# 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.

5,273
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### Weak extender models for supercompactness without choice

Assume ZFC and a supercompact cardinal $κ$. Is it consistent that there is a weak extender model $N⊨\text{ZF}$ for supercompactness of $κ$ such that the axiom of choice and well-ordering of $P_κ(λ)^N$...

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36
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### Diagrammatic representation of sets as irregular plane figures

I am trying to find the earliest use of plane diagrams of various shapes for representing sets. For example, this is snapshot is from the book called Some Modern Mathematics for Physicists and Other ...

4
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1
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117
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### What is the extent of a $\Sigma$-product of a (uncountable) power of a (countable) discrete space?

Recall that a $\Sigma$-product of a family of spaces $\{X_s:s\in S\}$ with a base point $a=(a_s)\in \prod_{s\in S} X_s$ is the subspace $$\Sigma(a)=\{x\in \prod_{s\in S} X_s: |\{s\in S:a_s\neq x_s\}|\...

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### Set-theoretic trees with ordering between siblings

In set-theoretic trees, we have the binary relation $<_T$ (which defines the "ancestor-descendant" ordering). This relation is partial, as children are incomparable in that ordering.
...

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113
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### Refinement-minimal intersecting covers

Motivation. Yesterday I was sitting idly in the train, contemplating the train network. I noticed that a lot of lines (not all) intersected, and some pairs of lines intersected in quite a few stations....

2
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1
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91
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### A problem with a $\Pi_1$ formula of the Lévy hierarchy

Let $M\equiv N$ means that $(M,\in_M)$ and $(N,\in_N)$ satisfy the same sentences of the language of set theory, with $\in_M$ and $\in_N$ being the standard membership relation restricted to $M\times ...

12
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2
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887
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### Why is inner model theory evidence for consistency of large cardinals?

I want to understand the viewpoint that existence of canonical inner model for a large cardinal notion is strong evidence for its consistency. For example, below is Trevor Wilson's answer to What &...

2
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1
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178
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### Can there be a minimal remote cardinal?

Working in $\sf Z- Reg.$ we can have sets bigger than every well-founded set, let's label such sets as "remote". Now, suppose we'll add the axiom of existence of transitive closures, and the ...

4
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91
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### Chromatic numbers realised by almost disjoint subsets of $\omega$

If $H=(V,E)$ is a hypergraph then the chromatic number $\chi(H)$ is defined to be the least cardinal $\kappa \leq |V|$ such that there is a map $c:V\to \kappa$ such that for all $e\in E$ with $|e| \...

4
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396
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### Can we have an axiom that refers to itself and the prior axioms of the theory it is an axiom of?

I know that this question is little bit imprecise, I'll try to present it in the best I can.
Can one have an axiom which is self referential with respect to itself and the theory in which it belongs?
...

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46
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### Is stratified Z - Infinity + there is a set as big as its powerset, consistent if NF is consistent?

The question of consistency of $\sf NF$ can be seen to be equivalent to the question of whether the theory "Stratified $\sf Z$ - Regularity - Infinity + There exists a set as big as its powerset&...

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156
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### Cardinality of maximal diverse families

Let $\kappa\geq \aleph_0$ be a cardinal. We say a collection ${\cal E} \subseteq {\cal P}(\kappa)$ is diverse if $|(A \setminus B) \cup (B \setminus A)| = \kappa$ whenever $A\neq B\in {\cal E}$. A ...

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### Can existence of uncountable sets be proved in a ZFC variant with mild definability restriction?

Starting with ZF[C], if we restrict Separation and Replacement to parameter free versions from Parameter free definable sets, re-write Infinity asserting existence of $\omega$. Add to it axioms of ...

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### Consistency of definability beyond P(Ord) in ZF

Is it consistent with ZF that the satisfaction relation of $L(P(Ord))$ is $Δ^V_2$ definable? More generally, is it consistent with ZF that there is a $Δ^V_2$ formula (taking $α$ as a parameter) that ...

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### Is the derived category of sheaves localised at pointwise homotopy equivalences locally small?

In order to define the cup and cross products in sheaf cohomology, Iversen makes computations in an intermediate derived category. If $K(X;k)$ is the triangulated category of cochain complexes of ...

6
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### Strengthening of a classical set mapping theorem of Lázár

We are writing a paper in which we need to use the following theorem to prove that a certain poset satisfies c.c.c. As far as I remember I learned this theorem from András Hajnal.
Theorem 1: If $\...

