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 ...

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Redundancy in definiton for forcing poset

The following definition appears in Kunen (2nd edition): For any sets $I,J$ and cardinal $\lambda$: $\text{Fn}_{\lambda}(I,J)$ is the set of all $p\in{[I\times{J}]}^{<\lambda}$ such that $p$ is ...
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19 views

What is the definition of the Feferman-Levy model?

Any (reference to) definition of Feferman-Levy model in set theory? I cannot find any... Though I know what is Levy collapse.
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1answer
13 views

Uniqueness in transfinite recursion.

Theorem of transfinite recursion: Given a well-ordered set $A$ let $\varphi(g,y)$ be a ZF formula such that for every $a \in A$ and every function $g$ with domain $I_a$ (where $I_a$ is the initial ...
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1answer
25 views

If a version of GCH holds for Chang's $\kappa$-constructibility, does a version of GCH hold for $L_{\infty}$?

In C.C. Chang's paper "Sets Constructible Using $L_{\kappa \kappa}$" one can "deduce a version of the GCH, theorem VI [(iv)--my comment], assuming that all sets are $\kappa$-constructible." Now ...
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0answers
117 views

Are there any good documentary films about the continuum hypothesis?

Are there any good documentary films about the continuum hypothesis? I'm looking for something slightly more serious than the usual "Cantor showed that infinity plus one equals infinity and then went ...
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1answer
92 views

Why is adding Cohen reals so “uninteresting”?

I read the following in this paper (Otmar Spinas, Proper products. Proceedings of the AMS, 137 (8), (2009), 2767–2772): $ \mathbb{L}^2 $ adds a Cohen real. Thus it was considered uninteresting ...
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2answers
83 views

Examples of figures whose area depend on axiom of choice

Many times I have heard that there are some 'figures' whose area is not fixed and they depend on axiom of choice. For example, In this answer. In this lecture about probabilty. I did not know ...
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6answers
514 views

Proof that aleph null is the smallest transfinite number?

The wikipedia page on the cardinal numbers says that $\aleph_0$, the cardinality of the set of natural numbers, is the smallest transfinite number. It doesn't provide a proof. Similarly, this page ...
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Coming from naive set theory, are the following understanding right in ZF system? [on hold]

In ZF system, a subset $S'$ of set $S$ is: If $S'$ is a set and for any object $b$ $b\in S'\implies x\in S$. That is, before being a subset, $S'$ must be a set by itself. The set of integer ...
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1answer
82 views

What would provable existence of a non-standard natural number imply about ZFC?

Assume that in ZFC we can define a set $x$ in some way and that the following are provable: $x$ is a finite ordinal, (i.e. $x\in\omega$) $x\neq0$ $x\neq1$ $x\neq2$ etc. So to say that the ...
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1answer
45 views

What are $\in$-well orderable classes?

$V$ is not $\in$-well ordered but $Ord$ is. Is $Ord$ the unique proper class of the universe which is $\in$-well ordered?
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Is the structure with sets and the ZFC axioms a model of the first order logic?

Wikipedia says ZFC is a one-sorted theory in first-order logic. The signature has equality and a single primitive binary relation, set membership, which is usually denoted ∈. The formula a ∈ b ...
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2answers
347 views

About translating subsets of $\Bbb R^2.$

I'm looking for a pair of sets $A,B$ of points in $\Bbb R^2$ such that $A$ is a union of translated (only translations are allowed) copies of $B;$ $B$ is a union of translated copies of $A;$ $A$ is ...
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0answers
55 views

Expository Papers on Extenders

Extenders are discussed in many set theory text books. Here I am looking for some expository "papers" which are focused on this subject and its connection with forcing and large cardinals. More ...
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46 views

Similarity of the Universe $V$ and its $\kappa$-Fragments

Intuitively, if a cardinal $\kappa$ is "large" then the $\kappa$-fragment of the universe $V_{\kappa}$ is so "similar" to the entire universe $V$. For example if $\kappa$ is supercompact then ...
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0answers
30 views

Is infinitary Levy hierarchy well-defined?

The well-known Levy hierarchy of formulas consist of two $\omega$-sequences of sets of formulas of different complexity $\langle\langle \Sigma_n:n\in \omega\rangle,\langle \Pi_n:n\in ...
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1answer
53 views

infinite Cartesian products

Quote from Wikipedia, infinite set: The Cartesian product of an infinite number of sets each containing at least two elements is either empty or infinite. I know that this is the regime where we ...
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0answers
33 views

Filters on a set of filters, are they equivalent to just filters?

