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

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 […]

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”?

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?

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.

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 […]

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 […]

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

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.

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?

Intereting Posts

Proving statement – $(A \setminus B) \cup (A \setminus C) = B\Leftrightarrow A=B , C\cap B=\varnothing$
Is the geometric dot product formula equal to the algebraic one and how can I get one from the other in a step by step fashion ?
Are there other cases similar to Herglotz's integral $\int_0^1\frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}\ \mathrm dt$?
The Limit of $x\left(\sqrt{a}-1\right)$ as $x\to\infty$.
Proving that a process is a Brownian motion
Homology of a simple chain complex
Algebraic numbers that cannot be expressed using integers and elementary functions
Limit of Lebesgue measure of interesection of a set $E$ with its translation
Closed-forms of infinite series with factorial in the denominator
Geometric interpretation of a complex set
Subadditivity of Lebesgue-Stieltjes measure
Can someone explain these strange properties of $10, 11, 12$ and $13$?
Should the domain of a function be inferred?
Constructing faithful representations of finite dim. Lie algebra considering basis elements
Show that if $B \subseteq C$, then $\mathcal{P}(B) \subseteq \mathcal{P}(C)$