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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Cardinality of a totally ordered union

Let $(X_{\alpha})_{\alpha \in A}, (Y_{\alpha})_{\alpha \in A}$ be families of sets such that for $\{X_{\alpha} \mid \alpha \in A \}$ and $\{Y_{\alpha} \mid \alpha \in A \}$ are totally ordered by ...
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60 views

Question on An Explicit Enumeration of Ordinals

Question: For any countable ordinal $X$, is there an ordered triplet of ordinals $(a,b,c)$ in the class of all ordered triplets of ordinals such that $X \in f((a,b,c))$ (where $f((a,b,c))$ is defined ...
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1answer
41 views

Why in ZFC any expressible set also exists?

I would like to ask you a few questions about the following quote from Wang (On Denumerable Bases, 1955): "Zermelo's set theory is, for example, a system in which every number set, if ...
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1answer
69 views

Families of subsets with “small” intersections

A classic problem in real analysis asserts the existence of $\mathfrak{c}$-sized $X \subset \mathcal{P} (\mathbb{N})$ with $|x \cap y|$ finite for any $x,y \in X$. The solution is fairly simple: swap $...
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46 views

Does the collection of countable subsets of real numbers has the same cardinality with the real numbers?Why?

The whole question has been displayed in the title. Is there a way to calculate such cardinality by using formalised methods such as we usually denote Aleph_1=2^Aleph_0. And thanks for sharing your ...
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1answer
29 views

Question Involving Sequences of Successive Limit Ordinals

This question centers around whether the following statement is true: If $X$ is some countable ordinal and $S^X = s_1, s_2, s_3, \dots$ is a (transfinite) sequence of successive limit ordinals ...
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1answer
43 views

$\Delta_0$-formula

How do I write a $\Delta_o$-formula $\phi(X,R)$ equivalent (in basic set theory) to "$(X,R)$ is a linear ordering"? I think I need to represent $(X,R)$ is a linear ordering in first order terms first ...
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1answer
61 views

Why does an inaccessible cardinal satisfy replacement?

If $\alpha$ is an inaccessible cardinal and $V_{\alpha}$ the corresponding von Neumann universe then $V_{\alpha}$ is supposed to be a model of ZFC. But the singleton $\{V_{\alpha}\}$ is not in $V_{\...
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1answer
38 views

What's the meaning of "the set of isomorphic classes of extensions of $\mathcal F$ by $\mathcal G$"?

I think an isomorphic class of extensions of $\mathcal F$ by $\mathcal G$ is a proper class, but a proper class cann't be an element of a set, so what's the meaning of "the set of isomorphic classes ...
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47 views

Ramsey cardinals lemma

I am reading the chapter on large cardinals in Jech's book and I found the following lemma: If $\kappa \rightarrow (\kappa)^{<\omega} $ and if $\lambda < \kappa $ is a cardinal, then $ \kappa ...
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1answer
91 views

Which set theory for category theory?

"A category $\mathfrak{C}$ is the data of the collection $\textrm{Obj}(\mathfrak{C})$ of objects of $\mathfrak{C}$ and for each objects $X,Y$ of a set of morphisms $\textrm{Hom}_{\mathfrak{C}} (X,Y)$, ...
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76 views

Schröder-Bernstein Theorem (Proof of existence of subsets) [closed]

So in the theorem, we have two sets and we want to partition each into 2 sets. Let $X$ and $Y$ be sets. Let $\mathscr F:X \rightarrow Y$ be $1$-$1$ Let $\mathscr G:Y \rightarrow X$ be $1$-$1$ ...
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1answer
27 views

Interpreting axiom of (naive) comprehension in a graph, $G(E,V)$

I am having some trouble interpreting axiom of (naive) comprehension in a graph, $G(V,E)$ Now suppose I define $x\in y \leftrightarrow xEy$, $x, y \in V$. So we now have a structure for set theory. ...
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1answer
78 views

Well-ordered system. [closed]

I would like to ask you the following question. Let $(S,<)$ is ordered system satisfying the condition: $\forall A\subset S:[\forall a\in S:S_a\subset A \implies a\in A] \implies A=S,$ where $S_a =\...
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47 views

How Vopenka's principle implies Semi-Weak Vopenkas principle?

Let's consider 2 large cardinal axioms, the Vopenka's principle (VP) which says that Ord cannot be fully embedded into Graphs and SWVP which says, that the equivalence $Hom(G(\alpha),G(\alpha'))=\...
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1answer
28 views

Is an algebra of sets on S containing all its finite subsets the power set of S?

Let S be a nonempty set and F a nonempty family of subsets of S defined as follows: 1. if A is in F, A' is in F; 2. if A, B in F, A+B is also in F. If one such family F contains every finite subset ...
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24 views

Decomposition of a subset in terms of linearly independent subspaces

Let's consider a subset $A$ of a vector space $\mathbb K^{n}$ over a field $\mathbb K$.Let's also assume that for every subset $B \subset A$ we have the following condition: $(1/d)|B| \leq r(B)$ , ...
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1answer
42 views

Cardinality of a connected compact Hausdorff space?

