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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3 questions around the Stone space of the free $\sigma$-algebra on $\omega_1$ free generators
During my studies, I came across several different Stone spaces, e.g.:
(i) The Cantor cube $X=\{0,1\}^{\omega_1}$, which is the Stone space of the free Boolean algebra on $\omega_1$ free generators;
...
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Existence of maximal analytic P-ideal
An ideal $\mathcal{I}$ on the positive integers $\mathbf{N}$ is a P-ideal if for every sequence $(A_n)$ of sets in $\mathcal{I}$ there exists $A \in \mathcal{I}$ such that $A_n\setminus A$ is finite ...
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3
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Non-isomorphic hypergraphs on $\omega$
Let $H_i = (V_i, E_i)$ be hypergraphs for $i=1,2$. Then we say that $H_1\cong H_2$ if there is a bijection $\varphi:V_1\to V_2$ such that $A\in E_1$ implies $\varphi(A) \in E_2$ and $B\in E_2$ implies ...
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Minimal dominating subsets in infinite graphs
Let $G=(V,E)$ be any simple, undirected graph. A dominating set is a set $D\subseteq V$ such that for all $v\in V\setminus D$ there is $d\in D$ such that $\{v,d\}\in E$.
Is there an infinite graph $G=...
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160
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Meeting a set of lines in a generalization of $\mathbb{R}^n$
I was wondering whether there is a generalization of Martin Sleziak's excellent answer to this question:
Suppose that $S$ is an infinite set, and let ${\cal L}$ be a collection of subsets of $S$ such ...
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315
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How slowly can the critical points of the Fibonacci terms grow?
Define the Fibonacci terms $t_{n}$ for all $n\geq 1$ by letting $t_{1}(x,y)=y,t_{2}(x,y)=x$, and $t_{n+2}(x,y)=t_{n+1}(x,y)*t_{n}(x,y)$ for all $n$. The Fibonacci terms are used in order to ...
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Self-containing graphs
[Second try, after this question failed.]
Let me sketch a notion of self-containing structures by a simple example. Consider the class $\Gamma$ of finite or countable digraphs ("graphs" for short) ...
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Infinite iterates of the contravariant hom endofunctors on sets
My recent answer to Is it possible to define higher cardinal arithmetics (about defining infinite tetrations) requires something I don't know. Here is the simplest case.
Take a set $S$ and consider
$$...
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A question on Sigma(n)-admissible ordinals
Are Sigma(n)-admissible ordinals for n>0, i.e. ordinals such that Gödelś constructible hierarchy at that level is a model of Sigma(n)-KP, recursively inaccessible? According to Wikipedia on large ...
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Chains of maximum cardinality in distributive lattices
It's quite straightforward to construct a (complete) lattice in which no chain has maximum cardinality: for each $n\in \omega\setminus\{0\}$ let $C_n$ be a copy of $n$ with the chain ordering ...
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Standard model of ZFC
Is ZFC+Con(ZFC) powerful enough to show there isn't any standard model of ZFC? What you think about it?
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Does Regularity schema imply $\in$-induction when added to first order Zermelo set theory?
That $\in$-induction fails to be a theorem schema of first order Zermelo + Foundation (see here), then it appears that it is more eligible to replace axiom of Regularity (Foundation) by a Regularity ...
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What is the consistency strength of this kind of iterating Berkeley cardinals?
[EDIT] After suggestion from Monroe Eskew, and after having an e-mail correspondence with Prof. Joan Bagaria, I'll re-present the older question as the second in a series of 4 questions. I've had an e-...
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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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Can Ackermann set theory find a natural interpretation in light\heavy class dichotomy?
I want to coin the notion of heaviness of a class as a function from classes to ordinals such that $$heaviness(x)=|TC(x)|$$, i.e. the heaviness of a class is the cardinality of its transitive closure.
...
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Can we state $\sf V=HOD$ using a single ordinal parameter(other than the formula code)?
$\sf V=HOD$ is stated as:
$\forall X \, \exists \theta \, \exists \alpha < \theta \, \exists \varphi : X=\{y \in V_\theta\mid V_\theta\models\varphi(y,\alpha)\}$
This use two ordinal parameters (...
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About having one axiom schema for ZFC motivated after the iterative conception of sets?
