Articles of elementary set theory

Infinity as an element

Can infinity be an element of a set? For example: {$1,2, \infty,4$} I understand $\infty$ is not a real number

Correct formulation of axiom of choice

In a paper, Asaf Karagila writes: Definition 1 (The Axiom of Choice). If $\{A_i \mid i ∈ I\}$ is a set of non-empty sets, then there exists a function $f$ with domain $I$ such that $f(i) ∈ A_i$ for all $i ∈ I$. Does this formally make sense? Shouldn’t it say If $(A_i)_{i\in I}$ be […]

Question about sets and classes

I know every set is a class and not the other way around, but can one consider the set of, say, two classes? Is this “well-defined”?

$\exists\text{ set }X:X=X^X$?

Given sets A and B, define the set $B^A$ to be the set of all functions A $\to$ B. My question is: Is there a set X such that X = $X^X$? Has this something to do with the axiom of regularity?

Is the supremum of an ordinal the next ordinal?

I apologize for this naive question. Let $\eta$ be an ordinal. Isn’t the supremum of $\eta$ just $\eta+1$? If this is true, the supremum is only necessary if you conisder sets of ordinals.

How many cardinals are there?

I’m trying to do the following exercise: EXERCISE 9(X): Is there a natural end to this process of forming new infinite cardinals? We recommend this exercise instead of counting sheep when you have trouble falling asleep. (This is from W. Just and M. Weese, Discovering Modern Set Theory, vol.1, p.34.) By this process they mean […]

Does there exist a non-empty set that is a subset of its power set?

While working through Velleman, I proved that if $A \subseteq P(A)$, then $P(A) \subseteq P(P(A))$. One example where this may be the case is when $A = \emptyset$. Another may be when $\emptyset \in A$. I cannot think of any other example though. Supposing that $x$ and $y$ are two arbitrary elements of $A$, then […]

Taking away infinitely many elements infinitely many times

This question already has an answer here: Mutually exclusive countable subsets of a countable set 6 answers

Alternatives to show that $|\mathbb{R}|>|\mathbb{Z}|$

Cantor’s Diagonal Argument is the standard proof of this theorem. However there must be other proofs, what are some of these proofs? I am asking this because whenever I think of this question, I immediately think of the Cantor’s Argument, which kills the possibility of other interesting finds.

Does Lowenheim-Skolem theorem depend on axiom of choice?

The proofs of Lowenheim-Skolem I have seen all depended on the use of choice functions. Is there any proof not dependent on axiom of choice? Or is Lowenheim-Skolem a result of axiom of choice?