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.
5,271
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When does the cardinality of a set equal the cardinality of an element of $V_\lambda$ for $\lambda$ being a limit ordinal?
Consider the following proposition.
Proposition: let $\lambda$ be a limit ordinal and $V$ be the cumulative hierarchy starting with the null set, and $S$ be a set with
$\vert S\vert<\vert V_\...
- 303
23
votes
3
answers
3k
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Half Cantor-Bernstein without choice
I had a discussion with one of my teachers the other day, which boiled to the following question:
Assume ZF. Let $A,B$ be sets such that there exist $f\colon A\to B$ which is injective and $g\colon A\...
- 60.4k
7
votes
1
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556
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What can be the measure of a Vitali set?
Suppose the continuum $\mathfrak{c}$ is real-valued measurable, i.e., there exists a countably additive probabilistic measure on $\mathfrak{c}$ that measures all subsets. Then by the construction on p....
- 4,032
1
vote
1
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248
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Adjunction, infinity and hereditarily finite sets
Is
$$\mathrm{U}_{\omega}=\Big\{x\mid\forall z\Big(\big(\emptyset\in z\wedge \forall u, v\;(u,v\in z\rightarrow\{w\mid w\in u\vee w=v\}\in z)\big)\rightarrow x\in z\Big)\Big\}$$
identical with the set $...
- 3,292
10
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5
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2k
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Extracting a common convergent indexing from an uncountable family of sequences
Let $\mathcal{A}$ be some uncountable index set and $X$ be some separable reflexive Banach space.
For each $\alpha \in \mathcal{A}$, let
\begin{equation}
\{ x_n^{\alpha} \}_{n=1}^\infty
\end{equation}
...
- 211
5
votes
1
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414
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Is there a class choice principle over MK that is equivalent to class well ordering over MK?
$\sf MKCWO$ is the theory obtained by adding a new primitive binary relation $\prec$ to the signature of $\sf MK$ and axiomatize that $\prec$ is a well order on classes, that is:
$\textbf{Transitive:}...
- 9,276
7
votes
2
answers
410
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On the existence of a real which is not set-generic over $L$
Recall that a real $r$ is set-generic over $L$ if there is a constructible forcing notion $\mathbb{P}$ and some $L$-generic filter $G\subset\mathbb{P}$ such that $r \in L[G]$.
I know that Jensen's ...
5
votes
1
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426
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Cardinality of subset of P(A) that contains all the sets with lesser cardinality than A [closed]
$\DeclareMathOperator\Card{Card}$Let $A$ be an infinite set, and let $B=\{X\subset A \mid \Card(X)<\Card(A)\}$. Can it be proven that $\Card(A)=\Card(B)$?
- 225k
3
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2
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196
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Existence of trees with height $\omega$, size $\aleph_1$ and $\aleph_2$ maximal branches
Definition A tree means a set-theoretic tree, that is a poset $(T,<)$ so that for each $x\in T$, the set $\{y\in T\mid y<x\}$ is well-ordered.
Question: I would like to know if it is consistent ...
- 60.4k
4
votes
3
answers
366
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Hyperarithmetically least elements in $\Pi^1_1$ sets
My question is: Do we have a hyperarithmetically $\le_H$-least real in any $\Pi^1_1$ set? That is
Question. Suppose that $A$ is a non-empty $\Pi^1_1$ set. Then can we find a real $a\in A$ such that $...
CommunityBot
- 1
5
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4
answers
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How short can we state the Axiom of Choice?
How short can we state a principle which is equivalent with the Axiom of Choice under $ZF$? The principle should be a sentence in the language of set theory with only $\in$ and$=$ as extralogical ...
- 4,618
10
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2
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577
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Is diamond consistent with 2nd order PA?
If $T$ is a theorem of ZF which says something only about reals, then one may want to prove $T$ using a theory like 2nd order PA or related theories like ZFC$^-$ or GBC$^-$ (minus accounts for the ...
