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 ...
47
votes
1answer
2k views
How do we know an $ \aleph_1 $ exists at all?
I have two questions, actually. The first is as the title says: how do we know there exists an infinite cardinal such that there exists no other cardinals between it and $ \aleph_0 $? (We would have ...
13
votes
1answer
638 views
About a paper of Zermelo
This about the famous article
Zermelo, E., Beweis, daß jede Menge wohlgeordnet werden kann, Math. Ann. 59 (4), 514–516 (1904),
available here. Edit: Springer link to the ...
49
votes
8answers
2k views
Different kinds of infinities?
Can someone explain to me how there can be different kinds of infinities?
I was reading "the man who loved only numbers" and came across the concept of countable and uncountable infinities, but ...
14
votes
2answers
609 views
Defining cardinality in the absence of choice
Under ZFC we can define cardinality $|A|$ for any set $A$ as
$$
|A|=\min\{\alpha\in \operatorname{Ord}: \exists\text{ bijection } A \to \alpha\}.
$$
This is because the axiom of choice allows any ...
16
votes
2answers
769 views
For every infinite $S$, $|S|=|S\times S|$ implies the Axiom of choice
how to prove the following conclusion:
[for any infinite set $S$,there exists a bijection $f:S\to S \times S$] implies the Axiom of choice.
Can you give a proof without the theory of ordinal ...
12
votes
2answers
522 views
Relationship between Continuum Hypothesis and Special Aleph Hypothesis under ZF
Special Aleph Hypothesis AH(0) is the claim $2^{\aleph_0}=\aleph_1$, i.e. there is a bijection from $2^{\aleph_0}$ to $\aleph_1$.
Continuum Hypothesis CH is the claim $\aleph_0 \leq \mathfrak{a}< ...
12
votes
5answers
1k 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 ...
8
votes
1answer
574 views
Polish Spaces and the Hilbert Cube
I've been trying to prove that every Polish Space is homeomorphic to a $G_\delta$ subspace of the Hilbert Cube. There is a hint saying that given a countable dense subset of the Polish space $\{x_n : ...
15
votes
1answer
2k views
Cardinality of Borel sigma algebra
It seems it's well known that if a sigma algebra is generated by countably many sets, then the cardinality of it is either finite or $c$ (the cardinality of continuum). But it seems hard to prove it, ...
19
votes
3answers
684 views
Set Theoretic Definition of Numbers
I am reading the book by Goldrei on Classic Set Theory. My question is more of a clarification. It is on if we are overloading symbols in some cases. For instance, when we define $2$ as a natural ...
9
votes
1answer
646 views
Nonnegative linear functionals over $l^\infty$
My purpose is a clarification of the role of the axiom of choice in constructing limits for bounded sequences. Namely, we want a linear functional of norm 1 defined on the space of all bounded complex ...
4
votes
3answers
343 views
Finite choice without AC
Can anyone explain how we choose one sock from each of finitely many pairs without the axiom of choice? I mean the following quote:
To choose one sock from each of infinitely many pairs of socks ...
4
votes
2answers
1k views
Proof of a Cantor-Bernstein-like theorem
I've been trying to find this proof:
If there exists $f \colon A\to B$ injective and $g \colon A \to B$ surjective, prove there exists $h \colon A \to B$ bijective.
I thought of using ...
11
votes
1answer
390 views
There's non-Aleph transfinite cardinals without the axiom of choice?
I can't find anything on this anywhere. The book I'm largely using at the moment is based around ZFC, so it makes no mention of anything other than the Aleph numbers, but according to Wikipedia on the ...
30
votes
9answers
3k views
Infinite sets don't exist!?
Has anyone read this article? Set theory
This accomplished mathematician gives his opinion on why he doesn't think infinite sets exist, and claims that axioms are nonsense. I don't disagree with his ...
8
votes
4answers
648 views
Foundation for analysis without axiom of choice?
Let's say I consider the Banach–Tarski paradox unacceptable, meaning that I would rather do all my mathematics without using the axiom of choice. As my foundation, I would presumably have to use ZF, ...
14
votes
2answers
572 views
The set of ultrafilters on an infinite set
After recently learning about filters and ultrafilters, we looked into further problems and properties. I am having trouble with this one:
If $X$ is an infinite set, then the set of all ultrafilters ...
