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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,...

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13 views

On a combinatorial set covering property

Let $\kappa < \lambda < \mu$ be infinite cardinals. Is there a collection ${\cal U}\subseteq {\cal P}(\mu)$ of subsets of $\mu$ with the following properties? for all $U\in {\cal U}$ we have $|...
2
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1answer
130 views

Another question concerning p and t

I refer to an article concerning p and t : edited Sep 14 '17 at 2:48 / Bjørn Kjos-Hanssen answered Sep 13 '17 at 21:50 / Mark Fischler I already asked a question December 14th 2018 and I received ...
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0answers
59 views

Can reflection in $V$ and reflection out of $V$ principles be used in the same theory?

I've noticed that most of the work in Ackermann set theory is primarily concerned with constructing sets in $V$, the rest of the classes are just excess material, carrying no comprehension over them. ...
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0answers
44 views

Can this reflective class theory interpret ZFC?

Reflective Set Theory $\mathsf{RfST}$ is formulated in first order predicate logic with extra-logical primitives of equality $``="$, membership $``\in"$, and a single primitive constant symbol $V$ ...
7
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1answer
141 views

Effective set= ordinal definable set

I just today realized that the concept of ordinal definability is defined in a different way by vopenka-Balcar-Hajek ``The notion of effective sets and a new proof of the consistency of the axiom of ...
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0answers
62 views

Why is $\omega^{\omega}$ countable? [migrated]

I'm confused as to why $|\omega^{\omega}| \neq \aleph_0^{\aleph_0}$. Since \begin{align} \omega^{\omega} = \left\lbrace \sum_{i < \omega}^{1} (\omega^i \cdot n_i) + n_0 : n_i,n_0 \in \mathbb{N}_0 \...
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0answers
40 views

What is the limit to iterating class comprehension, reflection and limitation of size?

In posting about a reflection principle coupled with a limitation of size axiom over Ackermann set theory, the answer is that the theory is blown up to a Mahlo cardinal. I'm here just wondering if ...
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0answers
117 views

My question is about an article concerning p and t [on hold]

you say concerning p and t: An example of an element of p is the family of sets (indexed by k∈ℕ) defined by {m to the power of k :m∈ℕ}. But the second condition is not met because the set {2 to the ...
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0answers
379 views

A new generalization of the dimension?

During my research, I came a cross on these notions : Definition 1: A structure $S$, is a pair $(X, \mathcal T)$ with $X$ a set and $\mathcal T$ a set of subsets of $X$, stable by arbitrary ...
3
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1answer
130 views

What is the strength of adding limitation of size and a simple version of reflection to Ackermann set theory?

The following theory is formulated in first order predicate logic with extra-logical primitives of equality $``="$, membership $``\in"$, and a single primitive constant symbol $V$ denoting the class ...
6
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0answers
222 views

measure of generic reals in forcing extensions

It is well-known that if $V[G]$ is a generic extension by adding a Cohen real, then the set $\{r \in V[G]: r$ is Cohen generic over $V\}$ has measure zero. On the other hand, if $V[G]$ is a generic ...
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0answers
101 views

What is the consistency strength of this kind of reflection principle?

If $\psi$ is a predicate that is definable in $FOL(\in,=)$ by a formula from parameters in $V$, then if $\psi$ hold of the class $\small ORD$ of all ordinals in $V$, then the class of all cardinals in ...
3
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0answers
90 views

Embeddability into $\beta\omega$ and $\omega^*$

It is well known that under CH every totally-disconnected compact F-space of weight at most $\omega_1$ can be embedded into the remainder $\omega^*=\beta\omega\setminus\omega$ of the Cech-Stone ...
2
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1answer
95 views

Non-discrete $T_2$-space $(X,\tau)$ with $2^{|X|}$ retracts

If $(X,\tau)$ is a topological space, we call $A\subseteq X$ a retract if there is a continous map $r:X\to A$ such that $r(a) = a$ for all $a\in A$ (we assume $A$ to be endowed with the subspace ...
9
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1answer
174 views

Rothberger property for finite covers

Let us recall that a topological space $X$ has the Rothberger property if for any sequence $(\mathcal U_n)_{n\in\omega}$ of open covers of $X$ there exists a sequence $(U_n)_{n\in\omega}\in\prod_{n\...
1
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1answer
130 views

Descending almost-contained subsets of $\omega$ [on hold]

Let $A$ be an infinite subset of $\omega$ such that $\omega\setminus A$ is also infinite. Under the Continuum Hypothesis is there a sequence $(A_\alpha)_{\alpha<\omega_1}$ of subsets of $\omega$ ...
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243 views
+200

A strong plus-one hypothesis

To make this more easily readable, I'll start with the question and then give the explanation/motivation. Question. Is the following principle (or its weakening, with "for every real $r$" replaced ...
10
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3answers
681 views

Are inclusions “canonical” injections?

