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.
1,112 questions
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Do these ordinals exist?
Given an ordinal $\alpha$, I define $F_{n}(\alpha)$ as follows:
$F_0(\alpha)=\alpha$
$F_{n+1}(\alpha)$ is the smallest $\beta$ such that no first-order $\phi$ in the
language of $\{\in\}$ has $(\...
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Which model is the minimal pointwise definable model of $\sf ZFC$?
Is the minimal transitive model of $\sf ZFC$ pointwise definable?
If not, then what is the minimal pointwise definable model of $\sf ZFC$?
Can we define that using Hamkins result for existence of ...
- 11.3k
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The universal algebra of a $\sigma$-algebra
I am searching for the 'dual' algebraic structure of a $\sigma$-algebra. The notion of duality is like in the case of the Boolean algebra and set algebra.
If $X$ is a set, the complement and ...
- 191
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Can we have this sequence where choice fails and returns?
Can we have a sequence of transitive sets $\langle\mathcal V_0, \mathcal V_1, \mathcal V_2,...\rangle$, all modeling $\sf ZF$, such that $\mathcal P(V_n) \subset \mathcal V_{n+1}$, and the cardinality ...
- 11.3k
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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, ...
- 11.3k
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274
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Forcing Martin's Axiom without cardinal arithmetic
We know that if $\kappa>2^{\aleph_0}$ and $\kappa^{<\kappa}=\kappa$, then there is a c.c.c. forcing which forces $\sf MA+2^{\aleph_0}=\kappa$. Traditionally, we even start with $\sf GCH$, which ...
- 39.8k
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505
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Upward Löwenheim–Skolem theorem for well-ordered models with/without measurable cardinals
Consider a complete first order theory $T$ whose language contains a binary predicate $\leq$. Assume that $T$ has an uncountable model that is well-ordered by $\leq$ so that this question isn't stupid ...
- 12.5k
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Smallest ordinal $\mu$ not embeddable in ${\cal P}(\omega)$
The motivation for this question is the startling fact that there is an order-preserving injective map (embedding) from $\mathbb{R}$ into ${\cal P}(\omega)$. (Think Dedekind cuts.) I am wondering how &...
- 48.9k
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When are all greater cardinals sharply greater?
Makkai and Paré introduced the following binary relation on regular cardinals: given $\kappa$ and $\lambda$, $\kappa \vartriangleleft \lambda$ (read, $\kappa$ is sharply less than $\lambda$) when $\...
- 15.9k
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Steinhaus number of a group
$\newcommand\Sn{\mathit{Sn}}$A subset $A$ of a group $X$ is called algebraic if $A=\{x\in X: a_0xa_1x\dotsm xa_n=1\}$ for some elements $a_0,a_1,\dotsc,a_n\in X$.
Let $\mathcal A_X$ be the family of ...
- 41.9k
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Is the statement that every convex complete metric space has midpoints equivalent to the axiom of dependent choice (DC)?
We say that a point $m$ is between points $p$ and $q$ of a metric space $(M, d)$ if $d(p, q) = d(p, m) + d(m, q)$ and $p ≠ m ≠ q$. Furthermore, if the equality $d(p, m) = d(m, q)$ holds for $m$, we ...
- 307
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Some questions about Ackermann set theory
In a comment on this site Andreas Blass stated:
"To fit this situation into my philosophical point of view, I'd say that what Ackermann's theory
calls proper classes are really certain sets. That ...
- 897
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Failure of Cantor-Bernstein for the Levy Collapse
Related to this question, is it possible to give an example of the failure of Cantor-Bernstein for complete embeddings of forcing notions involving the Levy Collapse $Col(\omega,<\kappa)$? Suppose ...
- 18.6k
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Parameter-free effective cardinals
In the paper "Effective cardinals and determinacy in third order arithmetic" by Juan Aguilera, effective cardinals is defined.
I'm curious about its little variation, parameter-free ...
- 1,490
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Independence over ZFC + CH
Acknowledging Woodin's result on $\Sigma^2_1$-absoluteness for forcing-models satisfying the continuum hypothesis (CH), it is natural to ask:
Are there examples of statements $\phi$ in "the usual ...
- 69
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Permutation on $\omega$ and Ramsey ultrafilter
let $\pi:\omega\to\omega$ be permutation and $\mathcal{F}$ is Ramsey selective ultrafilter on $\omega$. There are uncountable many increasing subsequences of $\pi$. Can one proof that one of them has ...
- 1,133
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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:}...
- 11.3k
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610
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Injection of the proper class of ordinals in every proper class
Is it possible to prove in the set theory NBG (with local choice but without global choice) that the proper class of ordinals injects in every proper class ?
- 2,655
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Is there an oracle that can compute something iff it is computable in every countable model that is equivalent to $(V, \in)$?
