On $\S$7b of David Foster Wallace's book "Everything and more", the author explains how Cantor derived the concept of transfinite numbers from P, a second-species infinite point-set.
"$P'$, can be "decomposed" or broken down into the union of two different subsets, $Q$ and $R$, where $Q$ is the set of all points belonging to first-species derived sets of $P'$, and $R$ is the set of all points that are contained in every single derived set of $P'$, meaning $R$ is the set of just those points that all the derived sets of $P'$ have in common."
Then, these two definitions are provided:
$(1) \ \ R = \cap (\ P^{(2)}, P^{(3)}, P^{(4)}, P^{(5)}, ...) ... $
$(2) \ \ R = \cap (\ P^{(n)}, P^{(n+1)}, P^{(n+2)}, P^{(n+3)}, ...) $
It is stated then that :
$(\alpha)$ "Notice how (1)'s got ellipses outside the right paren, too, meaning the sequence continues beyond the finite-superscripted progression. (2)'s ellipses are 100% intraparenthetical because n itself is infinite."
$(\beta)$ "What (1) and (2) together really are is a type of proof [...] called mathematical induction."
$(\gamma)$ "What Cantor's (1) and (2) allow him to do is to define R, as taken from P, as: $R = P^\infty $ - that is, R is the $\infty$th derived set of P."
Nonetheless, I don't get the idea of :
$(A)$ In $(\alpha)$ commentary: I just can't see why in $(1)$ ellipses are doubled but in $(2)$ they're singled. Is not $n$ an integer anymore, perhaps? But then, how it was introduced "a priori" in $(2)$?: this way it is more an axiom than a proof.
$(B)$ how (1) and (2) are related to mathematical induction, since (2) continues from $n$ instead of demonstrating that something works from $1$ to $n+1$, as induction should. Moreover, from $P'$ definition, $(1)$ and $(2)$ seem self-evident (if $n$ is an integer), so :
$(C)$ As it seems self-evident from the definition of $R$ ($R$ is the set of all points that are contained in every single derived set of $P'$) that $R = P^\infty $, how do these (1) and (2) definitions justify the then explained existence of $P^{\infty+1}$, $P^{\infty+2}$, ... $P^{{\infty}^{\infty+1}}$, ... ?
Maybe all the problem is that $n$ in $(2)$ is not an integer, but that is not clearly explained in the book, as said in $(A)$...
I'm looking for clarification in questions $A$ - $C$,
but an explanation of transfinite numbers in terms of P' would perfectly answer the question.