Questions tagged [set-theory]

This tag is for set theory topics typically studied at the advanced undergraduate or graduate level. These include cofinality, axioms of ZFC, axiom of choice, forcing, set-theoretic independence, large cardinals, models of set theory, ultrafilters, ultrapowers, constructible universe, inner model theory, definability, infinite combinatorics, transfinite hierarchies; etc. More elementary questions should use the "elementary-set-theory" tag instead.

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The proof of $\textbf{Lemma 10.1}$ in Nik’s book about forcing

I’m an undergraduate student trying to teach myself set-theory. And I have some trouble understanding the density of a constructed set. In Lemma 10.1 of Nik’s book, it states: Let $G$ be a generic ...
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Explaining Conditional Probability and the Urn problem with white and black balls

I am trying to interpret the meaning behind the conditional probabilities when setting up the problem. Can someone confirm with me if I am understanding the problem correctly? Problem Statement: There ...
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My question is related to real analysis [closed]

$L^\infty$$[0,1]$ is a complete metriç space.
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properties of symmetric difference of two set$|\Omega_1 \cap \Omega_2| |x_{\Omega_1} - x_{\Omega_2}| \le C(R) |\Omega_1 \Delta \Omega_2|$

how to show the following properties are holds in $\mathbb{R}^n$? for two bounded set $\Omega_1$, $\Omega_2 \subset \mathbb{R}^n$ and $\Omega_1$, $\Omega_2 \subset B_R$, $B_R$ is the Ball radius $R$ ...
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Do dense complements in $\mathbb{R}$, which are uncountable, both contain a rational number?

Suppose $D$ and $\mathbb{R} \setminus D$ are both dense in $\mathbb{R}$. If they are both uncountable, do both sets then necessarily have to contain a rational number?
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Definition of uniform $ω_1$-dense ideal

Can one define the term uniform $ω_1$-dense ideal without the use of Boolean algebras?
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Can $L$ supply models for every theory with a well founded model?

Does ZFC prove that if a theory has a well founded model, then there must be a model of that theory in $L$?
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Difference in definition of Reflexive Relation between Set Theory and Class Theory

The class-theoretical context of this comes from Smullyan and Fitting's Set Theory and the Continuum Problem (revised ed., 2010). Context: self-study. Consider a relation $R$ on a set: $R \subseteq S \...
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Notation for the set of conjonctions of two adjacent level of the levy hierarchy

Let $\Sigma_n$ and $\Pi_n$ be two levels of the levy hierarchy. We consider the set of formulas $$\Gamma = \left\{ \phi \wedge \psi, \phi \in \Sigma_n, \psi \in \Pi_n \right\}$$ Is there a common name ...
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Did I get this induction exercise on the natural numbers set-theory right?

Exercise. Prove that, $\forall n,m,k \in \omega$ \begin{equation*} n \leqslant m \Rightarrow n+k\leqslant m+k \end{equation*} My attempt of solving this. Using induction over $n$: For $n = \emptyset = ...
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The set $X$ can be seen as index set?

Let $X$ be a set, and the power set of $X$ be $\mathcal P(X).$ For $\mathcal B \subset \color{red}{\mathcal P(X)},$ does \begin{align} &\quad \ \left\{ U\subset X \mid \exists \Lambda : \mathrm{...
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The empty set in Ackermann set theory

Let $A$ denote Ackermann set theory, as laid out e.g. here. It is a standard result due to Levy and Reinhardt that $A$ and $ZF$ are mutually interpretable in conservative extensions of one-another. ...
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Laver's Proof that $V$ and $V[G]$ have the $\delta$-approximation property.

Let $V$ be a transitive model of ZFC, $\mathbb{P} \in V$ be a forcing poset such that $(|\mathbb{P}| < \delta)^{V}$ for some regular cardinal $\delta$ in $V$, and $G$ be a $V$-generic filter over $\...
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Bijection between the set of strictly increasing sequences of natural numbers and the set of all sequences of natural numbers

We have two sets: $A = \{f \in \mathbb{N}^\mathbb{N}:\ f \text{ is strictly increasing}\}$, $B=\mathbb{N}^\mathbb{N}$, i.e the set if all sequences of natural numbers. I'm looking for a bijection ...
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Motivating Grothendieck Toposes generated by Souslin Objects

I am currently reading about the independence of the continuum hypothesis from ZFC following the topos theoretic proof given in Chapter VI.3 of MacLane Moerdijk's Sheaves in Geometry and Logic. The ...
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Is this theory trying to capture the theory of the minimal model of ZFC correctly formalized?

