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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5
votes
3answers
901 views

Counting equivalence relations

I am aware that on a finite set the number of equivalence relations is the n-th Bell's Number. On the other hand, the only reference I could find on the web for infinite sets was this: On counting ...
7
votes
4answers
692 views

number of infinite sets with different cardinalities

I would like to know whether the number of infinity sets of different cardinalities is uncountable or countable. Is the proof simple? Thanks.
5
votes
2answers
504 views

large sets. set theory

I haven't studied properly the theory of infinities yet. Let $A_0$ denote the set of natural numbers. Let $A_{i+1}$ denote the set whose elements are all the subsets of $A_i$ for $i=0,...,n,...$ I ...
3
votes
2answers
644 views

Choice function on $\mathcal P (\mathbb R) \setminus \{ \emptyset\}$

When I was first learning about the axiom of choice, a very helpful and fun exercise was to try to find a rule that maps every nonempty subset of $\mathbb R$ to one of its members. No matter how hard ...
2
votes
1answer
137 views

dense in terms of order and in terms of the order topology

In a densely and totally ordered set, induce a order topology from the order. Are the dense in terms of the order and the dense in terms of the order topology equivalent? Thanks and regards!
12
votes
2answers
777 views

What are large cardinals for?

I've heard large cardinals talked about, and I (think I) understand a little about how you define them, but I don't understand why you would bother. What's the simplest proof or whatever that ...
4
votes
6answers
3k views

Cutting the Cake Problem

The tradition way for two children to divide a piece of cake fairly between them is "you cut, I choose", What process accomplishes this is intuitively clear. Is there a similar process which works for ...
1
vote
1answer
813 views

How to prove that nothing is a member of itself?

I'd like to prove that $\forall x\left(x\not\in x\right)$ in the context of Morse-Kelley set theory. Let's call $A=\left\{y:y\not\in y\right\}$. I can easily prove that $A\not\in A$. In fact, if you ...
8
votes
3answers
645 views

Is the Subset Axiom Schema in ZF necessary?

I am learning the axioms of Zermelo-Fraenkel (ZF) set theory. One axiom schema basically says that given any set S and any formula phi(x), there is a set T consisting of all those elements x of S ...
5
votes
5answers
605 views

What's so special with small categories?

Sometimes one encounter the requirement that the objects of a category needs to be a set. What if it was not, could you provide examples of what could go wrong? (One example per answer).
1
vote
4answers
431 views

Sets. Classes. …?

Classes can be considerd as "larger" than sets in the sense that any set is a class. Is there mathematical object which is "larger" than classes ?
8
votes
2answers
477 views

Definition of a set

What is a set? I know that results such as Russell's paradox mean that the definition isn't as straight forward as one might expect.
14
votes
5answers
2k 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?
3
votes
1answer
111 views

Is the set of all unique (convex) polygons countable? If so, by what bijection to the natural numbers?

Polygons are, in this question, defined as non-unique if they similar to another (by rotation, reflection, translation, or scaling). Would this answer be any different if similar but non-identical ...
33
votes
4answers
2k views

If all sets were finite, how could the real numbers be defined?

An extreme form of constructivism is called finitisim. In this form, unlike the standard axiom system, infinite sets are not allowed. There are important mathematicians, such as Kronecker, who ...
42
votes
13answers
14k views

Why is “the set of all sets” a paradox?

I've heard of some other paradoxes involving sets (ie, "the set of all sets that do not contain themselves") and I understand how paradoxes arise from them. But this one I do not understand. Why is ...
85
votes
10answers
4k 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" by Paul Hoffman and came across the concept of countable and uncountable ...