- 225k
12
votes
1
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472
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Is the Tukey order well-founded
Consider the Tukey order restricted to directed orders of the form $(U,\supseteq)$, where $U$ is an ultrafilter on $\omega$. It is defined as follows:
For two ultrafilters $U,W$ on $\omega$, we say ...
- 7,674
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votes
1
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389
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Precipitous ideal and inner model
Assume $\kappa$ is measurable, $U$ is its unique normal measure, $V=L[U]$. We levy collapse $\kappa$ to make it become $\omega_1$.
If we don't have the inner model condition, then we only know that $\...
- 8,752
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3
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671
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Is "the purely probabilistic version of Freiling's axiom of symmetry" disprovable in ZFC?
I'm trying to pinpoint the "intuitive argument" for Freiling's Axiom of Symmetry. It's meant to be a "probabilistic" argument, so thinking about what seems to me to be the ...
- 4,032
5
votes
0
answers
131
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When does an iteration not add functions $\eta\to V$ at the final stage?
I am interested in better understanding the following property:
Let us say that an iteration of forcings $\langle\mathbb{P}_\alpha,\dot{\mathbb{Q}}_\beta\mid\alpha\leq\gamma,\beta<\gamma\rangle$ is ...
- 1,251
4
votes
2
answers
196
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Does $\mathsf{ZF}$ prove $\operatorname{Col}(\lambda,\kappa)$ preserves cardinals below $\lambda$?
Let $\lambda<\kappa$ be cardinals and consider the forcing $\operatorname{Col}(\lambda,\kappa)$ adding a generic surjection $\lambda\to\kappa$. More formally, $\operatorname{Col}(\lambda,\kappa)$ ...
- 38.2k
3
votes
0
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113
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Is it consistent to have an infinite antitone sequence of elementary embeddings such that the involved models include iterated sharps?
$\DeclareMathOperator\crit{crit}$Background essays (the material I've tried to understand in leading up to this question):
Daghighi, et. al. [2014], "The foundation axiom and elementary self-...
- 60.4k
17
votes
3
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2k
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Why is an internal proof of consistency satisfactory for some systems?
I've only a shallow understanding of the relevant theory, but I don't understand how any internal proof of consistency is in any way satisfactory (even for systems that are so weak Gödel's ...
- 60.4k
15
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5
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In what sense does the sentence $\operatorname{con}(\mathsf{PA})$ "say" that $\mathsf{PA}$ is consistent?
It seems common amongst logicians to think of "truth" as being relative to a particular structure. Consider, for instance, the first-order theory of groups. The sentence $\forall x\forall y(...
- 60.4k
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1
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How to prove that increasing the number of constant symbols of a first-order logic by the number of formulas keeps the number of formulas the same [closed]
Let $S$ be a set of theory symbols for a first-order logic, and let $C$ be a set of constant symbols in $S$ such that $|C| = |L(S)|$, where $L(S)$ is the set of all formulas generated by $S$ in the ...
- 1,251
3
votes
2
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579
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Non-simple groups $G$ with only non-trivial quotient isomorphic to $G$
If $G$ is a group such that every non-trivial subgroup is isomorphic to $G$ itself, then $G= \mathbb{Z}$ is the only infinite group with that property (up to isomorphism). Amongst the finite groups we ...
- 45.5k
2
votes
0
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325
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What's the definition of a mouse in Mitchell's handbook article "the covering lemma"?
In the book "handbook of set theory", in the chapter "the covering lemma", definition 3.24, Mitchell defines what is mouse.
However he did not give any definition of $\mathcal{U}_\...
- 3,969
11
votes
2
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2k
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Is GCH useful in proving theorems?
By GCH I mean the Generalized Continuum Hypothesis. Let me give some context before presenting my question.
When the axiom of choice was introduced by Zermelo in his 1904 proof of Well-Ordering ...