[Background: I asked a vague question the other day, but as a result of the answers, particularly Andrej Bauer's, I now have a precise question] Summary of question: the inclusions are a particularly ...
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0answers
72 views

Extending permutation models

We know that within ZFC any structure may be extended to a rigid structure. My question is whether this holds also for models. I mean: can a permutation model be extended to a standard model where the ...
3
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0answers
99 views

A question about the products of power set sigma algebras

Let $\kappa$ be the least cardinal for which the sigma algebra generated by $\{A \times B: A,B \subseteq \kappa\}$ does not contain every subset of $\kappa \times \kappa$. It is known that $\kappa$ is ...
3
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0answers
129 views

Compactification of Tychonoff spaces without full axiom of choice

If $X$ is a Tychonoff space, then using the Tychonoff theorem and thus the full axiom of choice, it follows that $X$ admits a Hausdorff compactification. My question is : what remains true if we do ...
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0answers
188 views

Sacks property for higher cardinals

It is well known, that the Sacks forcing has the property that for any $f: \omega \rightarrow \omega$ in generic extension, one can obtain $F:\omega \rightarrow [\omega]^{<\omega}$ in ground model, ...
13
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0answers
262 views

Covering S2 with great circles

Let $S_2$ be the unit 2-dimensional sphere. Is there a way to cover it with great circles such that each point on $S_2$ has 1 or 2 great circles that go through it?
7
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1answer
380 views

Destroying Suslin, nothing special

Recall that a tree on $\omega_1$ is called Suslin if every chain and antichain are countable. If every level is countable and there are no cofinal branches, then it is called Aronszajn (in particular ...
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0answers
129 views

Is “ZF+ V=L” an upper limit theory for cardinal decidability (per its strength)?

{EDIT: this posting has been edited, the additional text is in italics} If $\varphi, \psi$ are two parameter free formulas in the language of set theory $T$ such that there is a theorem of $T$ that ...
3
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0answers
183 views

Formal foundations done properly [closed]

I would want to do mathematics properly, so that the proofs of results can be trusted on instead of them being just suggestions on which results could perhaps apply. This means formulating the math in ...
8
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1answer
126 views

Is there a minimal extension of $L$ that is not a forcing extension?

It's well known that Sacks forcing constructs a real of minimal constructability degree, i.e. a real $x$ such that for any $y\in L(x) \setminus L$, $L(y) = L(x)$. It's also well known that certain ...
7
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1answer
139 views

Does $\mathsf{MA}^+(\sigma-{\rm closed})$ imply there are no Kurepa Trees?

The question in the title is somewhat self contained but let me make some definitions and remarks to clarify. Recall that $\mathsf{MA}^+(\sigma-{\rm closed})$ is the statement that if $\mathbb P$ is ...
4
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0answers
282 views

The surreal numbers under a change of universe

Suppose we start with a model $\mathcal{M}$ of $ZFC$ (or $GBC$ or $MK$ if you prefer), and let $N_0^\mathcal{M}$ denote the surreals in $\mathcal{M}$. If we add some large cardinal assumptions $\{\...
5
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1answer
324 views

Base zero-dimensional spaces

Definition. A zero-dimensional topological space $X$ is called base zero-dimensional if for any base $\mathcal B$ of the topology that consists of closed-and-open sets in $X$, any open cover $\mathcal ...
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3answers
898 views

How many sigma algebras exist on $\mathbb{R}$?

On the one hand, there are at least $2^\mathfrak{c}$ sigma algebras on $\mathbb{R}$: one can take any subset $A$ of $\mathbb{R}$ and consider a sigma algebra $\{\emptyset, A, \bar A, \mathbb{R}\}$ On ...
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1answer
140 views

What is the strength of this strict constructible iterative hierarchy?

Begin with the empty set then construct the set of the empty set, then construct the set of all subsets of the latter set, then at each level of construction construct the next level as the set all ...
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1answer
178 views

Is Replacement motivated by ranked iterative conception of sets?

When one reads the Wikipedia article on the Von Neumann Universe, one gets the impression that the idea of "the cumulative hierarchy" serves as a motivation for $ZFC$. I don't see really how this is ...
16
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1answer
468 views

Is there a pair of non-isomorphic structures each of which is isomorphic to an ultrapower of the other?

Does there exist a pair of non-isomorphic structures $\mathfrak{A}$ and $\mathfrak{B}$ as well as sets $I$ and $J$ and ultrafilters $\mathcal{U}$ on $I$ and $\mathcal{F}$ on $J$ such that $\mathfrak{A}...
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0answers
127 views

Reference request: destroying saturation at an inaccessible?