Let us work in Kelly-morse set theory, so we can talk about $V$. For some model $M=(\mathbb N, \in_M)$ that is elementary equivalent $(V, \in)$, we can have an oracle that corresponds to $(\mathbb N, \...
- 6,353
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Is this set theory used by Gandy first-order with signature $(\in, \lambda)$?
In On the Axiom of Extensionality, Part II, The Journal of Symbolic Logic, Vol. 24, No. 4 (Dec., 1959), https://doi.org/10.2307/2963897, pp. 287-300, R. O. Gandy shows that a class theory X ...
- 3,574
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Relation between ultrafilters ${\scr U}$ and ${\scr U} \otimes {\scr U}$ [closed]
If ${\scr U}$ and ${\scr V}$ are ultrafilters on non-empty sets $A$ and $B$ respectively, then the tensor product ${\scr U}\otimes{\scr V}$ is the following ultrafilter on $A\times B$:
$$\big\{X\...
- 48.9k
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470
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Large cardinals and mild extensions
It is known that for many large cardinals $\kappa$ (like weakly compact, measurable,...) , $\kappa$ remains large of the same type after forcings of size $<\kappa$. Now the questions are:
Question ...
user36136
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2
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528
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Cardinality of a set of countable connected Hausdorff spaces
It is a non-trivial result that there is a countable connected Hausdorff space.
Let ${\cal T}$ be a set of connected Hausdorff topologies on $\omega$ such that whenever $\tau_1\neq\tau_2\in {\cal T}$ ...
- 48.9k
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655
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$C^n$ And Forcing: Reading a Recent Paper By Kunen
While reading a recent paper by Kunen arxiv.org/abs/0912.3733, which deals with PFA and the existence of certain differentiable functions, (defined on all of $\mathbb{R}$) which map certain $\aleph_1$-...
- 1,615
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Subsystems of Peano arithmetic and incompleteness theorem
I think everyone is familiar with Goedel's incompleteness theorems. In particular they imply that PA (Peano arithmetic) can not prove its own consistency. Now my question is what is the largest ...
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Natural examples of $\bf\Sigma^0_3$ equivalence relations
I have been reading about Borel equivalence relations and I have noticed that while $\bf\Sigma^0_3$ equivalence relations are mentioned, there is a conspicuous absence of natural examples (other than $...
- 1,052
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Showing a filter with a certain property on the power set of $\mathbb{Z}$ is a one point filter
Let $\mathcal{P}_0(X)$ the Power set of $X$ without the empty set and let $\dot{x}:=\{A\subseteq X: x \in A\}$ the one point filter generated by $x$. Furthermore let $$ \mathcal{A} := \{ f \in X^{\...
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Large cardinals without the ambient set theory?
In an attempt to understand a bit better large cardinals, I have been thinking along the following lines, which could be summarized under the slogan
Talk about cardinals without the
(ambient) ...
- 7,853
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If $0^{\sharp}$ exists, then every uncountable cardinal in $V$ is as large as it can be in $L$.
According to Wikipedia, if $0^{\sharp}$ exists, then every uncountable cardinal in $V$ satisfies every large cardinal property in $L$ that can be realized in $L$, e.g. weak compactness, total ...
- 3,982
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What is the relationship between non-existence of those kinds of singular sets and AC?
Let's say a set $A$ is super-singular, if and only if, there exists a set $x$ such that $|A|=|\bigcup x|$ and $| A| > |x|$, and for each $y \in x$ we have: $ |y| \not > |x|$ .
A set $A$ is ...
- 11.3k
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How many well-orders of reals are there?
It's commonly known that the cardinality of the set of all well-orders on $\aleph_0$ is the continuum (correct me if I'm wrong plz). What about that of all well-orders on $\mathbb{R}$? Is there a ...
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Does there exist a section of $\mathcal{P}(\kappa)\to\mathcal{P}(\kappa)/(\text{fin})$ that is "nearly Boolean"?
The following might be a somewhat esoteric question:
Does there exist an infinite cardinal $\kappa$ and a section $f$ of the quotient map $\pi:\mathcal{P}(\kappa)\to\mathcal{P}(\kappa)/(\text{fin})$ (...
- 2,830
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Strong chains of uncountable functions and cardinal characteristics
A family of functions $\langle f_\alpha:\alpha<\kappa\rangle$ from $\omega_1$ to $\omega_1$ is called a strong chain if $\alpha<\beta<\kappa\Longrightarrow \{\xi<\omega_1: f_\beta(\xi)\leq ...
- 7,125
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524
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A topologically transitive dynamical system without dense orbits
By a dynamical system I understand a pair $(K,G)$ consisting a compact Hausdorff space and a subgroup $G$ of the homeomorphism group of $K$.
We say that a dynamical system $(K,G)$
$\bullet$ is ...
