $HOD$ of $V[G]$ equals $V$.
This result is also stated and generalized ...
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A strengthening of Chang's conjectures
In the Handbook of Set Theory, Foreman has (essentially) the following proposition (3.9):
Suppose $\kappa_n>...>\kappa_0$ and $\lambda_n>...>\lambda_0$ are regular cardinals and $(\...
- 18.1k
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How distributive are the fake Laver tables?
The Laver table $A_{n}$ is the unique algebra $(\{1,...,2^{n}\},*)$ such that $x*1=x+1$ for $1\leq x<2^{n}$, $2^{n}*1=1$, and $x*(y*z)=(x*y)*(x*z)$.
Let's now replace the Laver table $A_{n}$ with ...
- 27.8k
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180
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rigidity of $\mathcal P(\omega_1) / NS$ under MA
In Woodin's book, Lemma 5.100 asserts that if $MA_{\omega_1}$ holds and there is an $\omega_2$-saturated ideal $I$ on $\omega_1$, then $\mathcal P(\omega_1)/I$ is a rigid boolean algebra, meaning it ...
- 18.1k
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$\alpha$-minimal degrees for singular $\alpha$
An important question in $\alpha$-recursion theory is whether there is a minimal $\alpha$-degree at $\alpha=\aleph_\omega.$
Question 1. Who first introduced the above question, and where can I find ...
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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 \...
- 9,781
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Core model for supercompact cardinals and iteration trees
I have a few somehow related questions:
Question 1. What do we expect for an inner model $\mathcal{K}$ to be a core model for a supercompact cardinal? What properties should it have, and what ...
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$\delta$-strong compactness and generalized strong tree properties
Are there non-trivial equivalent characterizations of $\delta$-strongly compact (and almost strongly compact) cardinals in terms of generalized tree properties?
Recall the definitions as per Joan ...
- 2,121
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Countable choice in $L(\mathbb{R}^*_G)$
Let $\lambda$ be a singular strong limit cardinal and let $G \subset \text{Col}(\omega,\mathord{<}\lambda)$ be a $V$-generic filter. Let $\mathbb{R}^*_G = \bigcup_{\alpha < \lambda} \mathbb{R}^{...
- 5,551
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Does $\mathsf{fReR}_0$ prove the existence of the cartesian product of two sets
$\mathsf{fReR}_0$ is the set-theoretical system whose axioms consist of:
(1) Axiom of extensionality: $\forall z\in x\ (z\in y)\wedge\forall z\in y\ (z\in x)\rightarrow x=y$
(2) Axiom of empty set: $...
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A question about cardinal numbers when the Axiom of Choice is absent
The Axiom of Choice constrains every set of cardinal numbers which is linearly ordered by size to be well-ordered. By contrast, does ZF-without the Axiom of Choice (but with the Axiom of Foundation)-...
- 6,105
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Extending a Ramsey filter
We take filter to mean filter on $\omega$ containing all cofinite sets. We say a filter $F$ is Ramsey if ZERO does not have a winning strategy in the following infinite game between the two players ...
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255
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Other variants of the Shelah's Weak Hypothesis
The paper
Menachem Kojman. Splitting families of sets in ZFC.
arXiv:1209.1307
presents these variants of the Shelah's Weak Hypothesis:
$$
(\textrm{SWH}_n) \textrm{ There are no infinite } \nu ...
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Impact of Supercompacts on Measurables
It is consistent that the least measurable cardinal can carry exactly one normal measure but in almost all models for this theory there is no supercompact cardinal. It seems existence of a ...
user42090
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Recent application of model theory in set theory by Shelah-Malliaris
Recently one of the oldest open problems in set theory about the cardinal invariants of the continuum (i.e the question of whether $\mathfrak{p}=\mathfrak{t}$) was solved by Shelah and Malliaris (see "...
user39869
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Why has Sacks' "Measure-theoretic uniformity" not been more influential?
In the 1969 paper "Measure-theoretic uniformity in recursion theory
and set theory," Trans. Amer. Math. Soc. 142 1969 381–420, Sacks gave
a measure-theoretic approach to several results previously ...
- 12.3k
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464
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Closure properties of familes of $G_\delta$ sets.
Given a family of sets $G\subset P(X)$, can one characterize by "closure properties" alone whether or not $G$ arises as the family of all $G_\delta$ for some topology on $X$? some Polish space ...
- 17.5k
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Infinity-Borel sets in ZFC
The notion of an $\infty$-Borel set of reals is useful in the study of AD. Under ZFC it becomes trivial: every set of reals is $\infty$-Borel. However, the notion of an $\infty$-Borel code is still ...
- 5,551
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377
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Sets of reals amenable to each L[x]
If $A$ is a set of reals such that $A \cap L[x] \in L[x]$ for each real $x$, is there a real $z$ such that $A \cap L[x] \in OD_z^{L[x]}$ for a cone of $x$?
This can be proved under the Axiom of ...
- 5,551
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What about replacing $\{0,1\}$ in Stone duality with another finite set?
Basically Stone duality or more general the duality between spatial locales and sober spaces is about enriching the set of morphisms $X \to \{0,1\}$ with an additional structure and then finding ...
- 61.5k
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Maps between forcing posets
We all know that forcing can be seen (if you like things that way) as a category of sheaves over the poset of forcing conditions equipped with the double negation Grothendieck topology. As such it is ...
- 33.9k
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From HODs to corresponding models of AD
If $M$ is HOD of a model $N$ of $\text{AD}^+ + V=L(P(ℝ))$, what kind of forcing construction in $M$ gives back such an $N$?
HODs for $\text{AD}^+ + V=L(P(ℝ))$ are conjectured (and under anti-large ...
- 6,102
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Do maximal compact logics exist?
By "logic" I mean regular logic in the sense of abstract model theory (see e.g. the last section of Ebbinghaus/Flum/Thomas' book). My question is simple:
Is there a logic $\mathcal{L}$ ...
- 19.1k
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148
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Complexity of transfinite 5-in-a-row and other games
Suppose that 5-in-a-row is played on an infinite board, and after an infinite number of moves, if no one won yet and there is an empty square, the game just continues. At limit steps, it is the first ...
- 6,102
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What are these non-classical versions of ZFC defined by realizability?
See Kleene realizability in Peano arithmetic for a similar question, but about PA instead of ZFC.
In the context of constructive set theory, consider two ways of defining realizability.
The first is $\...
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