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.

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Is definability in $V$ in $\sf Ack+MK$ expressible in its language?

Recall Ackermann set theory. If we extend Ackermann's set theory by adding all axioms of $\sf MK$ to it. We shall denote the universe of all elements by $W$. Then can we express pure set theoretic ...
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Terminology for the property: "Each uncountable disjoint open family is locally countable"

Suppose that a topological space $X$ satisfies the following property (P): "Each uncountable disjoint open family is locally countable", where a family $\mathcal U$ of subsets of $X$ is ...
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3 votes
1 answer
124 views

Existence of a particular function that maps an arbitrary set of ordinals to a single ordinal

Does there exist a function $f$ that satisfies all of the following three properties? The function converts an arbitrarily large (empty, finite, countably/uncountably infinite) set of ordinals to a ...
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1 vote
1 answer
82 views

Minimal cardinality of non-bipartite sub-family of $[\omega]^\omega$

Let $[\omega]^\omega$ the collection of infinite subsets of $\omega$. We say that $E\subseteq [\omega]^\omega$ is bipartite if there is $d\subseteq \omega$ such that for all $e\in E$ the intersections ...
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5 votes
2 answers
464 views

Can Category theory be founded in set theory using worldly cardinals instead of inaccessibles?

What is exactly demanded for a set theoretic foundation of Category theory? I saw generally two main approaches. One is Muller's who did the work in Ackermann's set theory, but his criteria seem to ...
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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 $\...
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11 votes
0 answers
330 views

Definition of "galaxy", due to Sabbagh/Samuel

Gabriel Sabbagh, a PhD student of Pierre Samuel, called his thesis Ensembles artiniens, univers et galaxies. I learned from a recent talk of Colin McLarty that Grothendieck, in his 1973 Buffalo ...
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3 votes
1 answer
130 views

Weak form of $\text{CH}$ in $L(\mathbb{R})$

I was wandering whether this weak form of $\text{CH}$ holds in $L(\mathbb{R})$ provably in $\text{ZF}+\text{DC}$ $(\text{ZF}+\text{DC}) \ L(\mathbb{R})\vDash \forall X\subseteq\mathbb{R} ( X \text{ ...
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-1 votes
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58 views

" for all x in E , we have p(x) " [migrated]

Let E be an empty set . why the assertion "$(\forall x \in E) , p(x)$ " is true ??
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1 vote
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145 views

Can Set theory be interpreted in Relational Mereology?

In a posting to MathStackExchange, I've presented a theory of rudimentary relations having a rudimentary kind of membership together with a primitive ordered pairing with the aim for it to capture the ...
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4 votes
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Does second-order logic satisfy Craig interpolation for second-order languages?

(For simplicity, all languages are relational.) In analogy with first-order languages, say that a second-order language is a set of relation symbols of two kinds: first-order relation symbols and ...
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2 votes
1 answer
97 views

A continuous map relating co-constructible reals

My question is the following: Given $x,y \in \omega^\omega$ such that $x\equiv_c y$ is there an $L$-definable continuous map $\varphi: \omega^\omega\rightarrow \omega^\omega$ such that $\varphi(x) = ...
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5 votes
0 answers
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Understanding descending intersections of generic extensions

Let $B_{0}\supseteq B_{1}\supseteq\dots\supseteq B_{\alpha}\supseteq\dots\,\,\left(\alpha<\kappa\right)$ be a descending sequence of complete Boolean algebras, $B_{\kappa}:=\bigcap_{\alpha<\...
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1 vote
1 answer
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Optimal partitions amongst a given set of partitions

Let $X\neq \emptyset$ be a set. By $\text{Part}(X)$ we denote the set of all partitions of $X$ not containing $\emptyset$ as an element. First, note that $\bigcup{\frak P}$ is the collection of ...
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6 votes
1 answer
243 views

Consistency strength of an attempt at higher order set theory

Work in a theory with (deep breath) a countable number of primitives denoted with capital letters from the end of the alphabet with numerical subscripts $\{X_n,Y_n,Z_n,\dots\}_{n<\omega}$ ...
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5 votes
0 answers
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Choicelessly assigning ordinals to fast growing functions

Consider functions from the natural numbers to themselves. One can define a partial ordering on them by saying that, for $f, g: \mathbb{N} \to \mathbb{N}$, $f>g$ if there exists an integer $N$ such ...
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2 votes
0 answers
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Can this predicative kind of type-set theory reach the consistency of ${\sf Z}_2$?

Add a primitive total one place fuction symbol $\tau$, and a primitive binary relation $<$, to the language of set theory. Add the following axioms: Extensionality: $\forall z \, (z \in x \iff z\in ...
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2 votes
0 answers
58 views

How closely do ordinal collapsing functions relate to Mostowski collapse?

Ordinal collapsing functions (such as Rathjen's $\psi_\pi$-functions, not the Levy collapse function) name large countable ordinals by mapping larger ordinals below some "large" ordinal, ...
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3 votes
0 answers
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Why are the sharps of sets of big ordinals not in $\mathcal{P}(\omega)$?

