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
2answers
150 views

How to interpret $1 \to 0$ in ${\bf Set}^\mathrm{op}$, and ${\bf Set}^\mathrm{op}$ itself?

How to interpret the morphism $1 \to 0$ in ${\bf Set}^\mathrm{op}$, dual to $\bf Set$, with the standard meanings of the initial and terminal objects? Since the objects have the same interpretation ...
5
votes
3answers
354 views

Is there a branch of mathematics that requires the existence of sets that contain themselves?

I notice that Russell's paradox, Burali-Forti's paradox, and even Cantor's paradox, all depend on our tolerance of sets that contain themselves (at one level of depth or another). Thus, I was thinking ...
2
votes
2answers
502 views

Paradox of General Comprehension in Set Theory, other than Russell's Paradox

As is well known, the General Set Comprehension Principle (any class is a set) leads to the Russell Paradox (the class $x \notin x$ cannot be a set). As a result, set theories must restrict the ...
3
votes
1answer
312 views

Given an injection $\mathbb{N}\to\mathcal{P}(X)$, how can we construct a surjection $X\to\mathbb{N}$?

I goofed on my earlier post, here Given a surjection $f:\mathbb{N}\to\mathcal{P}(X)$, how can one construct an injection $X\to\mathbb{N}$? I am trying to show that there is an injection ...
1
vote
1answer
174 views

Given a surjection $f:\mathbb{N}\to\mathcal{P}(X)$, how can one construct an injection $X\to\mathbb{N}$?

I'm proving that $|X|\leq^*\mathbb{N}$ iff $\mathbb{N}\leq\mathcal{P}(X)$. One direction is trivial, but I'm not sure how to proceed for the other direction. Any suggestions? Update My apologies. ...
32
votes
7answers
8k views

Difference between a class and a set

I know what a set is. I have no idea what a class is. As best as I can make out, every set is also a class, but a class can be "larger" than any set. (A so-called "proper class".) This obviously ...
7
votes
1answer
359 views

Epsilon induction

I recently came upon this technique called epsilon induction, and was searching for some proof using the same. But I saw no such proof. Does someone know of any proof using this technique?
1
vote
1answer
81 views

Maximal set of pairwise disjoint elements of a dense subset.

Let $B$ be a complete Boolean algebra, and suppose $D \subseteq B$ is a dense subset. How is it possible to construct a maximal set of pairwise disjoint elements of $D$? Is it true that $\sum S = \sum ...
-4
votes
1answer
656 views

A counter-example for a set-theoretic problem?

I have proved the below conjecture for the special cases $n\in\{0,1,2\}$. The cases $n\ge 3$ (finite and infinite) are unknown. If the following conjecture is true, I don't expect that you will be ...
0
votes
1answer
42 views

(final) episinks preserved under pullbacks in SET

How do I prove the following: We work in SET. If $(f_{i}:Y_{i} \rightarrow Y)_{i \in I}$ is a (final) episink, $f: X \rightarrow Y$, and $h_{i}: X_{i} \rightarrow Y_{i}$ are functions such that for ...
3
votes
2answers
166 views

Existence of a particular well-ordering of [0,1]

How do you show, assuming the Axiom of Choice and the Continuum Hypothesis, that there exists a well-ordering on $[0,1]$ such that for all $x$, there are only countably many $y$ such that $y \leq x$?
4
votes
3answers
116 views

Can the truth value of an independent property be changed at will by enlarging the model?

Let $\phi$ be a property that's independent of $ZFC$, so that there are strcutures ${\mathfrak A}=(A,{\in}_A)$ (where $A$ is a set or class and ${\in}_A$ is a binary relation on $A$) that are models ...
2
votes
2answers
946 views

set-theoretic function definition; recursion theorem

I am an undergraduate student, currently studying axiomatic set theory (I am reading Halmos' Naive Set Theory as an overview, and consulting other sources recommended to me to supplement the sparser ...
7
votes
1answer
198 views

Shelah's proof of diamond

This paper is http://www.cs.elte.hu/~kope/ss3.pdf . In Remark 1 : I want to prove that the set $$D=\{\delta<\lambda^+:\pi\restriction \delta \text{ is a bijection onto }\lambda\times\delta\}$$ is ...
2
votes
2answers
68 views

Elementary Question

Let be $f:X\to X$ a bijection, an $A\subset X$ a invariant subset of $X$, i.e $f(A)\subset A.$ How can see that $$f(A)=A$$ I'm trying to show that $$f(A^{c})\subset A^c$$ but I can not.
0
votes
1answer
87 views

enumeration of subsets

Good evening. Let $\lambda>\omega$ a cardinal. We know that there is a bijection $\pi$ between $\lambda^+$ and $\lambda\times\lambda^+$. I don't understand in Remark 1 of the paper Shelah's proof ...
5
votes
2answers
266 views