Let $F(X)$ be the set of all filters (including the improper filter) on a poset $X$, ordered reversely to set-theoretic inclusion of filters. Let $U$ be a set. Is $F(F(\mathscr{P}U))$ order ...
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0answers
51 views

Existence of a generic ultrafilter over constructible universe

I am referring to the Jech's "Set Theory" book. He wants to show that, if $(P,<)$ is the notion of forcing defined as: $p \in P$ iff $p$ is a finite sequence of ordinals less than $\omega_{1}^{L}$ ...
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2answers
538 views

The Continuum Hypothesis & The Axiom of Choice

Does anyone here know of a reference to an analysis on a proposed relationship between The Continuum Hypothesis and The Axiom of Choice?
22
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1answer
493 views

Model existence for infinitary logics

One of the problems of infinitary logic is that it is possible for compactness to fail in a spectacular way: for example, one can concoct an inconsistent set of axioms whose proper subsets are all ...
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0answers
56 views

Zuckerman's “Sets and Transfinite Numbers”

I am beginning a study in set theory and I found an old book in my school's library by Martin Zuckeman called Sets and Transfinite Numbers which was published in the 1970's. Has anyone used this text ...
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5answers
572 views

Do we need Axiom of Choice to make infinite choices from a set?

According to the answers to this question, we do not need choice to pick from a finite product of nonempty sets, even if each of the sets is infinite. The axiom of choice is required to ensure that a ...
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0answers
32 views

Flaw in the proof that a set is countable [duplicate]

Q: Let S be the set containing all sequences of 0's and 1's. i.e $S = \{(a_1,a_2,a_3,a_4,\ldots) : a_i = 0 \text{ or } 1\}$ Show that S is countable. Proof(Flawed) : Let $A_i$ be the ...
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106 views

Partitions of the Cantor space into parities

Call a partition of $2^\mathbb{N} = A\cup B$ a parity partition if, for any $n\in\mathbb{N}$, flipping the $n$th bit of any element of $A$ results in an element of $B$, and vice-versa. Given a choice ...
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0answers
61 views

Algebraic constructions to add a real to a sub-group of $\langle \mathbb{R}^+,.\rangle$

Consider the group $\langle \mathbb{R}^+,.\rangle$ and let $r$ be a real number (e.g. $r=\sqrt{2}$). I would like to know about all known algebraic constructions to build a sub-group of $\langle ...
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1answer
45 views

Exact $\Delta$ - System Lemma

The well-known $\Delta$-system lemma states: Each uncountable family of finite sets has an uncountable sub-family with a root. Precisely: $$\forall F~~~(|F|>\aleph_0~\wedge~\forall X\in ...
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1answer
49 views

A question about $cov(I)$

Given a set $X$ and an ideal $I \subseteq \mathcal P(X)$, $cov(I) := \min \{ |\mathcal A| | \mathcal A \subseteq I (\bigcup \mathcal A = X)\}$. My question is: isn't it possible that for each ...
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1answer
78 views

difference of infinities [closed]

So I haven't studied this subject In a while, and when i get home, I'm going to consult my book on the Theory of computation to reinforce my understanding, but here is my question. So I understand ...
3
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1answer
75 views

$\aleph_\omega$ and large ordinals?

Assuming the GCH, each time you take the powerset of an aleph number, the subscript increases by one. So there is $\aleph_0, \aleph_1, \aleph_2\ldots \aleph_\omega\ldots \aleph_{\epsilon_0}\ldots$ and ...
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1answer
200 views

Product forcing and generic objects

If we start with a model of $\sf ZFC$, $M$ and $(P,\le)\in M$ is a notion of forcing, $G\subseteq P$ a generic filter, then in $M[G]$ we can define some generic object from $G$. For example, if $P$ is ...
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0answers
77 views

A Question Regarding Consistent Fragments of Naive (Ideal) Set Theory

It is well known that Naive (to some, otherwise known as Ideal) set theory, that is, the set theory generated by the axioms: (EXT) (x)(x $\in$ A iff x $\in$ B) iff A=B (COMP) ($\exists$y)(x)(x ...
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1answer
52 views

Could induction give us an infinite sequence of sets $X_1 \subseteq X_2 \subseteq \cdots$ or do we need the axiom of choice?

Suppose at each step $n$ of a proof by induction, we have a set $X_n$. Also, $X_{n} \subseteq X_{n+1}$. Is induction enough to give us $\bigcup_{n \in \mathbb{N}} X_n$? Or rather, when is it enough? ...
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109 views

Is it true that $\omega=\{0,1,2,3,\ldots\}$ in ZFC?

This is a bit of a philosophical question. According to "Set Theory" by Jech, the set $\omega$ of natural numbers is defined as the least nonzero limit ordinal. After thinking about this definition ...
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1answer
30 views

What is the meaning of the notation $cf(add(\mathcal I))$?

