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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ineffable, weakly compact, and ? cardinal

In the online book, page 312, http://projecteuclid.org/download/pdf_1/euclid.pl/1235419485 what cardinal notion do we get, by requireing that X in the bottom of the page, is not only stationary ...
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0answers
37 views

Does this version of Schröder-Bernstein-Cantor imply choice? [duplicate]

Consider the following statement: $(*)$ For all sets $A$,$B$ and surjections $f\colon A \rightarrow B$, $g\colon B \rightarrow A$ there is a bijection $h\colon A \rightarrow B$ Given choice, this ...
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1answer
28 views

Generic in Boolean-Valued-Models

Let $M$ be a transitive $\in$-interpretation of a extension $T$ of $ZF$ in $ZF$,and let $B$ such that $$T\vdash B\in M\wedge B\text{ is a complete Boolean algebra}$$ Then, using the fact that any set ...
2
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1answer
60 views

Equivalent condition for CH

I would like to know why the following condition $\otimes$ is equivalent with the Continuum Hypotheses. $\otimes$ There exists a sequence $<A_{\alpha} | \alpha < \omega_1>$, such that for ...
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2answers
32 views

Are the hyperreal numbers densely ordered?

Using the construction mentioned in this post are the hyperreals densely ordered? If not, is there a construction in which they are? This should be a rather straightforward question, but my brain ...
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4answers
183 views

Partition of $\mathbb{R}^+$ into two semigroups

Can the semigroup $(\mathbb R^+ ,+)$ be partitioned into two semigroups? I have been trying but haven't found anything, please help.
2
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0answers
65 views

Kanamori's proof that Vopenka's principle implies the existence of extendibles.

In The Higher Infinite, Kanamori gives a proof that Vopenka's principle implies the existence of (many) extendibles. The relevant theorem is Proposition 24.15 on p.337. Working in a $V_\kappa$ which ...
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1answer
54 views

What is the formal way to define “class” in ZFC?

Unlike axiomatic systems deal with classes such as NBG, the term "class" is not a word in ZFC. How do I formally treat classes? Here is an example of what I'm exactly talking about. Example ...
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1answer
99 views

There are any known example of independence proofs about independence result?

(This question is inspired by deleted question, and the questioner who write the deleted question wrote new question.) It is well-known that consistency of "ZF + DC + every set of reals are Lebesgue ...
2
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2answers
145 views

Different models of ZF disagree on equality of explicit recursively enumerable sets

Assuming that ZF is consistent, are there two recursively enumerable sets defined by explicit enumerators that are the same in one model of ZF+Con(ZF) but different in another model of ZF+Con(ZF)? If ...
0
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1answer
39 views

Normal tree with $\aleph_1$ nodes and no branch of cardinality $\aleph_1$

I am trying to understand how a normal tree with $\aleph_1$ node can fail to have a branch of cardinality $\aleph_1$. Consider the tree of height $\omega_1$ whose nodes are $\mathbb{Q}$-valued ...
5
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1answer
158 views

Second reading on set theory? Any recommendations?

I have in past six-ish months studied through the Herbert Enderton's Elements of set theory book. Up to the point the book is great,I loved most parts of it and learned almost everything up to the ...
10
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2answers
354 views

Words in the Category of Sets

I was wondering about free objects in different categories and the "words" in those categories. I think I have a generally good grasp on the idea, but I started to think about stranger free objects ...
4
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1answer
133 views

Consistency strength of 0-1 valued Borel measures

The following is an overly fancy way of asking a question suggested in Borel Measures: Atoms vs. Point Masses Let $\phi$ be a property that topological spaces can have (such as "compact", "$T_1$", ...
2
votes
2answers
64 views

Cardinality of the set of all numbers that modern math can define?

I have recently learned that the algebraic numbers are countably infinite, and that very few transcendental numbers are known. Are enough transcendentals known to make up an uncountable set, or is ...
1
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1answer
36 views

Prove that $\omega_{\alpha+1}$ is regular

Here, under the section 'Regular and Singular Cardinal', there is this sentence 'Assuming the Axiom of Choice, $\omega_{\alpha+1}$ is regular for each $\alpha$' . May I know how to prove this? Also, ...
0
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1answer
30 views

Haudorff Formula Set Theory

For every $\alpha$ and every $\beta$, $$\aleph_{\alpha+1}^{\aleph_{\beta}}=\aleph_{\alpha}^{\aleph_{\beta}} \cdot \aleph_{\alpha+1}$$ Proof: If $\beta \geq \alpha+1$, then ...
3
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0answers
77 views

Intuition for the choice of background (set) theory

Problem From the formalist point of view, any mathematical statement should ultimately be an assertion about the derivability of a certain formula in a certain formal system, call it the background ...
2
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1answer
39 views

Todorcevic's proof on existence of Aronszajn Trees

I'm currently using Kanamori's book, The Higher Infinite, and am trying to understand the proof of Theorem 7.10 on page 79. However, I am unable to follow the construction of the $\beta^{\alpha}_i$. ...
2
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1answer
64 views

Graph of all sets

Working with set theory of ZFC, would it be possible to construct a graph (as in graph theory) with the class of all sets as its nodes? With "is it possible" I mean, would it lead to a contradiction ...
4
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2answers
145 views

What is the opposite category of $Set$?