3
votes
5answers
1k views
Cardinality of all cardinalities
Let $C = \{0, 1, 2, \ldots, \aleph_0, \aleph_1, \aleph_2, \ldots\}$. What is $\left|C\right|$? Or is it even well-defined?
23
votes
3answers
2k views
First-Order Logic vs. Second-Order Logic
Wikipedia describes the first-order vs. second-order logic as follows:
First-order logic uses only variables that range over individuals (elements of the domain of discourse); second-order logic ...
21
votes
4answers
724 views
Is Banach-Alaoglu equivalent to AC?
The Banach-Alaoglu theorem is well-known. It states that the closed unit ball in the dual space of a normed space is $\text{wk}^*$-compact. The proof relies heavily on Tychonoff's theorem.
As I have ...
15
votes
2answers
572 views
Continuity and the Axiom of Choice
In my introductory Analysis course, we learned two definitions of continuity.
$(1)$ A function $f:E \to \mathbb{C}$ is continuous at $a$ if every sequence $(z_n) \in E$ such that $z_n \to a$ ...
15
votes
1answer
613 views
Infinite Set is Disjoint Union of Two Infinite Sets
A finite set is a set such that there exists a bijection from it to some finite ordinal. An infinite set is a set that is not finite.
In ZF, can you prove that every infinite set is the union of two ...
6
votes
3answers
181 views
What can I do with proper classes?
There are standard tricks, constructions and techniques in ZFC when working with proper classes; for instance one can form the cartesian product of a pair of classes without difficulty, or more ...
1
vote
2answers
297 views
Advantage of accepting non-measurable sets
What would be the advantage of accepting non-measurable sets?
I personally feel that non-measurable sets only exist because of infamous Banach-Tarski paradox...
46
votes
4answers
3k views
What are the Axiom of Choice and Axiom of Determinacy?
Would someone please explain:
What does the Axiom of Choice mean, intuitively?
What does the Axiom of Determinancy mean, intuitively, and how does it contradict the Axiom of Choice?
as simple ...
51
votes
11answers
4k views
Why did mathematicians take Russell's paradox seriously?
Though I've understood the logic behind's Russell's paradox for long enough, I have to admit I've never really understood why mathematicians and mathematical historians thought it so important. Most ...
11
votes
5answers
710 views
What is a good text in intermediate set theory?
I've been working my way through Enderton's Elements of Set Theory for a while, and I feel I have a decent grasp on some of the basics of elementary set theory. My question is, where should I look to ...
43
votes
6answers
2k views
Why is $\omega$ the smallest $\infty$?
I am comfortable with the different sizes of infinities and Cantor's "diagonal argument" to prove that the set of all subsets of an infinite set has cardinality strictly greater than the set itself. ...
12
votes
1answer
531 views
Proofs given in undergrad degree that need Continuum hypothesis?
Or alternative you need to assume CH is false.
I know several proofs that use axiom of choice. Heine Borel theorem is the best example I can think off. Zorns lemma is heavily used in the non ...
-5
votes
1answer
891 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 ...
11
votes
2answers
463 views
Is there an algebraic homomorphism between two Banach algebras which is not continuous?
According to wikipedia, you need the Axiom of Choice to find a discontinuous map between two Banach spaces.
Does this procedure also apply for Banach algebras yielding a discontinuous multiplicative ...
9
votes
2answers
634 views
Uncountable subset with uncountable complement, without the Axiom of Choice
Let $X$ be a set and consider the collection $\mathcal{A}(X)$ of countable or cocountable subsets of $X$, that is, $E \in \mathcal{A}(X)$ if $E$ is countable or $X-E$ is countable. If $X$ is ...
7
votes
2answers
248 views
Implications of continuum hypothesis and consistency of ZFC
I've been a bit confused whilst doing some reading. I think the confusion arises because I am trying to read Wikipedia on topics without being able to work along. Anyway,
I have read that, of course, ...
7
votes
3answers
540 views
Is the class of cardinals totally ordered?
In a Wikipedia article
http://en.wikipedia.org/wiki/Aleph_number#Aleph-one
I encountered the following sentence:
"If the axiom of choice (AC) is used, it can be proved that the class of cardinal ...