Analytic versus Analytical Sets

Browsing MathOverflow I came across a question about analytical sets. Through the discussion following a comment made by our very own Asaf, I learned that bold face $\mathbf{\Sigma^1_1}$ and light ...
5
votes
1answer
222 views

How do we justify functions on the Ordinals

In ZFC there seem to be two ways to define a function, as an ordered triple of domain, codomain and the set of ordered pairs that make its graph, or a relation over the two sets. Whichever we use ...
1
vote
0answers
67 views

$\Delta_0$-formulas in the Boolean-valued model $V^B$

I want to show that $\Vert \check x = \check y \Vert = 1$ implies $x = y$, where $\check x$ is the canonical name for $x$ in $V^B$. I'd like try induction over the ranks of $\check x$ and $\check y$ ...
4
votes
1answer
246 views

$\kappa$-Suslin (Aronszajn, Kurepa) subtree of the complete binary tree, $2^{\lt \kappa}$

This is from chapter 2 of Kunen, Set theory: an introduction to independence proofs. Given $\kappa$ a regular cardinal, and the existence of a $\kappa$-Suslin (Aronszajn, Kurepa) tree, show that ...
8
votes
2answers
1k views

Difference between ZFC & NBG

Can someone tell me what's the advantages and disadvantages of using NBG rather than ZFC and what's the advantages and disadvantages of using ZFC rather than NBG?
2
votes
1answer
129 views

Question about Cuts in Boolean Algebras

Let $A$ be a Boolean algebra, and let $A^+$ denote the set of non-zero elements of $A$. A cut $U \subseteq A^+$ is a set such that if $q\in U$, then $p\le q \implies p \in U$, for all $p \in A^+$. A ...
3
votes
1answer
276 views

Provability and truth

The following is quoted from Set Theory and the Continuum Problem by Raymond M. Smullyan and Melvin Fitting. So far, no attempts have been the slightest bit successful in determining whether the ...
5
votes
1answer
900 views

Union of ordinals

From Jech (pg. 20): If $X$ is a nonempty set of Ordinals, than $\bigcup X$ is an ordinal. This should be easy enough to prove but I don't see how; also, I guess this is not the case if $X$ is a ...
1
vote
2answers
451 views

Example of total order with some properties that is not well ordered

Is there an example of a total order with properties there is a least element and every element has a (unique) successor not is not also a well ordering?
5
votes
1answer
205 views

Cardinal power $\kappa^\kappa$. When is it equal to $2^\kappa$?

Under what assumptions on an infinite cardinal $\kappa$ we have $$\kappa^\kappa= 2^\kappa?$$ Please delete this question. I know the answer.
2
votes
1answer
235 views

Forcing questions

I have been looking at a proof of a technical forcing lemma, and I have a couple of questions. Here is the setup: $(N, \epsilon) \prec (\mathbf{H}( \chi ), P, \epsilon ) $ is a countable submodel, ...
2
votes
1answer
153 views

Cardinal exponentation

Let $\lambda, \kappa$ be infinite cardinals. Does it follow from the inequality $\kappa\lt2^\lambda$ that $2^\kappa\lt2^{2^\lambda}$?
2
votes
1answer
155 views

Is the notation for the definition of a set stricly formalized?

In the 800 pages set theory book by Jech, he uncommented starts using $$Y=\{ u\in X : \phi(u)\}$$ as equivalent to $$Y=\{ u:u\in X \wedge\phi(u)\}$$ on the first few pages. The fact that, in the ...
1
vote
1answer
271 views

Do the proofes in set theory rely on the semantics of the formulas used in the axioms?

Motivation: The Axiom of separation $$\forall w_1,\ldots,w_n \, \forall A \, \exists B \, \forall x \, ( x \in B \Leftrightarrow [ x \in A \wedge \phi(x, w_1, \ldots, w_n, A) ] )$$ is used to ...
3
votes
1answer
206 views

Measurable cardinal existence condition

We know that a cardinal $k > \omega$ is measurable if there is a measure function $\mu :2^k\mapsto \{0, 1\}$ that satisfies the following 3 conditions: 1.$<k$ -additive: for every set I of ...
13
votes
1answer
1k views

Can one construct a non-measurable set without Axiom of choice?