In $Set$ the initial object is the empty set, and it has an unique morphism to each other object, namely $f=\emptyset$. However I find it difficult to think about the category ${Set}^{op}$, is there ...
0
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1answer
67 views

Infinite Cardinal Addition Without the Axiom of Choice

In the book 'Introduction to Set Theory' by Hrbacek and Jech, cardinal addition is defined as $$\sum_{i \in I}{\kappa_i}=\left|\bigcup_{i \in I}{A_i} \right|$$ where $|A_i|=k_i$ for all $i \in I$ ...
3
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3answers
64 views

$|A\times B|= \text{max}(|A|,|B|)$ for infinite sets

I am fairly sure, given examples $\Bbb{R}\times \Bbb{R},\Bbb{R}\times \Bbb{Q},\Bbb{Q}\times \Bbb{Q} $, that this is correct, but do not know how to prove it. In my cited examples the proof has ...
3
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1answer
74 views

Aronszajn Trees and König's Lemma

I'm looking at König's Lemma and Aronszajn Trees. I've seen the following link: König's Infinity Lemma and Aronszajn Trees Long story short, I'm still left with two questions and I was ...
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1answer
57 views

ZFC, NBG and Naive set theory

I want to discuss about 2 systems: ZFC and NBG, and Naive set theory. I know that we can prove some class is a set in NBG. Can we prove some class is a set in ZFC? (I know that every thing is a set ...
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0answers
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Transitive proof

I am trying to prove: Let $Trans(Z)\wedge x\subseteq Z$. Then $Z \cup \{x\}$ is transitive. Or prove: $$ Trans(Z)\wedge x\subseteq Z \rightarrow Trans(Z\cup \{x\}) $$ My attempt used the identity: ...
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0answers
71 views

Is the notion of “small cardinal” well definable?

When we talk about large cardinals, at least for many of them, we usually isolate a particular property expressing their "relative largeness with respect to cardinals below them". For example being ...
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1answer
72 views

Problem about infinitary combinatorics

I faced following problem: Prove that there is an s.d. family $\mathscr{A}\subset\mathcal{P}(\omega)$ of size $\omega_1$ which has a subfamily $\mathscr{B}\subseteq\mathscr{A}$ s.t. ...
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1answer
132 views

Platonist research on the cardinality of the reals

Apologies to any formalist! Here's the basic thought: $\mathbb{R}$ is a well-defined concept with unambiguous meaning in reality. Everyone can imagine an infinite series of digits (signifying the ...
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0answers
46 views

Cardinal Arithmetic Example Wikipedia

Hello I am studying cardinal arithmetic, and found out that I found that $\mathfrak{c}^{\aleph_0} = (2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0 \aleph_0} = 2^{\aleph_0} = \mathfrak{c} $. However I found ...
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1answer
63 views

If $X$ is a subset of $\omega_{\alpha}$ such that $|X| < \aleph_{\alpha}$, then $|\omega_{\alpha} - X| = \aleph_{\alpha}$

If $\alpha=0$, then $\omega_0=\mathbb{N}$ and $\aleph_0=$ countable. So $|\omega_{\alpha} - X| = \aleph_{\alpha}$ becomes $|\mathbb{N}-X|=|\mathbb{N}|$ which is true (the function $f:\mathbb{N} ...
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3answers
114 views

Showing a set of indexes where restriction is bijection is a club

I'm trying to show some general statement: I have a regular cardinal $ \kappa $ and an increasing continuous family of sets $ \langle X_\alpha \mid \alpha < \kappa \rangle $ with $ ...
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1answer
74 views

Godel Universes [closed]

Can somebody give me a nice and clear definition of what these are at different levels and different ordinals. I have read the wikipedia page and talked with peers, but am still confused about the ...
3
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1answer
92 views

ZF Set Theory and Law of the Excluded Middle

I know that the law of the excluded middle is implied in ZFC set theory, since it is implied by the axiom of choice. Taking away the axiom of choice, does ZF set theory (with axioms as stated in the ...
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1answer
35 views