- 41.9k
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Are all models of ZF + DC + "All set of reals are lebesgue measurable" also models of CH? [duplicate]
Possible Duplicate:
Lebesgue Measurability and Weak CH
I have studied a little set theory and I found that Solovay constructed a model of ZF+DC+"All set of reals are Lebesgue measurable" and I ...
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Calculate the $\downarrow$, $\downarrow\uparrow$ and $\uparrow\downarrow$ cofinalities of the poset of nontrivial finitary partitions of $\omega$
Let $(P,\le)$ be a poset. For a point $x\in P$ let
$${\downarrow}x=\{p\in P:p\le x\}\quad\text{and}\quad{\uparrow}x=\{p\in P:x\le p\}$$be the lower and upper sets of the point $x$, and for a subset $...
- 41.9k
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Mathias forcing with Ramsey ultrafilters, and Cohen reals [closed]
Edit/update: One reason this question never received an answer is because it was founded on a faulty premise! The Blaszczyk-Shelah paper I mentioned below does not prove that Mathias forcing with a ...
- 2,701
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Comparability implies well-orderability?
I am trying to prove a small proposition that got me completely stumped, and I cannot find a single counterexample.
(ZF) Suppose that $E$ is such that for every $A\subseteq\mathcal P(E)$ either $|...
- 39.8k
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Can the Knaster-Tarski theorem be proved using the Schroeder-Bernstein theorem?
The reverse can be done easily and the proof is well known I am wondering if the exact same argument can be used to prove reverse as well.
- 51
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Sequences of projecta in the constructible hierarchy
For $n$ a natural number, $\alpha$ an ordinal, let $\rho(n,\alpha)$ be the $n$-th projectum of $J_\alpha$, where $J$ is the Jensen hierarchy for $L$.
Call a finite sequence $s:=(x_1,\dots,x_m)$ of ...
- 521
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How many Dedekind-finite sets can $\mathbb{R}$ be partitioned into?
Building off Asaf Karagila's answer to my previous question (Can $\mathbb{R}$ be partitioned into dedekind-finite sets?) on partitioning $\mathbb{R}$ into strictly Dedekind-finite sets:
(1) What ...
- 20.5k
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Universally Baire Tree Representation of Projective Sets
In Feng, Magidor, and Woodin "Universally Baire Sets of Reals", they show that if $A$ is a $\mathbf{\Pi}_2^1$ set and $U$ and $V$ are any pair of trees witnessing the universal baireness of $A$, then ...
- 1,750
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(Weakly) minimal subcovers of linear covers
Motivation. The starting point of this question is the trivial observation that if we cover $\mathbb{N}$ with $$\big\{\{0,\ldots n\}: n\in \mathbb{N}\big\},$$ then this cover doesn't have a minimal ...
- 48.9k
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Absoluteness and the scale property for $Π^2_2$ or $Σ^2_2$
Under the diamond principle $◊$ and large cardinal axioms, which of the two pointclasses $Π^2_2$ or $Σ^2_2$ is expected to have the scale property?
Because conditional $Σ^2_2$ absoluteness under $◊$ ...
- 7,545
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3
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Are there first-order statements that second order PA proves that first order PA does not?
Are there first-order statements that second order PA proves that first order PA does not? Is this known one way or the other? Could you share an example? (edit: to clarify, by 'second order PA' I don'...
- 117
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367
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"Intersection number" of a cardinal
Let $\kappa$ be an infinite cardinal. We call a cardinal $\lambda \leq 2^\kappa$ intersecting if there is ${\cal C}\subseteq {\cal P}(\kappa)$ such that
for every $A\in {\cal C}$ we have $|A|=\kappa$,...
- 48.9k
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2
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361
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Stationary sets and $\kappa$-complete normal ultrafilters
Let $\kappa$ be a measurable cardinal, and let $u$ be a normal $\kappa$-complete ultrafilter over $\kappa$. It is a standard easy fact that every closed unbounded set must belong to $u$ (notice that ...
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521
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A question about Mitchell/Steel Fine Structure and Iteration Trees
In chapter 8 of Mitchell's and Steel's FSIT, they prove a central fine structural result, which basically states that if $\mathcal{M}$ is 1-small, $k$-sound, $k$-iterable premouse then the $k+1$-...
- 3,804
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627
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Can there be no "surprisingly averageable" second-order sentences?
Say that a second-order sentence $\varphi$ is averageable iff there exists some infinite cardinal $\kappa$ and some nonprincipal ultrafilter $\mathcal{U}$ on $\kappa$ such that for every $\kappa$-...
- 20.5k
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3
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"name" for the ground model
I am a beginner of forcing, often I read from some articles something like "$p \Vdash \dot{G}$ is $P$-generic over $\check{M}$" (where $M$ is a countable transitive model, for instance).
Q1. I ...
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