I'm trying to capture theory $T_0$ written by Noah's answer to a prior posting of mine. First we add a constant symbol $\mathcal M$ to the language of set theory. Now we add all axioms of $\sf ID$ and ...
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Measurable cardinals, elementary embeddings, and Kunen's theorem

Suppose $\kappa$ is a measurable cardinal. Then if $U$ is the ultrafilter on $\kappa$, we can use this to generate an ultrapower of the entire universe. We can then embed $V$ into this ultrapower in ...
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2 votes
1 answer
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Definitions of nice recursive analogues of cofinality?

The cofinality of an ordinal is the minimal order type of a cofinal subset of that ordinal. In the study of admissible ordinals, some concepts that mimic the idea of cofinality appear. One common ...
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Baire class one and Borel functions

The picture below is page 190 of Classical Descriptive Set by Kechris, there is Theorem 24.3, I just need that theorem for Baire class 1 case, in other words, I just need to proof for the case when $\...
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What exactly is zero? [duplicate]

I'm learning set theory and the construction of set of natural numbers by Peano's axioms, which says "$0$ is a natural number", and everything starts at $0$. Then what is $0$? In set theory, ...
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What is the rigorous definition of a large cardinal axiom?

I have read in a lot of set theory textbooks about large cardinals and large cardinal axioms. But what is the precise definition of a large cardinal axiom? Is there a rigorous definition of when an ...
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Defining convergence in topological spaces

In topology, we define continuity (I should be defining convergence to a point here, however) of a function at a point through for any (first) open neighbourhood containining the value of the function ...
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4 votes
1 answer
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Cohen reals satisfying a formula

Consider the Cohen forcing $\mathbb{C} =Fn(\omega,2)$, the one that adds a Cohen real, and now suppose that for a Cohen real $r$ generic over $V$ we have $$V[r]\models \exists x (x \in \mathbb{R} \...
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A question on Kanamori's Book (Proposition 10.20)

I don't see a case involved in the proof of Proposition 10.20 in Kanamori's book. Let $P$ be a separative poset with $|P|\leq|\alpha|$, for an ordinal $\alpha$. Suppose $P$ forces that there is a ...
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What does $U^{-N}$ means? Where N is the natural numbers set.

I was reading this paper called; "The echo state approach to analysing and training recurrent neural networks-with an erratum note." In the paper, I found the expression: $\left(u(n)\right)_{...
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5 votes
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Examples of Substructures that "do not know they are that substructure"

Just learned $\mathbb{L}\vDash \mathbb{V}=\mathbb{L}$ and was warned that this property is not obvious with the counterexample mentioned being $HOD$. I can think of a few examples of definable ...
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How to express formally all possible 2-tuples of a set into one set

Let be a triplet of sets $(A, B, R)$, where: $A = \{a, b, c, ...\}$ $B = \{1, 2, 3, ...\}$ $R$ denoted the set for relations for the elements of the sets $A$ and $B$, $R = \{(a, 1), (b, 2), (c, 3), .....
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Kolmogorov's construction of real numbers cardinality of functions that represent real numbers

Hi i am reading about lesser know construction of real numbers by Kolmogorov. In his construction real numbers are defined as a set $\Phi$ of functions $\alpha: \mathbb{N} \rightarrow \mathbb{N}$ that ...
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7 votes
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Finding where in Ramsey's theorem one uses the Axiom of choice

Ramsey's Theorem for infinite graphs requires some choice but when looking at the proof it is not evident how choice is exactly used. Sketch of the proof: Given $c:[\omega]^2\rightarrow 2$ a ...
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1 vote
1 answer
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Understanding a proof about $\lambda$-supercompact cardinal

I have trouble understanding the proof of this Lemma 20.15 from Jech's Set Theory, could someone explain why is $(2^\alpha)^M = (\alpha^+)^M = \alpha^+$?
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Set that is equal to its union

Is there any set $X$ such that $X=\bigcup X$?. I think that there is not any, but I don't know how to prove it, any ideas?
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Is there a minimal constructible model for every first order theory that has some?

If a first order theory $T$ has a constructible model, then does that entail that $T$ must have a minimal constructible model, i.e. one that is homomorphic to every constructible model of $T$. By &...
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Subtle point in Class Theory: is the intersection of two classes a set of a class?