This posting is related to this posting, and builds its motivation from this answer to it.
Define: $\operatorname {History}(x) \iff \\\forall y \in x: y=\{c \mid \exists z : z \in y \cap x \land (c \...
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Is every formula of LΩ equivalent to a formula of L1 modulo T1?
Q: Is every formula of $L_\Omega$ equivalent to a formula of $L_1$ modulo $T_1$?
The notation comes from the following question: Is the following theory countably axiomatizable?
Edit: I mean $T_\...
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Is the following theory countably axiomatizable?
Let $L$ be the language of set theory, and for each countable ordinal $\kappa$, define $L_\kappa$ as follows: $L_0 = L$, and $L_{\alpha+1}$ is obtained by adding countably many new class constants $c_\...
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158
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On cardinality of generating subsets of some submodules
Let $R$ be a commutative ring with unity. Let $\alpha$ be an infinite cardinal . Let $M$ be an $R$-module such that $\mu(M)< \alpha$ . Let $N$ be a submodule of $M$ and $m\in M$ and $r\in R$ be ...
user111524
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getting one tower from two (stronger hypothesis than a previous question with same title)
Suppose that $(L,\leq_L,0,1)$ is a Boolean algebra that is dense as an order (i.e. if $a<_L b\in L$ then there exists $x\in L$, s.t. $a<_L x<_L b$) s.t all non trivial closed segments are ...
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612
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Can this kind of Mereology be synonymous with Set Theory?
This question is about synonymy of Morse-Kelley set theory "$\sf MK$" with the following Mereological theory:
Language: first order logic with equality. Extra-logical primitives: $\subseteq$ ...
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196
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Infinite graph with degrees given
Let $\kappa$ be an infinite cardinal and suppose $$n, d: \kappa \to \big((\kappa+1)\setminus \{0\}\big) = \{1, \ldots, \kappa\}$$ are arbitrary functions.
Is there $E \subseteq \big\{\{x,y\}: x\neq y ...
- 48.9k
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Where do models of false theories exist?
I have some difficulties in understanding the [uni]verse platonic view. How are we to understand the existence of a model of a false theory? what is the relationship of this model to the platonic ...
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How do you prove that the category of weak equivalences of sSet is accessible?
I am trying to prove that the category of simplicial sets is a combinatorial model category by using Proposition A.2.6.15 of Lurie's book, and this requires proving that the weak equivalences are ...
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171
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How do chains of elementary extensions compare to shrewdness?
I thought of the following large cardinal axiom, extending the notion of $\theta$-upliftingness:
Let $\eta$ be be an ordinal, and $X$ be a class of ordinals. $\kappa$ is called $\eta$-iteratively ...
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Are countable models constructible?
Suppose a first order set theory $S$ has a countable model. Does it follow that there is a countable ordinal $\alpha$ so that $L_\alpha$ in the constructible hierarchy is a model of $S$?
- 3,574
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Sign of the permutation which brings a subsequence back to its original form [closed]
I have the following question, which I am thinking about for days now and can't get the answer right. I have a sequence of elements in this order $x_{1},x_{2},...,x_{2n}$, $n \ge 1$ and then I perform ...
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What's the consistency strength of adding this inference rule to Ackermann's set theory?
Working in the language of Ackermann set theory:
Let $\phi(\vec{P},\vec{x})$ be a formula not using the symbol $V$, where $\vec{P}$ are predicate symbols definable in the language of set theory, and $\...
- 11.3k
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273
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A Borel perfectly everywhere surjective function on the Cantor set
Does there exist a Borel (or even continuous) function $f:\mathcal{C}\to\mathcal{C}$, where $\mathcal{C}$ is the Cantor set (or Cantor space $2^\omega$) such that for every nonempty closed perfect set ...
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Group graphs and Ramsey theory. Sub-question 1
Question: Find/compute relations between the classical Ramsey numbers and their variations (described below) -- exact or asymptotic.
A graph is a set $\ X\ $ together with a (coloring) function
$\ c:\...
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Can (how) one distinguish germs of continuous functions by a countable set of params?
Continuous functions can be distinguished by their values at say rational points of [0 1].
Germs of analytic functions can be distinguished by derivatives at a point.
So in both cases we see ...
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Is the Rudin-Keisler order of ultrafilters linear?
