Request for Review of ZF Inconsistency Proof

Hi All:

I am in the process of learning the subtle details of the basics of
set theory, a little deeper than textbooks go into. In studying
Cantor's Theorem, I've run across what I believe is a proof that P(N)
is countable infinite. I would like to ask the group to review it and
point out any problems, fallacies, or other areas to focus on in my
studies. Thanks in advance.

Let us begin.

In this thread, Cantor's Theorem will be applied only to the set of
natural numbers -- the general case is left for the reader. Let S
denote the infinite set of denumerable binary strings with alphabet
{0,1}. An individual string s is denoted as s[0],...,s[i-1],s[i]. Let
n[j],n[j-1],...,n[0] denotate a natural number n in radix 2. Let P[x]
denote an element of P(N).

Proposition 2.1: There exists a function f from N into S.

Proof: Let f be f(n) = s[0],...,s[i-1],s[i] where s[x] = ((n/2^x) mod
2) using integer division. By construction, every s[x] is either 0 or
1 and so every natural number n has some corresponding string s.
Suppose there exists f(n)=s and f(n)=t where s != t. Then there exists
some y such that ((n/2^y) mod 2 ) != ((n/2^y) mod 2). Contradiction.
Therefore, f is an injective function from N to S.

Comments/Feedback so far?

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