- 15.4k
4
votes
1
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357
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Does limitation of size imply axiom of powerset in Morse-Kelly if the generalized continuum hypothesis is included in Morse-Kelley set theory?
Suppose Morse-Kelley set theory consists of class comprehension, class foundation, class extension, axiom of infinity , limitation of size, and the general continuum hypothesis. Can the axiom of ...
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2
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Writing a function on $\mathbb{R}$ as a sum of two injections
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. It is well-known that, using transfinite recursion with a well-ordering of $\mathbb{R}$, one can construct two injective functions $g,h: \...
- 225k
6
votes
5
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2k
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Standard models of N and R: An Alice/Bob approach
This is a question about a comment in a recent publication by Roman
Kossak. Kossak wrote:
"Nonstandardness in set theory has a different nature. In
arithmetic, there is one intended object of ...
- 2,321
47
votes
10
answers
5k
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What is the most "concrete-feeling" equivalent formulation of the Continuum Hypothesis that you can think of?
There are many equivalent formulations of the Continuum Hypothesis, but I think the most standard one is that
there is no infinite cardinality lying strictly between the cardinality of the natural ...
- 4,123
23
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4
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3k
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A Löwenheim–Skolem–Tarski-like property
I am interested in the following Löwenheim–Skolem–Tarski-like property.
Given a cardinal $\kappa$, what (if any) is a property $\phi(x)$ such that if $\phi(\kappa)$ holds, then we can prove the ...
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15
votes
2
answers
873
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Measuring big stuff
Often during informal discussion with colleagues, the following pattern emerges when we are stuck trying to prove a theorem about $x \in X$.
A: "let's assume this hypothesis $H$ on $x$"
B: "most ...
- 225k
7
votes
1
answer
429
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Can we have more malleable proper classes without sacrificing conservativity?
NBG is a conservative extension of ZFC that includes a concept of "proper class." Now I like the conservativity, since it means anytime I want to prove something in ZFC, I am free to work in NBG. ...
- 225k
6
votes
1
answer
604
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Forcing in Ackermann's Set Theory
How would one do forcing in Ackermann's set theory? C. Alkor published an article entitled "Forcing in Ackermanns Mengenlehre" (Zeitshcr. f. math. Logik und Grundlagen d. Math., Bd. 25,8.265-286 (...
- 225k
-1
votes
1
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235
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Is $\text{Sym}(\omega)/\text{(fin)}$ embeddable in $\text{Sym}(\omega)$? [duplicate]
Let $\omega$ denote the set of natural numbers, let $\text{Sym}(\omega)$ be the collection of bijections $\psi:\omega\to\omega$, and let $\text{(fin)}$ be the set of members of $\text{Sym}(\omega)$ ...
3
votes
1
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Is Morse-Kelley set theory with Class Choice bi-interpretable with itself after removing Extensionality for classes?
Let $\sf MKCC$ stand for Morse-Kelley set theory with Class Choice. And let this theory be precisely $\sf MK$ with a binary primitive symbol $\prec$ added to its language, and the following axioms ...
- 225k
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votes
3
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Are all free ultrafilters 'the same' in some sense?
Consider the set of ultrafilters $\beta(\mathbb N)$ on $\mathbb N$.
Any function $f\colon\mathbb N\to\mathbb N$ extends to a function $\beta f\colon \beta \mathbb N \to \beta\mathbb N$. We say that ...
- 4,618
6
votes
3
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307
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Group such that factors in any product-decomposition are reducible
Motivation. Let us call a group $G = (G,\cdot)$ (product-)reducible if there are groups $H_1, H_2$, each having more than $1$ element, with $G \cong H_1\times H_2$. Otherwise, $G$ is said to be ...
- 60.4k
35
votes
3
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5k
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Counterintuitive consequences of the Axiom of Determinacy?
I just read Dr Strangechoice's explanation that if all subsets of the real numbers are Lebesgue measurable, then you can partition $2^\omega$ into more than $2^\omega$ many pairwise disjoint nonempty ...