An ideal $I$ on $P(\kappa)$ is said to be $\kappa^+$-saturated if there is no sequence $\langle S_\alpha \mid \alpha<\kappa^+\rangle$ of $I^+$ sets such that $\alpha<\beta<\kappa^+\implies S_\...
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3answers
338 views

How do we formally construct the successor universe $\mathscr{U}^+$ of a universe $\mathscr{U}$ in $\mathsf{ZFC}$?

A set $\mathscr{U}$ is a universe if the following conditions are met: For any $x \in \mathscr{U}$ we have $x \subseteq \mathscr{U}$ For any $x,y \in \mathscr{U}$ we have $\{x,y\} \in \mathscr{...
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1answer
354 views

Translating Grothendieck axiom UB into ZFC

In SGA 4, Grothendieck introduced a set-theoretic device called a Grothendieck universe. He and his collaborators worked in Bourbaki set theory, which is practically similar to $\mathsf{ZFC}$ but ...
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0answers
54 views

Can MK be interpreted in a class theory about an abstract hierarchy principle + an accessibility principle?

The following is a first order MONOSORTED class theory, that is primarily motivated by an abstract hierarchy principle. It extends first order logic with equality, its language has only two extra-...
6
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1answer
334 views

Action of infinite symmetric groups on iterated power sets

Let $X$ be an infinite set, and $k \ge 1$ be a natural number. We work without the axiom of choice. Let $G_0$ be the full symmetric group on $X$, and let $G_1$ be the full symmetric group on ${\cal ...
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1answer
117 views

Enumeration Hierarchies

A set $M$ of real values is said to be enumerable if there is a bijection between the elements of $M$ the elements of $\mathbb{N}$. That definition does however not impose any restrictions on the ...
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0answers
319 views

What happens when you iterate Cohen reals?

There are a few classical theorems in set theory: The finite support iteration of ccc forcing is ccc. The countable support iteration of proper forcing is proper. The finite support iteration of ...
10
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1answer
316 views

Looking for “Set theory for a small universe” by Ketonen

In the paper Partition theorems for systems of finite subsets of integers, Pudlák and Rödl show a Ramsey-type result. The main feature of this result is that the sizes of sets in such systems are not ...
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0answers
143 views

stationary reflection in $[\kappa]^\omega$

It is well-known that the following reflection principle is consistent relative to a supercompact: For all $\kappa \geq \omega_2$ and all stationary $S \subseteq [\kappa]^\omega$, there is $X \...
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142 views

Singular strong generator

Let $V$ be a model of $\mathrm{ZFC}$ and let $j\colon V \to M$ be an elementary embedding with a critical point $\kappa$ ($M$ is transitive). A strong generator of $j$ is an ordinal $\zeta \geq \kappa$...
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1answer
141 views

A proof of recontruction of Sacks generic filter from it's Sacks real (M[G] = M[f])

Given the Sacks forcing $ (\mathbb{S} = \{T \subset 2^{<\omega} : T \text{ is perfect}\},\subset) $ and $G$ generic over M, we have $f = \bigcup \bigcap G = \bigcup_{T \in G}stem(G) $ a path ...
11
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1answer
214 views

The Parovichenko cardinal, is it equal to $\max\{\aleph_2,\mathfrak p\}$?

Let us define the Parovichenko cardinal $\mathfrak{P}$ as the largest cardinal $\kappa$ such that each compact Hausdorff space $K$ of weight $w(K)<\kappa$ is the continuous image of the remainder $...
3
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1answer
101 views

Function $f:\kappa\to\alpha$ with small fibers where $\alpha\in\kappa$

Is it consistent in $\mathsf{ZF}$ that there is an infinite cardinal $\kappa$, cardinals $\alpha, \beta\in\kappa$ and a function $f:\kappa\to \alpha$ such that for each $x\in\alpha$ there is an ...
14
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0answers
276 views

O-minimality and forcing

It is well-known that the structure $(\mathbb{R}, +, \cdot, <, 0, 1)$ is an o-minimal structure and hence the set of integers $\mathbb{Z}$ is not definable in it. In an ongoing project with Will ...
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2answers
134 views

Are Regularity schema and $\in$-induction schema equivalent in intuitionistic logic?

In posting "Does Regularity schema imply $\in$-induction when added to first order Zermelo set theor?" the answer was that they are equivalent in classical first order logic with membership "$\in$". ...
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0answers
393 views

Is it still an open problem whether $\mathbb{R}^\omega$ is normal in the box topology?

On page 205 of his Topology book, James Munkres makes an interesting remark: It is not known whether $\mathbb{R}^\omega$ is normal in the box topology. Mary-Ellen Rudin has shown that the answer ...