7
votes
1answer
180 views
Cardinal equality question
The question contains 2 stages:
1. Prove that ${\aleph_n} ^ {\aleph_1} = {2}^{\aleph_1}\cdot\aleph_n$
This one is pretty clear by induction and by applying Hausdorf's formula.
2. Prove ...
3
votes
2answers
348 views
How do I choose an element from a non-empty set?
Suppose I have a non-empty set $A$.
How do I choose an element $x\in A$?
More precisely, I believe I would like to find a formula $P(x,y)$ of ZF such that for every non-empty set $y$ there is ...
13
votes
4answers
663 views
Algebraic closure for $\mathbb{Q}$ or $\mathbb{F}_p$ without Choice?
I know the usual proof of the existence of an algebraic closure for any field using Zorn's Lemma. The answer to this previous question makes it clear that in general, some nonconstructive axiom (not ...
13
votes
2answers
246 views
Does $2^X \cong 2^Y$ imply $X \cong Y$ without assuming the axiom of choice?
A friend of mine told me that $X \cong Y \Rightarrow 2^X \cong 2^Y$ ($X$ and $Y$ being sets), which is very easy to prove, but he was wondering about the converse in ZF, i.e., can one take logarithms? ...
8
votes
4answers
473 views
What's the cardinality of all sequences with coefficients in an infinite set?
My motivation for asking this question is that a classmate of mine asked me some kind of question that made me think of this one. I can't recall his exact question because he is kind of messy (both ...
7
votes
2answers
351 views
Uncountability of countable ordinals
According to Wikipedia, there are uncountably many countable ordinals. What is the easiest way to see this? If I construct ordinals in the standard way,
$$1,\ 2,\ \ldots,\ \omega,\ \omega +1,\ \omega ...
4
votes
1answer
167 views
Strictly associative coproducts
Background. This question belongs to evil mathematics. It is motivated by this question which links to a paper in which it is claimed that it is an open problem whether there exists strictly ...
9
votes
3answers
722 views
Axiom of choice, non-measurable sets, countable unions
I have been looking through several mathoverflow posts, especially these ones http://mathoverflow.net/questions/32720/non-borel-sets-without-axiom-of-choice , ...
6
votes
1answer
214 views
Is there a way to define the “size” of an infinite set that takes into account “intuitive” differences between sets?
The usual way to define the "size" of an infinite set is through cardinality, so that e.g. the sets $\{1, 2, 3, 4, \ldots\}$ and $\{0, 1, 2, 3, 4, \ldots\}$ have the same cardinality. However, is this ...
6
votes
3answers
466 views
Unary intersection of the empty set
In MK (Morse-Kelley) set theory life is easy: $\forall X\forall y\left(y\in\bigcap X\leftrightarrow\forall x\left(x\in X\rightarrow y\in x\right)\right)$. If $X=\left\{\right\}$ then $\bigcap X=U$, ...
15
votes
8answers
892 views
Where to begin with foundations of mathematics
I would like to know more about the foundations of mathematics, but I can't really figure out where it all starts. If I look in a book on axiomatic set theory, then it seems to be assumed that one ...
12
votes
7answers
668 views
Applications of ultrafilters
I'm looking for some interesting applications of ultrafilters and also everything of interest involving ultrafilters. Do you know some applications or interesting things involving ultrafilters?
I'm ...
33
votes
4answers
4k views
Can you explain the “Axiom of choice” in simple terms?
As I'm sure many of you do, I read the XKCD webcomic regularly. The most recent one involves a joke about the Axiom of Choice, which I didn't get.
I went to Wikipedia to see what the Axiom of ...
22
votes
6answers
807 views
Why is the Continuum Hypothesis (not) true?
I'm making my way through Peter J. Cameron's seminal text "Sets, Logics and Categories" where he makes the statement that the Continuum Hypothesis (There does not exist a set with a cardinality less ...
9
votes
1answer
730 views
Axiom of choice and calculus
I thought many results in calculus need axiom choice.
For example, I thought one needs AC to prove that a bounded sequence in the real line has a convergent subsequence.
Recently I was taught that one ...
20
votes
6answers
1k views
What are natural numbers?
What are the natural numbers?
Is it a valid question at all?
My understanding is that a set satisfying Peano axioms is called "the natural numbers" and from that one builds integers, rational ...