- 4,618
13
votes
1
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Are Berkeley cardinals easier to refute in ZFC than Reinhardt cardinals?
Kunen showed that Reinhardt cardinals are inconsistent in ZFC. But his proof is a bit technical for a non-set-theorist to follow. Berkeley cardinals are stronger than Reinhardt cardinals. You can ...
- 7,674
2
votes
1
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74
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Finite pair-splitting family of $\mathbb{N}$
This is a kind of "dual" of an older question.
Is there a finite family ${\frak F}\subseteq {\cal P}(\mathbb{N})$ such that for all $a\neq b\in\mathbb{N}$ there is $S\in{\frak F}$ with $|S\...
- 118k
2
votes
1
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196
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Finite $k$-set-respecting splitting of $\mathbb{N}$
Motivation. My sons participated in a large football tournament recently; everyone wanted to be in a team with everyone else at least once. Tricky!
Formulation of the question. For any positive ...
- 11.9k
7
votes
0
answers
181
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Strengthening Determinacy in constructive set theory?
Recall how games work. Let $X$ be a set (the "game space") and $\alpha$ an ordinal (the "game clock"). Alice and Bob take turns naming elements of $X$. We write them down in order ...
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votes
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Defining the standard model of PA so that a space alien could understand
First, some context. In one of the comments to an answer to the recent question Why not adopt the constructibility axiom V=L? I was directed to some papers of Nik Weaver at this link, on ...
- 15.4k
3
votes
2
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328
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Are there $2^{\aleph_0}$ pairwise non-isomorphic countable groups containing every finite group?
Let us call a group $(G,\cdot)$ finitarily complete if $G$ is countable, and every finite group is isomorphic to a subgroup of $(G,\cdot)$.
Is there a collection of $2^{\aleph_0}$ pairwise non-...
- 54.4k
8
votes
1
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1k
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Ill-founded models of set theory with well-founded ordinals
Let $(\mathcal{M},E)$ be an internally non-well-founded model of set theory i.e of $ZFC^{\neg f}=ZFC\setminus \mathrm{foundation}+\neg \mathrm{foundation}$, then $\mathcal{M}$ includes an infinite ...
- 17.1k
2
votes
1
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582
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"Potency set" for power set?
Cross-posted at HSM.
Has the term "potency set" been used in English language mathematics for power set, and, if so, what are good references?
It is relevant that for historical reasons, &...
- 11.4k
12
votes
0
answers
347
views
Can Friedman's property fail at or above a supercompact cardinal?
If $\kappa>\omega_1$ is a regular cardinal, $FP_\kappa$ is the assertion that every stationary subset of $\kappa$ consisting of ordinals of countable cofinality has a closed subset of order type $\...
- 435
35
votes
3
answers
2k
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How much choice is needed to show that formally real fields can be ordered?
Background: a field is formally real if -1 is not a sum of squares of elements in that field. An ordering on a field is a linear ordering which is (in exactly the sense that you would guess if you ...
21
votes
2
answers
1k
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Gently changing measure
This question was asked and bountied on MSE without answer, so I'm porting it here:
There's an easy way to change the measure of a set of reals by moving to a larger universe: simply make $\mathbb{R}$...
- 4,032
4
votes
1
answer
121
views
$\aleph_0$-uniform non-bipartite linear hypergraph
A hypergraph $H= (V,E)$ is said to be bipartite if there is $S\subseteq V$ such that for all $e\in E$ with $|e|>1$ we have $$S\cap e\neq \emptyset\neq (e\setminus S).$$ A hypergraph is said to be ...
- 11.9k
80
votes
5
answers
6k
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How do the compact Hausdorff topologies sit in the lattice of all topologies on a set?
This question is about the space of all topologies on a
fixed set X. We may order the topologies by refinement, so
that τ ≤ σ just in case every τ open set is open in σ.
...
- 1,861