Cantor's Diagonalization Proof (in 1891), N(Natural numbers set) is countably infinite, not Reals.
Channel: Somenath Roy Choudhury
Date: 29/04/2022

Cantor's diagonal argument
https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

An illustration of Cantor's diagonal argument (in base 2) for the existence of uncountable sets. The sequence at the bottom cannot occur anywhere in the enumeration of sequences above.

An infinite set may have the same cardinality as a proper subset of itself, as the depicted bijection f(x)=2x from the natural to the even numbers demonstrates. Nevertheless, infinite sets of different cardinalities exist, as Cantor's diagonal argument shows.

In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began.

The diagonal argument was not Cantor's first proof of the uncountability of the real numbers, which appeared in 1874. However, it demonstrates a general technique that has since been used in a wide range of proofs, including the first of Gödel's incompleteness theorems and Turing's answer to the Entscheidungsproblem. Diagonalization arguments are often also the source of contradictions like Russell's paradox and Richard's paradox.

Su, Francis E., et al. “Cantor Diagonalization.” Math Fun Facts. https://www.math.hmc.edu/funfacts
So are all infinite sets countable?

Cantor shocked the world by showing that the real numbers are not countable… there are “more” of them than the integers! His proof was an ingenious use of a proof by contradiction. In fact, he could show that there exists infinities of many different “sizes”!

By the way, a similar “diagonalization” argument can be used to show that any set S and the set of all S’s subsets (called the power set of S) cannot be placed in one-to-one correspondence. The idea goes like this: if such a correspondence were possible, then every element A of S has a subset K(A) that corresponds to it. Now construct a new subset of S, call it J, such that an element A is in J if and only if A is NOT in K(A). This new set, by construction, can never be K(A) for any A, because it differs from every K(A) in the “A-th element”. So K(A) must not run through the entire power set of A, hence the 1-1 correspondence asserted above must not be possible.

This proof shows that there are infinite sets of many different “sizes” by considering the natural numbers and its successive power sets! The “size” of a set is called is cardinality.