Nate Lourwell

You seem that you could independently come up with that proof yourself on page 261 in Set Theory and Its Philosophy (Theorem. The alephs do not form a set). It's about cardinality. Best hopes for you as well!

It's stated on page 261 of Set Theory and Its Philosophy (Potter 2004), "The alephs do not form a set," with a proof. I'll have to try to understand it later, but this makes my set constructions an unlikelihood. Although disappointing, I'm glad for a firm answer to my inquiry this early on.

To continue coming up with new names for the sets would be difficult. The hierarchy of alephs is sometimes referred to as an ascending tower, so I selected [math]Tower[/math][0] = { [math]\aleph_{i}[/math] | [math]i \in \mathbb{N}[/math] } and, again, observed [math]\wp[/math]([math]Tower[/math][0]) = [math]Tower[/math][1] to grasp where it could lead. Without having to come up with a new name, there's [math]Tower[/math][1, 0] = { [math]Tower[/math][[math]i[/math]] | [math]i \in \mathbb{N}[/math] }, indicating a base-[math]\mathbb{N}[/math] positional system. In general, [math]Tower[/math][[math]1 _{n+1}[/math], [math]0_{n}[/math], . . . , [math]0_{2}[/math], [math]0_{1}[/math]] = { [math]Tower[/math][[math]i_{n}[/math], . . . , [math]i_{2}[/math], [math]i_{1}[/math]] | ([math]i_{n}[/math], . . . , [math]i_{2}[/math], [math]i_{1}[/math]) [math]\in base[/math] [math]\mathbb{N}[/math] }.

Hey, all. I don't have much background yet in pure mathematics. In the meantime, I'm wondering if it would make any sense in set theory to make the set [math]kaph_{0}[/math] = {[math]n \in \mathbb{N}[/math] | [math]\aleph_{n}[/math]} and then make [math]\wp[/math]([math]kaph_{0}[/math]) = [math]kaph_{1}[/math], analogous to how [math]\wp[/math]([math]\aleph_{0}[/math]) = [math]\aleph_{1}[/math]. If that's possible, could it be continued without bound, next with [math]yodh_{0}[/math] = {[math]n \in \mathbb{N}[/math] | [math]kaph_{n}[/math]}? Merged post follows: Consecutive posts mergedI learned that a correct way to describe [math]kaph_{0}[/math], if there is one, would be [math]kaph_{0}[/math] = {[math]\aleph_{i}[/math] | [math]i \in \mathbb{N}[/math]}. This set would fully capture the hierarchy of alephs, [math]yodh_{0}[/math] = {[math]kaph_{i}[/math] | [math]i \in \mathbb{N}[/math]} would fully capture the hierarchy of kaphs, and so on. My inquiry is about whether that's sensibly possible or not and, if it is, about whether it's at least as mathematically meaningful as it intuitively appears.