A filter $G$ on $P$ is generic iff whenever $W$ is a partition of $P$, $G \cap W \not = \emptyset$

I'm finally struggling a little bit with the notion of forcing, so I decided to do some of the exercises in Jech's book (2nd Edition) in order to sharpen a little bit my intuitions about the whole ...
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2answers
60 views

Show that $\aleph_{\alpha} +\aleph_{\alpha} = \aleph_{\alpha}$ and $n \cdot \aleph_{\alpha} = \aleph_{\alpha}$

a) Give a direct proof of $\aleph_{\alpha} +\aleph_{\alpha} = \aleph_{\alpha}$ by expressing $\omega_{\alpha}$ as a disjoint union of two sets of cardinality $\aleph_{\alpha}$. b) Give a direct proof ...
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1answer
51 views

Introducing a new element to make a new model of set theory

Say we have a model of set theory $V$ and a partially ordered structure $\mathbb{P}$, and I want to talk about a $V$-generic filter $G$. A $V$-generic filter is a filter such that for every $D\in V$ ...
5
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1answer
39 views

What's a good introductory text to ZF set theory? [duplicate]

I've done the usual undergraduate coursework and am interested in learning about ZF set theory. What are some texts that would be accessible to me, and what are the most popular texts in this ...
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0answers
18 views

Logarithms of Cardinals [duplicate]

Given any infinite cardinal $\lambda\neq\omega$, is it the case that there's a cardinal $\kappa$ such that $2^{\kappa}=\lambda$? Does this depend on whether the Continuum Hypothesis is true? Clearly, ...
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0answers
37 views

Proving Replacement in $ZF^-$ without Replacement but with Collection

I was told in lecture, that in $ZF^-$ the replacement axiom scheme follows from adding the collection axiom scheme (without proof). So I tried proving it, but since I'm new to set theory, I need ...
6
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2answers
110 views

Does $\sf GCH$ imply that every uncountable cardinal is of the form $2^\kappa$?

I think that this is a popular fallacy that GCH implies that every uncountable cardinal is of the form $2^\kappa$ for some $\kappa$. I think it does imply that for successor cardinals only. It cannot ...
3
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2answers
62 views

Proving that a set that's not finite is infinite.

Call a set finite if there is a bijection of the set with some natural number, and call a set infinite if there is an injection of the set of natural numbers into that set. How do you prove that sets ...
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0answers
56 views

How to make a large cardinal unique?

A general form of questions regarding large cardinals is the following: Let $A(x)$ be the formula asserting "$x$ is a large cardinal of type $A$" then is the following true? ...
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1answer
53 views

Confusion in set-theory: Definition of formulas needs set

I am confused about some definitions in logic/ axiomatic set theory: We stated in our logic lecture the ZFC axioms and called the members of a ZFC-model "sets". But to define formulas and structures, ...
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1answer
56 views

Prove that $|A| < |A| + h(A)$ for all $A$

Prove that $|A| < |A| + h(A)$ for all $A$, where $h(A)$ is the Hartogs number of $A$. Attempt: By definition, $h(A) > 0$ because it is the least ordinal number which is not equipotent to any ...
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0answers
75 views

Do We Need Non-Constructible Sets?

I was reading about Godel's Constructible Universe in which the Continuum Hypothesis and Axiom of Choice are true. It made me wonder, what kind of math would we be unable to do if every set were ...
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1answer
35 views

For an infinite set $S$ , is $|S| < |$Sym $(S) |$?

Let $S$ be an infinite set ; does there exist any surjection of $S$ onto $A(S)$ ( the set of all bijections on $S$ ) ? I have atmost been able to prove that if $C( S)$ is the set of all countable ...
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1answer
77 views

Continuous functions on a Suslin line

This question is motivated by Brian Scott's answer in this thread. It looks to me that continuous functions on Suslin lines may have remarkable properties (from my perspective). Convention. I am ...
3
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1answer
67 views

If $2^{\aleph_0}$ is weakly inaccessible, can every cardinal $\kappa$ in the interval $[\aleph_0,2^{\aleph_0})$ satisfy $2^\kappa = 2^{\aleph_0}$?

Question. Is the following consistent with ZFC? $2^{\aleph_0}$ is weakly inaccessible Every cardinal $\kappa$ in the interval $[\aleph_0,2^{\aleph_0})$ satisfies $2^\kappa = ...
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1answer
92 views

When are extensional equivalence classes still sets?

Let $\sim$ denote extensional equivalence. That is, $y\sim x \Leftrightarrow \forall z(z\in y \leftrightarrow z\in x)$. Given a set $x$, let $[[x]] := \lbrace y:y\sim x\rbrace$. Clearly, ...