# 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,...

3,315 questions

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### 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 $|...

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**1**answer

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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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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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 $V$ ...

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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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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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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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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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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 ...

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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 ...

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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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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 ...

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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 ...

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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 ...

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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\...

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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

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### 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 ...

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**3**answers

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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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 ...

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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 ...

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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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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, ...

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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?

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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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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 ...

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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 ...

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**1**answer

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 ...

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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 ...

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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 $\{\...

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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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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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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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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 ...

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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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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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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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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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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-...

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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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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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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 ...

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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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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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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 ...

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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 $...

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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 ...

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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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**2**answers

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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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 ...