I need help on understanding this with the proper notation because I have been told I am not using the notation correctly and have not understood this concept correctly.
I thought everyone “knew” that when you form the Power Set from an Infinite Set that was countable the original Infinite Set had a Cardinality of Aleph Null and the Power Set had a One to One Correspondence with the base two Continuum interval [0,1] (so had the Cardinality of the base two interval of the number of points in the interval [0,1] or 2^(Aleph Null) for the cardinality of the Continuum) because the binary choice that a specific element in the countable Infinite Set in any subset was a 1 if the element was included in its related "decimal" or binary digit to the right of the "decimal" or binary point and a 0 if it was not included, so if you have EVERY possible subset of a countable infinite set there was a one to one with EVERY possible base two representation of a point in the base two continuum [0,1] and thus the base two binary continuum had a one to one with the power set of any countable infinite set.
Question: So, how is that interpretation? Is it correct? Can it be improved? Is this well known or not? I also need to get the terminology down and I learned countable and one to one first but I read about denumerable and bijection so what is the preferred terminology? The way I understand it Cantor used this property of base two and power sets to define the Continuum Hypothesis, is that correct?
The powerset of $\aleph_0$ indeed has the same cardinality as the so-called continuum $\mathfrak{c}=\|[0,1]|$, and both can be seen as the set of all countably infinite sequences of $0$ and $1$, denoted by $\{0,1\}^{\Bbb N}$.
The fact that the powerset has the same size as $\{0,1\}^{\Bbb N}$, is indeed the argument that any subset $A$ of $\Bbb N$ (as a set of size $\aleph_0$) is determined uniquely by its characteristic fucntion (which maps $n$ to $1$ if $n \in A$ and to $0$ otherwise), and that function is just such a binary sequence.
Given a binary sequence we can form a base-$2$ real number in $[0,1]$ (put a $0.$ before the sequence) and this gives a surjection, not a bijection, as (countably!) many reals have two ways of representing them as such binary expansions. But with some extra arguments this is fixable, and we can see that indeed the sequences (so the powerset) has the exact same cardinality as $[0,1]$ (or $(0,1)$ or $\Bbb R$ etc.). Some Schröder-Bernstein goes a long way..
This is well-known and classical.
Denumerable and countable are synonyms, sometimes denumerable is only used for infinite sets and countable includes all the finite sets too. Basically a set that has a bijection with (a subset of) $\Bbb N$.
Cantor didn't really consider base 2 arguments at all, but just wondered if there is any distinct cardinality between $\aleph_0$ and $\mathfrak{c}=2^{\aleph_0}$. He did define $\aleph_1$ (the first uncountable cardinal) and knew (diagonal argument) that $2^{\aleph_0}$ was also uncountable and that every set in the reals that he could "imagine" was either countable or size $\mathfrak{c}$ so it is a natural question to wonder whether $\mathfrak{c}$ is actually equal to the first uncountable cardinality. Hence the Continuum Hypothesis (CH) was born, which turned out to be undecidable (independent of ZFC, more precisely).
Cantor would have been disappointed, had he known, I imagine. He was more of a Platonist, philosophically, and believed in a definite truth value, from what I've read in his biography.