Is it known (in ZFC) that every connected compact Hausdorff space has cardinality at least $\mathfrak c=|\mathbb R|$?
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1answer
37 views

Proving Isomorphism Linear SubSpaces

$X$ & $Y$ are two subspaces of $W$. Show $(X+Y)/Y$ is isomorphic to $X/(X∩Y)$. So to show that I believe I have to show a bijective homomorphism of linear spaces $ T: W\to V$. Now there is the ...
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1answer
34 views

what it means to fail the axiom of comprehension

I am (re)reading Kunen's book on set theory, and I find myself curious to know what it means exactly for a structure to fail the comprehension schema. In essence, it means every definable sub-...
2
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1answer
71 views

Uncountable family of pairwise disjoint non-stationary subsets

Suppose $\mathcal A$ is an uncountable family of pairwise disjoint non-stationary subsets of $\omega_1$. Show that there exists $\mathcal B \subseteq \mathcal A$ such that $\mathcal B$ is uncountable ...
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58 views

Can an example of a non-measurable set be exhibited with non-well-founded set theories?

On wikipedia it says an example of a non-measurable set cannot be exhibited, although they do exist. I was curious what premises lead to this conclusion, and whether there are ways to exhibit non-...
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1answer
88 views

Sets with an infinite number of {{{…}}} [closed]

In pure mathematics, is there such a set with an infinite number of {{{...}}}? Is there a name for them? What are their properties? Letting $$R=\{x∶x\notin x\},$$ does Russell's paradox hold for this ...
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1answer
37 views

Descriptive Set Theory: Totally/well ordering nodes of a tree with infinite branches?

Here is the problem I am trying to solve: Let $T \subseteq \mathbb{N}^{\leq \mathbb{N}}$ be a tree. Define a total ordering $<$ on $T$ such $<$ is a well ordering if and only if $T$ doesn't ...
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3answers
66 views

What’s ‘Function’ in Axiom of Choice?

In ZF Axiom a function is defined by a formula $\phi$ i.e. $x \in A \leftrightarrow \phi (x)$. But what is the function in AC. Is it a particular set or a undefined notion like set, membership and ...
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2answers
34 views

Using axiom of replacement to form a function $f$

So I was (re)reading Kunen's set theory textbook, and in it there is a lemma (I.6.9) that illustrate using Axiom of replacement to define a function. Assume that $\forall x \in A\ \exists!y\ \varphi(...
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1answer
35 views

Cores in Boolean valued models

The following is taken from Bell's Set Theory: Boolean-Valued Models and Independence Proofs. $V^{(B)}$ is a Boolean valued model, where $B$ is some (complete) Boolean algebra. I'm stuck in verifying ...
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1answer
67 views

Isomorphism of $V_\lambda$ and the ultraproduct of $V_{\lambda_{\mathrm{otp}(x)}}, x \in \mathcal{P}_\kappa({\lambda})$ by a normal fine measure.

This question is actually exercise (20.5) from Set Theory by Thomas Jech. The original statement is: Let $\lambda \ge \kappa$ and let $U$ be a normal measure on $\mathcal{P}_\kappa({\lambda})$. The ...
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1answer
73 views

A graded definition of class using types.

I think we can define class using types as follows. First, prepare the type $\mathbb{S}$ for sets.There are initial elements of type $\mathbb{S}$ such as the empty set. If you don't want numbers $0, ...
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3answers
107 views

Permutation Models of Set Theory - meaning of “ZFA does not distinguish between atoms”

Halbeisen Combinatorial Set Theory on page 169 states "ZFA does not distinguish between the atoms and so a permutation of the set of atoms induces an automorphism of the universe". How does the ...
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0answers
61 views

What if we suppose continuum hypothesis is false? [duplicate]

Something I have puzzled about is the demonstrated reality continuum hypothesis can be neither proved nor disproved from Zermel-Fraenkel Set Theory. I'm not sure if my question makes sense, but I ask "...
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0answers
53 views

Existence of finite measure on uncountable set

Suppose you have some uncountable set $Y$ and a finite measure $\mu$ on the power set of $Y$ such that $\mu(Y)>0$ and $\mu(X)=0$ for all countable subsets. Does such a measure exist? I have read ...
2
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1answer
90 views

Transversal duality for upwards-closed subsets of ${\cal P}(X)$

Let $X$ be a set. Recall that an $U\subseteq {\cal P}(X)$ is upwards-closed if $A\in U, A\subseteq B \Rightarrow B\in U$ for any $A,B \subseteq X$. I'll write "ucs" as a shorthand for upwards-closed ...
2
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0answers
28 views

Prove that $\{x\in X\mid x\cap\kappa$ is $<\gamma$-closed$\}\in\mathcal I^*$.