Context: self-study from Smullyan & Fitting's Set Theory and the Continuum Problem (revised ed., 2010). Suppose that all classes are subclasses of a "basic universe" $V$, that is: every ...
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$\alpha<\beth_\omega\implies 2^\alpha<\beth_\omega$

"Let $\alpha$ be a cardinal such that $\alpha<\beth_\omega$. Then $2^\alpha<\beth_\omega$". I'm stuck, any suggestions or bibliographies of the $\beth$ function? :(
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Constructing natural numbers from nothing

I found that many of us (mathematicians) try to construct natural numbers defined from the intuitive concept 'size of the set'. They take $\emptyset$, the empty set as the starting, then define and ...
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How could we use the continuum hypothesis if we are able prove its truth?

I'm not sure if this is an appropriate question to ask, but if we are able to agree on the truth of the continuum hypothesis, then what problems could we solve using this newfound knowledge? I'm ...
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Why is ZFC incapable of interpreting second-order logic?

Why is ZFC incapable of interpreting second-order logic? Also, when we say this, are we talking about ZFC as a background theory or using ZFC in some different way? I am interested in this answer ...
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Terence Tao' analysis 3.4.6. Needed help with exercise specifications

I've been reading Terence Tao's Real Analysis. Sometimes I don't understand what exactly I am supposed to do with exercise. Exercise 3.4.6. Prove Lemma 3.4.9. (Hint: start with the set {0, 1}$^X$ and ...
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How Gödel's first incompleteness theorem can be used for proving statements true or false

If Gödel's first incompletness theorem states $$\exists S: g(S)=g(\neg P(g(S)))$$ Where $g$ is the Gödel numbering of the statement. Since there is a proof that this statement is true but has no proof,...
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4 votes
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Where do I use finiteness in this proof of: In ZF, the compactness theorem implies the Axiom of Choice for collections of finite sets?

Work in ZF, and assume the compactness theorem. Let $\mathsf{AC}^\text{fin}$ be the sentence "every collection of finite non-empty sets has a choice function". UPDATE: Thank you to the ...
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Is a rational number a real number? [duplicate]

$\newcommand{\weakpartial}[1]{\leq_{#1}}$ $\newcommand{\strictpartial}[1]{<_{#1}}$ $\newcommand{\rationalset}{\mathbb{Q}}$ $\newcommand{\realset}{\mathbb{R}}$ $\newcommand{\powerset}[1]{\mathcal{P}\...
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Is the set of Turing Machines decidable in ZFC non-recursive?

Let S be the set of all the TMs which halting is decidable in ZFC (for each TM in S, we can find one algorithm in ZFC that determines whether the machine halts or not). Is S recursive? Is there one ...
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Clarification on Paragraph in 'Introduction to Set Theory' By J. Donald Monk

Background: I have just started reading J. Donald Monk's "Introduction to Set Theory," and I would like to double-check my understanding of the following paragraph: The fundamental idea in ...
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3 votes
1 answer
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A property of product forcing

I've read in an article the following statement (which is said to be a standard fact in forcing) Given $\mathbb{P}$ a forcing notion and $G_1,G_2\subseteq\mathbb{P}$ such that $G_1\times G_2$ is $\...
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What is the purpose of a witness in existential projection?

I am reading my class notes and have come across the following terms - The existential projection of a language $B \subseteq \Sigma^{*}$ is the language, $\exists B \subseteq \Sigma^*$ defined by $\...
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Letting $\{a_i\}$ and $\{b_i\}$ be families of cardinal numbers $a_i < b_i$, prove there is an injection between $\sum_{i}a_i$ to $\prod_i b_i$

Letting $\{a_i\}$ and $\{b_i\}$ be families of cardinal numbers $a_i < b_i$, prove there is an injection between $\sum_{i}a_i$ to $\prod_i b_i$. I need it to complete the solution to Halmos' ...
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Could we regard $\aleph_0\in \mathbb Z$? [closed]

I hope that $\aleph_0\in \mathbb Z$, however if so, $\aleph_1={\aleph_o}^{\aleph_0}\in\mathbb Z$ should also be true, since $\mathbb Z$ is closed for multiplication, but $\aleph_1$ is for uncountable ...
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Are Set theoretic constructions of mathematical objects unique

It is generally believed that mathematics (at least a considerable portion) can be embedded in ZFC set theory. But one would wonder what this statement really means, and whether the set theoretic ...
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the union of a chain of well ordered sets is well ordered

I have difficulty to understand the following claim in "Naive set theory" of Halmos on page 68. Let $\mathcal{C}$ be a continuation chain of well-ordered sets, and $U$ be the union of these ...
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Are the open sets of $\{0, 1\}^I$ measurable?

Let $I$ be an uncountable set and let $2^I=\{0, 1\}^I$ be the set of all functions from $I$ into $\{0, 1\}$. Consider the product measure $\mu$ on $2^I$. The domain of this measure is the $\sigma$-...
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