I am trying to prover the following claim: Assume that $\mathcal I$ is an ideal over some infinite set $X$. Then, $cf(add(\mathcal I))=add(\mathcal I)$. Doe anyone know what is the meaning of the ...
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0answers
54 views

Other Types of Order Types

Consider following fundamental fact of ZFC. Fact ‎‎$‎‎‎\forall ‎‎~\text{well-ordering}~\langle ‎A,<‎\rangle~~~‎\exists ‎!\alpha~~~‎‎‎‎‎\langle A,<‎‎\rangle‎\cong‎\langle‎\alpha‎,\in‎‎\rangle‎$‎ ...
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1answer
102 views

Constructing a Banach space of cardinality $\beth_{\omega+1}$

This is related to yesterday's question Constructing a vector space of dimension $\beth_\omega$; it's the next exercise (I.13.35 (a)) in Kunen's Set Theory. Let $B_0 = \ell^1$ and let $B_{n+1} = ...
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1answer
129 views

Can't prove Continuum Hypothesis

My teacher said (and I checked on Wikipedia too), that there is a theory, the Continuum hypothesis, that can neither be proved nor disproved. However, the theory states that: There is no set ...
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0answers
47 views

Exercise on preservation of cardinals

I'm trying to solve exercise 12.1 in Prof. Monk's lectures on set theory which asks me to show that $Fn(\omega_1, 2, \omega_1)$ preserves cardinals larger than $\omega_2$. Now, the only method in the ...
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2answers
98 views

Constructing a vector space of dimension $\beth_\omega$

I'm trying to solve Exercise I.13.34 of Kunen's Set Theory, which goes as follows (paraphrased): Let $F$ be a field with $|F| < \beth_\omega$, and $W_0$ a vector space over $F$ with $\aleph_0 ...
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1answer
114 views

cov(meager) strictly between $\aleph_1$ and $2^{\aleph_0}$

Is is consistent that $\aleph_1 < \text{cov(meager)} < 2^{\aleph_0}$? I can only seem to find references for results that assert it is consistent that it (or other cardinal characteristics) is ...
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1answer
35 views

Model of a system of set theory

I was trying to understand Easton's theorem's proof. But I am not able to understand a few terms- model of a system of set theory, ranked variables, free variables, abstraction term, set constant, ...
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1answer
38 views

Functions with cardinals as domain

How do we actually define a function with cardinals as domain? For example, take domain of the function as $\aleph_1$. Do we define it for all cardinal numbers strictly less than $\aleph_1$?
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0answers
37 views

The notations $ M[X] $ and $ V[A] $ and intermediate models

I am confused about Jech's notations $ M[X] $ and $ V[A] $. Let us begin with exercise 13.34 (see [Jec02, p. 199]). Let $ \mathbf{M} $ be a transitive model of $ \mathsf{ZF} $ containing all ...
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1answer
30 views

Cardinal Arithmetic, Regular Cardinals, and Exponentiation

I cannot solve a simple cardinal exponentiation/regularity exercise. Let $\kappa$ be a regular cardinal. Why is the cardinal $\kappa^{\lt \kappa}=\sum_{\alpha<\kappa}\kappa^\alpha$ regular as ...
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1answer
25 views

Union of Dedekind-finite sets

$F$ is Dedekind-finite if for every $A\varsubsetneq F$ we have $A<_cF$. Need help to prove that if $F,G$ are Dedekind-finite sets, $F\cap G=\emptyset$ then $F\cup G$ is also Dedekind-finite. ...
2
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1answer
57 views

In $\sf{ZF}$, is every finite set also hereditarily finite?

The following is an excerpt from Wikipedia's Hereditarily finite set page: A recursive definition of well-founded hereditarily finite sets goes as follows: Base case: The empty set is a ...
6
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1answer
77 views

Construction of Ultrafilters

I've been doing a lot of work with ultrapowers and saturation recently. In particular, I am reading chapter 6 of Chang and Keisler as well as Keisler's paper on "Ultraproducts which are not ...
3
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1answer
83 views

Definition in Kunen

In Kunen's second edition of set theory he gives the following definition Let $(\mathbb{Q},\leq_\mathbb{Q},\mathbb{1}_\mathbb{Q})$, and $(\mathbb{P},\leq_\mathbb{P},\mathbb{1}_\mathbb{P})$ be forcing ...
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2answers
113 views

Is the “Most Important Property a Set S has” Necessary and Sufficient to Define a Paradox-Free Notion of Set?

About a year and a half ago, while I was looking on the Web for papers regarding the Russell paradox, I chanced to find an interesting concept. This concept was contained in what (for want of a ...