Suppose that $\mathcal I$ is a $\kappa$-dense and normal ideal in $X$ (that is $\mathcal P(X)/\mathcal I$ has a dense set of power $\leq\kappa$), where $\kappa\subset X$ and $\kappa>\omega$ is ...
2
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2answers
84 views

Difficulty understanding Axiom of Choice

The Axiom of Choice is defined by Mcdonald, A Course in Real Analysis as follows: "Suppose that $C$ is a collection of non-empty sets. Then there exists a function $f: C \rightarrow \bigcup\limits_{...
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2answers
87 views

Can $(X, τ)$ be a nontrivial topological space such that $(τ,X)$ is also a topological space and a metric space under the same metric?

Can $(X, τ)$ be a nontrivial topological space such that $(τ,X)$ is also a topological space and a metric space under the same metric? I'm looking at the Collatz graph as a surjection from a subset ...
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1answer
55 views

Another ordinals question $\sum_{i\in\omega +3} i\cdot\omega + \omega\cdot i$

So I have to calculate sum $$ \sum_{i\in\omega +3} i\cdot\omega + \omega\cdot i$$ If I got it right, $1\cdot\omega+2\cdot\omega+\dots+(\omega-1)\cdot\omega = \omega\cdot(\omega-1)$ Then I have to ...
1
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1answer
32 views

Defining certain injections on non-stationary sets

Suppose $S\subset \omega_1$ is a stationary set of limit ordinals. For each $\alpha\in S$, let $\langle \beta_n^\alpha:n\in\omega\rangle$ be an increasing sequence cofinal in $\alpha$, i.e. $\beta_n^\...
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1answer
26 views

$\kappa$-dense implies $\kappa^+$-saturated

We say that and ideal $\mathcal I$, in a boolean algebra $\mathcal B$ $\kappa$-dense if $\mathcal B/\mathcal I = \{[x]\mid x\in\mathcal B \wedge [x] = \{y\mid x\triangle y \in\mathcal I\}\}$ has a ...
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0answers
104 views

Condensation lemma in Devlin

This is probably a dumb question, but I'm stuck at a step of the proof of the Condensation Lemma in Devlin's Constructibility. Context: we have $X\prec_1 L_\alpha$ for some limit $\alpha$ and want ...
1
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1answer
30 views

showing chain-completeness or conditional completeness of functions

Q: Given a nonempty set $X$ and a poset ($Y$, $\succcurlyeq$), define the partial order $\trianglerighteq$ on $Y^X$ by $f \trianglerighteq g$ if and only if $f(x) \succcurlyeq g(x)$ for each $x\...
3
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2answers
287 views

What is the definition of belonging in axiomatic set theory?

In Hungerford's Algebra it says that Intuitively we consider a class to be a collection $A$ of objects such that given any object $x$ it is possible to determine whether or not $x$ is a member (or ...
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1answer
53 views

Hungerford's statement in what is class in Gödel-Bernays axiomatic set theory. [duplicate]

For a collection $C$ define an object $b \in C$ iff $b$ is a class and $b \notin b$ and otherwise $b \in C$. Then we can decide any object whether it belongs to $C$. In Hungerford's Algebra p.2 it ...
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1answer
40 views

Defining functions of arbitrary arity and higher abstraction in set theory

I am wondering what is the best / agreed upon way to define functions of arbitrary arities in formal ZFC set theory, allowing also for higher levels of abstraction (e.g. arbitrary sequemces of ...
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1answer
57 views

Inequality with cardinal numbers

I want to check if the following statement holds: If $a, b, c $ are cardinal numbers and $a<b$ then $a^c<b^c$. If the statement would hold, given that there is a bijective funcion $f: a \to b$ ...
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0answers
53 views

Is any proper subclass of a proper class a set?

Am I allowed in ZFC / other ZFC-style systems to write: (1) C ≝ { x | x ∉ x} (2) S ⊂ C ∧ S ≠ C And use S as a set? As far as I see, Russell's paradox doesn't go through for S, since it is not ...
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1answer
78 views

Reference request: definition of class

I am looking for a reference on mathematically strict definition of class. For example, how do you construct the class of all ZFC-sets?
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1answer
138 views

What is the cardinality of the set of equivalence classes of infinite sequences in $\Bbb{Q}$ that converge in $\Bbb{R}$?

This started out as a question about Dedekind cuts and determining which of two interpretations of the definition of "Dedekind cut" is correct. While writing that question, I realized that I had ...
2
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1answer
42 views

Axiom of Choice for finite vs infinite product of sets

Why we don't need Axiom of Choice to prove the following statement Let $S_{\alpha}, \alpha \in A$ be a family of disjoint nonempty sets, and consider $P = \bigcup_{\alpha \in A} S_{\alpha}$. If $|A|...
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0answers
49 views

What are the normal subgroups of symmetric groups? [duplicate]

Let's for any cardinal $\alpha$ define $S_\alpha$ as the group of all permutations of a set of cardinality $\alpha$. Is there some sort of classification of normal subgroups of $S_\alpha$? For $\...