Is the Rudin-Keisler order of ultrafilters linear?
Is it a well ordering?
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Is there an effective way to generalize this approach of affinely extending the number line?
The general approach here is a follow up of the approach outlined in a prior posting on extending the projectively extended real line. In particular arithmetic operators break down to ternery ...
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Can we have a bijection between a set and its powerset with the following properties?
This question is related to a question Is this internalization of a bijection between a set and its powerset possible? lately posted to $\cal MO$. Similarly, we add one primitive unary partial ...
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Is this reflection schema equivalent to second order Bernays reflection?
This posting comes as a possible salvage to this earlier presented reflective theory which was proved inconsistent. Attesting the particulars of the underlying language in reflection axioms.
Working ...
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Coloring non-principal ultrafilters on $\omega$
Let ${\cal U}$ be a non-principal ultrafilter on $\omega$. If $\kappa>0$ is a cardinal, we say that a function $c:\omega \to \kappa$ is a coloring for ${\cal U}$ if for all $U\in{\cal U}$ the ...
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Is a function needed here?
This question is related to my question Can we choose an element from a class?.
However, I decided to create a separate question.
Let $H$ be a complex Hilbert space and $H_1,\dotsc,H_n$ be closed ...
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Is acyclic ZF consistent?
I need to know if the following system is consistent, because I want to use it in presenting automorphisms over stratified versions of it.
The system I'd label as "Acyclic ZF", which is $\...
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Is Burgess' ST theory an algebraic set theory? If it is, what kind of algebras are described by this theory?
I found the mention of Burgess' ST set theory in the article https://en.wikipedia.org/wiki/General_set_theory about George Boolos' generalized set theory (GST). It sounds like this: "ST is GST ...
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Can ZFC commit cardinality errors?
Add a primitive a one place function symbol $c$ to represent "True cardinality" of a set, to the first order language of set theory.
Add the following axiom schema:
1. Cardinal Equality: If $\phi(x,...
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Is there a known shorter axiomatization of NF than this?
Is there an already known axiomtization of $NF$ that is shorter than the following axiomatic system in first order logic with equality $``="$ and membership $``\in"$? And what is exactly meant by ...
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Can Godel's incompleteness theorems be in some sense circumvented this way?
New foundations "NF" (formulated in the language of $\small \sf FOL(\in)$), can define a kind of ordered pair relation $``\rho"$ such that we can have a set $E$ of those pairs where NF proves the ...
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643
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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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315
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Graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$
Let $V$ be the set of all functions $f:\mathbb{N}\to\mathbb{N}$. Let two distinct functions $f,g:\mathbb{N}\to\mathbb{N}$ form an edge if and only if they differ in exactly one input $n\in\mathbb{N}$.
...
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Can Foundation be captured for some $\mathcal L_{\omega_1, \omega}$ theories?
This posting is a continuation to an earlier one titled "Can Foundation be captured in $\mathcal L_{\omega_1, \omega}$ ?"
It appears that capturing foundation is problematic at every $\...
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133
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Large complete minors of $\mathbb{Z}^\omega$
Let $x,y\in \mathbb{Z}^\omega$ and let $x,y\in\mathbb{Z}^\omega$ form an edge if there is $i\in\omega$ such that $|x_i - y_i|=1$ and $ x_k = y_k$ for all $k\in \omega\setminus\{i\}$.
$K_\omega$, the ...
- 48.9k
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121
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Subset of $[\kappa]^{<\kappa}$ with linear intersection
For any cardinal $\kappa$, let $[\kappa]^{<\kappa}$ denote the collection of subsets of $\kappa$ with cardinality $<\kappa$. Is there an infinite cardinal $\kappa$ and ${\cal C}\subseteq [\kappa]...
- 48.9k
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737
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How can I force the continuum to be weakly compact?
Since the continuum has to be a regular/singular uncountable cardinal, and since by Easton Theorem it can be anything we want it to be, what is the forcing that makes the continuum a weakly compact ...
- 3,804
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A ``1-soft'' improvement of the Parovichenko theorem
This is a ``1-soft'' modification of this problem. We start with the necessary definitions.
Definition 1. A compactification $c\mathbb N$ of the discrete space $\mathbb N$ is called 1-soft if for any ...
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