Is axiom of choice required to show the existence of non-measurable sets? Is there a Lebesgue non-measurable set that can be constructed without axiom of choice? Related question on MO says it is ...
6
votes
1answer
514 views

Cantor's Back-and-Forth method

I'm interested in Cantor's back-and-forth method so I searched online and surprisingly it was on the Wikipedia. My question is about the paragraph that starts saying: If we iterated only step (1), ...
2
votes
2answers
286 views

Is there a way of defining the notion of a variable mathematically?

I know that the notion of "set" is one that cannot be defined mathematically since it is the fundamental data type that is used to define everything else (and the definition which says that "sets" are ...
0
votes
1answer
211 views

Moving a union of functions with disjoint domains

I will denote $(\lambda x\in D: f(x)) = \{(x;f(x))\,|\,x\in D\}$ for every set $D$ and a form $f$ dependent on the variable $x$. For a function $z$ and a function $a$ we define $$\operatorname{Move} ...
3
votes
3answers
261 views

An elementary question regarding the uniqueness of a set, viewed with different cardinality

Does the cardinality of sets, like for example the real numbers, depend on the fundamental axioms one is working with? If so, what does it mean to speak of such a set if it is not really one single ...
12
votes
6answers
2k views

Axiom of Choice and finite sets

So I am relatively familiar with the Axiom of Choice and a few of its equivalent forms (Zorn's Lemma, Surjective implies right invertible, etc.) but I have never actually taken a set theory course. I ...
2
votes
1answer
87 views

Cardinality of a club

I proved that a club $C$ in $\kappa$ has the same cardinality as $\kappa$. Is it really true ? Thanks.
4
votes
3answers
342 views

How do I get the existence of a set in ZFC following Jech?

I am learning some set theory and logic on the side and am looking Jech's book, "Set Theory". At the moment, I am learning the basic axioms, and what I can and cannot do with them. Most of the axioms ...
5
votes
2answers
342 views

Can non-existent set still be infinite?

Can we claim that there are infinite number of objects when the set of the objects does not exist? For example, there is no set of all sets, but can we still say that there are infinitely many sets ...
-4
votes
1answer
1k views

Formalizing an idea [closed]

All rational numbers of the unit interval [0, 1] can be covered by countably many intervals, such that the $n$-th rational is covered by an interval of measure $1/10^n$. There remain countably many ...
20
votes
4answers
2k views

Why is the axiom of choice separated from the other axioms?

I don't know much about set theory or foundational mathematics, this question arose just out of curiosity. As far as I know, the widely accepted axioms of set theory is the Zermelo-Fraenkel axioms ...
3
votes
1answer
488 views

Can the real numbers be forced to have arbitrary cardinality?

Let $\mathbb R$ be the real numbers in a given model of set theory. Given an arbitrary cardinal number $\kappa$, does forcing produce a larger model in which the cardinality of $\mathbb R$ is ...
1
vote
1answer
182 views

Currying for dependent functions

Currying and uncurrying is defined between functions in $Z^{X \times Y}$ (the first set) and $\left( Z^Y \right)^X$ (the second set). But what if $Y$ is not a constant but is dependent on $X$? The ...
11
votes
3answers
3k views

Is Foundational Research a Dead Field?

I'm a second year mathematics major at a pretty good school. Ever since I became a math major I have been most interested in set theory and logic, which I guess can be lumped into the category of ...
3
votes
1answer
169 views

Aronszajn tree which cannot be a Suslin tree

I am doing some exercises from Kunen's book "Set Theory", I'm having problems with exercise 39 from page 90, the exercise goes like this: Show that any Aronszajn tree which is a subtree of $\{ s \in ...
20
votes
3answers
742 views

Conflicting definitions of “continuity” of ordinal-valued functions on the ordinals

I've encountered the following definition in Kunen, Levy, and other places: A function $\mathbf{F}:\mathbf{ON}\to\mathbf{ON}$ is continuous iff for every limit ordinal $\lambda$, we have ...
3
votes
2answers
198 views

The Product of Suslin Trees under the Diamond Axiom

I am familiar with the result that $\Diamond$ implies that there exists a Suslin tree, but does it also imply that, if S, T are Suslin trees, then S x T is a Suslin tree? If not, perhaps there is a ...
0
votes
1answer
353 views

Set terminology and symbols in optimization

Coming from engineering background, I'm getting a little lost in terminology and symbols, but I still want to be mathematically precise. In engineering optimization, I often have say two design ...
3
votes
2answers
158 views

Bijections in ZF

Let $X$ be an infinite set. Is it provable in ZF that there is a bijection between $X$ and the family of finite subsets of $X$? If no, does the existence of such bijection imply (in ZF) that there ...