In his talk A Condensed History of Condensation, Welch presents the following recursive sharp function, that is total when all sharps exist: \begin{align*} \# \colon ON &\to \mathcal{P}(ON) \\ \...
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6 votes
0 answers
268 views

Are there quantifiers that require multiple "steps" to define?

(Below I conflate quantifiers and quantifier symbols in a couple places for readability; I can change that if that actually makes things less readable.) For the purposes of this question, an $n$-ary ...
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-1 votes
0 answers
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A dumb axiom of collection [migrated]

Add to $ZFC$ the following axiom: Dumb Collection. For any predicate $\phi$ such that there exists a set $y$ satisfying $\phi$, there exists a nonempty set $x$ whose members are precisely the sets $y$...
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1 vote
0 answers
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Does ${\sf ZC + Universes + ZFC}^V$ meet Muller's criteria for a founding theory of Mathematics?

I was re-thinking Muller's criteria in Sets, Classes and Categories: page 14 for a theory that founds Mathematics. To him, it should be able to be a foundation of Category Theory. He lays down six ...
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2 votes
1 answer
132 views

Given $\pi$ permutation on $\{1,\dotsc,n\}$, what is the sign of a permutation of $\{2,\dotsc,\hat\jmath,\dotsc,n\}$?

This question is related to my other question Sign of the permutation which brings a subsequence back to its original form. Suppose I have a complete ordered set $\{a_{1},\dotsc,a_{2n}\}$ and take $\...
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1 vote
0 answers
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Can all stages of the cumulative hierarchy beyond $V_\omega$ violate the weak partition principle?

This question is a follow up of this. Is it consistent for ALL infinite stages $V_{\alpha > \omega}$ of the cumulative hierarchy of $\sf ZF$, to violate the weak partition principle? That is, each ...
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5 votes
1 answer
132 views

Can a stage of the cumulative hierarchy violate the partition principle?

If we violate the partition principle and add to $\sf ZF$ the axiom that there exists a set $X$ that has a partition on it that is greater in cardinality than the set of singleton subsets of $X$. Can ...
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4 votes
0 answers
141 views

Construction of a function by transfinite recursion

Let $\mathcal F:=\{f_\xi\colon \xi<\mathfrak c\}$ be a family of functions from $\Bbb R$ to $\Bbb R$ and $K$ be a nonempty perfect set. The question is Construct a function $g\colon \Bbb R \to \...
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1 vote
1 answer
83 views

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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4 votes
0 answers
200 views

How much "finitary combinatorics" can be emulated by an infinite Dedekind-finite set?

Throughout, we work in $\mathsf{ZF}$. Let $[n]=\{1,...,n\}$. Given a set $X$ and a first-order sentence $\varphi$, let $M_X(\varphi)$ be the set of isomorphism types of models of $\varphi$ with ...
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9 votes
1 answer
449 views

Number of Laurent monomials of n variables with degree at most d

Introduction: We have a question of how to calculate the number of $n$-variables Laurent monomials of degree at most $d$. For example: If $n=2$, $d=2$ then we have 19 monomials, which are: $x^{-2}$, $...
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3 votes
2 answers
213 views

MAD family with the choosability property

By $[\omega]^\omega$ we denote the collection of infinite subsets of $\omega$. Two sets $A,B\in[\omega]^\omega$ are said to be almost disjoint if $A\cap B$ is finite. An almost disjoint family is a ...
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8 votes
2 answers
657 views

Why can we assume a ctm of ZFC exists in forcing

Following Kunen's book, it makes clear that countable transitive models (ctm) exist only for a finite list of axioms of ZFC. So, why can we assume a ctm of the whole ZFC axioms exists and use it as ...
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1 vote
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Can all relations and functions be implemented as sets in some fragments of set theory?

Define wholly stratified $\sf NF$ to be $\sf NF$ with its language restricted to stratified expressions. In this theory we can arrive at a general implementation of tuples, that is: $\langle x_1,..,...
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2 votes
0 answers
102 views

Weak form of CH in $L(\mathbb{R})$, reference

I've heard someone saying that in $L(\mathbb{R})$ the following form of $\text{CH}$ holds: $L(\mathbb{R})\vDash \forall X\subseteq \mathbb{R}(X \text{ countable or } \mathbb{R}\le^* X)$, i.e. every ...
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2 votes
2 answers
131 views

Diagonalization over a normal function and its derivatives on transfinite ordinals

Let $\Phi(0,\beta)$ a normal function from $On$ to $On$, and let $\Phi(\alpha,\beta)$ be the $\alpha$-th derivative of $\Phi(0,\beta)$. For example, let $\Phi(0,\beta)=\aleph_\beta$. Then, all ...
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3 votes
0 answers
65 views

Is $\mathfrak q_0$ equal to the smallest cardinality of a second-countable $T_1$-space which is not a $Q$-space?

