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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3
votes
1answer
27 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
votes
1answer
24 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
52 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
vote
1answer
26 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
votes
1answer
22 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
votes
0answers
55 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
votes
1answer
36 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 $\beta^{\alpha}_i$. ...
1
vote
1answer
58 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 ...
-2
votes
0answers
32 views

Every subset of a well ordered set is well ordered [on hold]

Can anyone give me the proof that every subset of a well ordered set is well ordered.
4
votes
2answers
129 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
votes
1answer
59 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
votes
3answers
58 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 ...
1
vote
2answers
157 views

Friends and Enemies of Infinities [on hold]

Infinity is a dividing line in the community of mathematicians. There is a long standing contest between those who believe in rich theory of infinite mathematics and large infinite numbers and those ...
3
votes
1answer
58 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 ...
-1
votes
1answer
52 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 ...
1
vote
0answers
22 views

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: ...
0
votes
0answers
61 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 ...
1
vote
1answer
67 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. ...
0
votes
1answer
90 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 ...
0
votes
0answers
41 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 ...
1
vote
1answer
62 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} ...
5
votes
3answers
113 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 $ ...
0
votes
1answer
67 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
votes
1answer
78 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 ...
1
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1answer
31 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 ...
0
votes
2answers
57 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 ...
2
votes
1answer
48 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
votes
1answer
36 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 ...
0
votes
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, ...
1
vote
0answers
34 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
votes
2answers
99 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
votes
2answers
58 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 ...
1
vote
0answers
54 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? ...
1
vote
1answer
49 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, ...
0
votes
1answer
54 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 ...
4
votes
0answers
70 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 ...
0
votes
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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0answers
36 views

Interesting Characterizations of Woodin Cardinals

Woodin cardinals are very important large cardinals by many technical reasons. In this big list question I would like to start a thread for collecting all known/interesting characterizations of Woodin ...
1
vote
1answer
72 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
votes
1answer
62 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 = ...
4
votes
1answer
91 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, ...
3
votes
1answer
93 views

Undefinable Real Numbers

Disclamer: I'm sure my definition of "definable" may be different than the/a established mathematical one, I am more than interested in learning why/how this is so, but that is not my question Part ...
1
vote
1answer
88 views

Consistency of restricted forms of Martin's Axiom with the negation of the Continuum Hypothesis

Consider $\mathsf{MA}(S)$, the forcing axiom for all ccc posets which preserve a Souslin tree $S$. is $\lnot \mathsf{CH}$ consistent with $\mathsf{ZFC}+\mathsf{MA}(S)$? Does there exist a model for ...
2
votes
2answers
77 views

Embedding of linear order into $\mathcal{P}(\omega)/\mathrm{fin}$

I try to prove following problem (in Kunen): Assume $\mathrm{MA}(\kappa)$ and $(X,<)$ be a total order with $|X|\le\kappa$, then there are $a_x\subset \omega$ such that if $x<y$ then ...
2
votes
0answers
48 views

Large Cardinal Consequences of $\kappa$-Suslin Hypothesis

$\kappa$-Suslin Hypothesis ($\kappa$-SH) for the infinite regular cardinal $\kappa$ says that every tree of height $\kappa$ either has a branch of length $\kappa$ or an antichain of cardinality ...
0
votes
1answer
141 views

Category theory? Logic? Anyone experienced this like me? [closed]

Mathematics is not logic, but if one uses Zorn's lemma and stuff he should accept logical impact on mathematics. I'm the one who cares a lot about logics. It seems like Category theory is inevitable ...
1
vote
1answer
61 views

Product of a Suslin tree with itself

From Kunen "Set Theory", Chapter II, Exercise 36: If $T, T'$ are $\kappa$-trees, the product, $ T \times T' $ is the $\kappa$-tree whose $\alpha$-th level is $\mbox{Lev}_\alpha(T) \times ...
3
votes
2answers
63 views

$0^\sharp$ and the regularity of $\aleph_\omega$

I'm sure I'm missing something trivial, and the most likely of it is that I'm simply wrong on my understanding of the constructible universe $L$, or maybe one of the Wikipedia entries I'm about to ...
2
votes
1answer
65 views

How big can the continuum be without choice?

I've heard an argument before (although I can't remember where) that the continuum hypothesis is false, since the powerset operation is a something much more 'powerful' than the mere cardinal ...
4
votes
1answer
63 views

Are there any large cardinal properties of the critical point of a $j: L \longrightarrow L$?

I've recently been thinking a bit about $L$ and $0 \sharp$. As is well known, the existence of $0 \sharp$ is equivalent to the existence of a non-trivial elementary embedding $j: L \longrightarrow ...