5
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1
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263
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### Would strengthening Foundation and Choice in NBG, make it equi-consistent with MK?

This is a follow up to an earlier question about strengthening of foundation in relation to proving the consistency of ZFC. It was shown that it would achieve that, but it may fail short of MK. Here, ...

5
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1
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200
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### Long chains of Dedekind finite sets

This is a variation on this question with amorphous cardinals replaced with dedekind finite sets.
Dedekind finite sets are sets that have no countable subset, and it is well known that this is a ...

2
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1
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307
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### Transversal of $\mathbb{N}\times\mathbb{N}$

Motivation. I am trying to make an interesting infinite version out of this fascinating problem from the Russian mathematical olympiad:
There are $c$ flavours of cookies, we are given $n$ cookies of ...

6
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2
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469
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### Is strengthening Foundation in NBG sufficient to make it prove Con(ZFC)?

Can $\sf NBG$ class theory prove the foundation scheme:
Foundation schema: if $\Phi(X)$ is a formula in which "$X$" occurs free and only free, and in which "$Y$" doesn't occur, ...

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### What happens if we restrict inputs in Separation and Replacement axioms to definable sets?

If we replace the axiom of Foundation by
Foundation schema: if $\varphi(x)$ is a formula in which "$x$" occurs free and only free, and in which "$y$" doesn't occur, whose free ...

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1
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532
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### Long chains of amorphous cardinalities

An amorphous set is an infinite set that cannot be partitioned into 2 infinite subsets. An amorphous cardinality is the cardinality of an amorphous set. Working in $\sf ZF$, it is consistent that ...

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1
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280
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### Do precipitous ideals "always" come from collapsing?

It's well-known that if $\kappa$ is a measurable cardinal, then there is a poset $\mathbb{P}$ that forces $\kappa$ to carry a precipitous ideal.
Suppose that $\omega_1$ carries a preciptous ideal $I$.
...

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87
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### Uniformization and functions on Turing degrees

Assuming Martin's Conjecture on functions between Turing degrees, is AD + DC consistent with existence of an $f:\mathcal{D}_t → \mathcal{D}_t$ of rank $Θ$ ?
$\mathcal{D}_t$ is the set of Turing ...

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600
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### Can uncountable sets be proved to exist in this variant of ZFC with definability restrictions?

If we restrict all parameters in the set building axioms of $\sf ZFC$ to definable sets, would the celebrated Cantor's theorem still apply? Can existence of uncountable sets be proven at all?
The set ...

11
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1
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392
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### Building the real from Dedekind finite sets

It is well known that the real numbers can be countable union of countable sets by starting with GCH and taking a finite support permutations while collapsing all of $\aleph_n$ for natural $n$.
The ...

10
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1
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468
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### Does proper forcing preserve properness under PFA?

I'm interested in forcing classes $\Gamma$ which preserve membership in themselves, i.e. for all posets $\mathbb{P}, \mathbb{Q}\in \Gamma$, we have $\Vdash_{\mathbb{P}}\check{\mathbb{Q}}\in\Gamma$. ...

6
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376
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### Second-order ordinal definability

As is familiar, a set $S$ is ordinal definable ($S \in \mathsf{OD}$) just in case there exists a formula $\varphi(x, \vec{z})$ of the first-order language of set theory with parameters $\vec{z} = z_1, ...

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336
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### Consistency strength of strongly compact cardinal

Where can I find a proof that strongly compact cardinal has higher consistency strength than Woodin cardinal, or even just strong? Recall that a strongly compact cardinal itself may not be strong, ...

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147
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### What is the strength of adding this de-schematizing inference rule to Ackermann's set theory?

Language: first order logic with equality, membership, and a constant symbol $W$.
Axioms:
Extensionality: $\forall z \, (z \in x \leftrightarrow z \in y) \to x=y$
Comprehension: $\exists x \forall y \,...

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86
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### Can ur-elements be used as/to construct infinitesimals?

Background material:
Truss[95], "The structure of amorphous sets."
Harrison-Trainor and Kulshreshtha[22], "The Logic of Cardinality Comparison Without the Axiom of Choice."
...

2
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323
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### Convergence of distance

Consider these sets
$$
A\equiv \bigcap_{\delta>0} \liminf_{n\rightarrow \infty} \{x \in X: d(p_n, [\ell(x), u(x)])\leq \delta\}
$$
$$
C_n(L_n)\equiv \{x \in X: d(p_n, [\ell(x), u(x)])=0\}
$$
where:
...