A topological space $X$ is a $Q$-space if every subset of $X$ is of type $G_\delta$. The smallest cardinality of a metrizable separable space which is not a $Q$-space is denoted by $\mathfrak q_0$ and ...
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5 votes
1 answer
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Product topology from two premetric spaces induced by sum of premetrics?

For metric spaces $(M_1, d_1)$ and $(M_2, d_2)$, it is an exercise that the product topology on $M_1\times M_2$ is induced by the metric $d((x_1, y_1), (x_2, y_2)) =d_1(x_1, x_2) + d_2(y_1, y_2)$. Do ...
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3 votes
1 answer
115 views

Is every first-countable Lindelof space of cardinality $<\mathfrak c$ a $Q$-space under MA?

Definition. A topological space $X$ is a $Q$-space if every subset of $X$ is of type $G_\delta$. It is clear that every $Q$-space has countable pseudocharacter (= all singletons are $G_\delta$) and is ...
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5 votes
0 answers
87 views

What's the purpose of $\mathsf M\text-\mathsf P$-expressions?

In ordinal notations such as Stegert's (Ordinal Proof Theory of Kripke–Platek Set Theory Augmented by Strong Reflection Principles) and Rathjen's (An Ordinal Analysis of parameter free $\Pi_2^1$-...
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  • 470
13 votes
2 answers
1k views

Set theory without the empty set

Has there ever been a set theory without an empty set? Is this possible? I ask because we usually take the empty set to exist axiomatically or obtain it through separation and a nonempty set together ...
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1 vote
1 answer
81 views

Hypergraphs with finite matching / covering balance

Let $H=(V,E)$ be a hypergraph such that $\emptyset\notin E$. We say that $C\subseteq V$ is a (vertex) cover if for all $e \in E$ we have $C\cap e\neq \emptyset$. The minimum size that a cover can have ...
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7 votes
1 answer
304 views

What is an example of a meager space X such that X is concentrated on countable dense set?

A topological space $X$ is concentrated on a set $D$ iff for any open set $G$ if $D\subseteq G$, then $X\setminus G$ is countable. What is an example of a separable metrizable (uncountable) meager (...
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9 votes
0 answers
334 views

Locally small categories in ZFC

This question is primarily a reference request. It arose from a personal coding/formalization project. I am using a particular form of a definition of a category in ZFC. According to this definition, ...
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4 votes
1 answer
172 views

Is König's Property for graphs inheritable from finite subgraphs?

Let $G = (V,E)$ be a simple, undirected graph. A set $C \subseteq V$ is said to be a (vertex) cover if $C \cap e \neq \emptyset$ for all $e\in E$. A matching is a set $M\subseteq E$ of pairwise ...
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3 votes
0 answers
148 views

Is this recursion theoretic analogue of a criterion of weakly compact cardinal true?

Jensen proved that, if V=L, and $\kappa$ is a regular cardinal, then if for any stationary $A\subseteq \kappa$, the set $\{\alpha\mid A \text{ is stationary below }\alpha\}$ is stationary in $\kappa$, ...
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25 votes
2 answers
2k views

Can we interpret arithmetic in set theory, with exactly PA as the ZFC provable consequences?

There are many interpretations of arithmetic in set theory. The Zermelo interpretation, for example, begins with the empty set and applies the singleton operator as successor: $$0=\{\ \}$$ $$1=\{0\}$$ ...
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1 vote
0 answers
77 views

Given a partition of a field, construct a partition of its extension

The motivation for my question is the following algebraic consequence of the Continuum Hypothesis ($2^{\aleph_0}=\aleph_1$) by Zoli: (T1) Assume the Continuum Hypothesis holds. Then $\mathbb{R}^{\...
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1 vote
1 answer
91 views

(Maximal) almost disjoint families of true cardinality ${\frak c}$

Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$. We say $a,b\in [\omega]^\omega$ are almost disjoint if $a\cap b$ is finite. A subset $A\subseteq [\omega]^\omega$ is said ...
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7 votes
1 answer
382 views

Set theory / Formal logic of Baba is You

''Baba is You'' is a recent puzzle game in which the player builds a set of rules by pushing squares with words written on them. If we leave aside the combinatorial difficulty of how to move the ...
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6 votes
1 answer
176 views

Gaps in cardinalities of MAD families

Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$. We say $a,b\in [\omega]^\omega$ are almost disjoint if $a\cap b$ is finite. A subset $A\subseteq [\omega]^\omega$ is said ...
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3 votes
0 answers
77 views

The set of ground model reals arbitrarily close to a new real in the forcing extension

Consider a forcing notion $\mathbb{P}$, a condition $p\in\mathbb{P}$, a $\mathbb{P}$-name $\dot{r}$ and a formula (with ground model parameters) $\varphi(x)$ such that $$p \Vdash \dot{r} \in \omega^\...
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