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0
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### Is this modified H. Friedman theory bi-interpretable with ZFC + ORD is Mahlo?

The following theory is a modification of Harvey Friedman $\sf K(W)$ theory.
Language: first order logic with equality, membership, and a constant symbol $W$.
Axioms:
Extensionality: $\forall z \, (z ...

6
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113
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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 ...

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181
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### CH vs Not CH, What is the Consequence?

EDIT: Altered (3) to say that $\beth_1$ can't be $\aleph_{\omega}$ or $\aleph_0$ but removed statement that it must be less than $\aleph_{\omega}$.
Let us assume ZFC. We now consider 2 transfinite ...

6
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1
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161
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### An iteration of proper forcing without proper iterands

Famously, a countable support iteration with proper iterands is again proper. This is mostly stated as follows: Suppose $(\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\alpha})_{\alpha<\lambda}$ is a ...

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### Friedman's proof of covering lemma for $L$

There is a two-page proof of the covering lemma for $L$ using $\Sigma_n$ substructures (Theorem 3.10) in Sy Friedman's Fine Structure and Class Forcing, compared to the proof that spans about twenty ...

6
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139
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### Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective?

Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective ?

2
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2
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197
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### Name for a certain type of cardinal

I'm not a set-theorist, but I hope this question is appropriate. This is just a question about names:
Fix a cardinal $\lambda$. I'd like to know if there is a name for regular cardinals $\kappa$ such ...

12
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1
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704
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### Can proper classes have different sizes?

I'm presently working in a non-ZF set theory, where there are proper classes. (Think MK or VNBG.) And I'm interested in how to think about the possibility (or impossibility) of proper classes with ...

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0
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### How does the cardinality of a set and its powerset compare in the hereditarily rank-concordant constructible world?

Working in the constructible universe "$L$", if we define two kinds of ranks for any constructible set $x$, one being the ordinal index of the first $L_\alpha$ where $x$ appears as a subset ...

5
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1
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209
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### Is the partition tiling relation transitive?

The following is motivated by an (as of yet) unanswered question on optimal colorings of graphs. I am convinced that the question below has a positive answer in $\newcommand{\ZF}{{\sf (ZF)}}\ZF$, but ...

4
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1
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197
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### Simplified method of building an Aronszajn tree

There is a very interesting method to build an Aronszajn tree in Judith Roitman's "Introduction to Modern Set Theory", on pages 100-102. In short, we build a tree $T$ in which the nodes are ...

8
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187
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### A reference for forcing projections

The idea of a projection $\pi\colon\mathbb{Q}\to\mathbb{P}$ of forcing notions is something like a combinatorial stand-in for the fact that forcing with $\mathbb{Q}$ produces a generic for $\mathbb{P}$...

3
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1
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171
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### Is there a metric separable space with the following properties...?

Let $\omega_1<\mathfrak{q}_0$ where $\mathfrak{q}_0:=\min\{|Y|:Y\subseteq \mathbb{R}$, $Y$ is not a $Q$-space$\}$.
Is there a metric separable space $X$ with the following properties:
$|X|\geq\...

6
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0
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219
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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 $\mathcal{L}$ ...

5
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1
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243
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### How far does the intuition "non-stationary = null sets (of certain measure)" go?; what position do non-stationary ideals take in measure theory?

When I was taught undergraduate set theory I was told that the idea for club sets is that they are "of full measure" and the non-stationary sets are "of null measure". (There were ...

2
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### (MK) universal class = von Neumann universe?

I am studying MK set theory. By the axiom schema of class comprehension, which roughly states that
Given a monadic predicate $\phi$ of MK, then there is a class $C = \{x: \phi(x)\}$,
the universal ...

11
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1
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207
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### Is ground model $\Sigma_n$ truth $\Sigma_n$-definable in every forcing extension?

It is well-known (and quite useful) that for any formula $\phi$, forcing poset $\mathbb{P}$, $p\in\mathbb{P}$, and $\mathbb{P}$-name $\dot{a}$, $p\Vdash_{\mathbb{P}}\phi(\dot{a})$ is definable as a ...

7
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2
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620
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### Ideals generated by Turing independent sets

Recall that $X \subseteq 2^{\omega}$ is Turing independent if no $y \in X$ is computable from the Turing join of any finite subset of $X \setminus \{y\}$.
Question 1. Can we construct a Turing ...