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Cantor's diagonal argument: why the reals cannot be listed
SUPPOSED COMPLETE LIST r1 = 0. 4 1 5 9 2 6... r2 = 0.7 8 2 4 3 1... r3 = 0.31 4 1 5 9... r4 = 0.271 8 2 8... r5 = 0.1415 9 2... ... (infinitely many rows) DIAGONAL d = 0.4849... Change each digit: 4→5, 8→9, 4→5, 8→9 d* = 0.5959... NOT on the list! Any list of reals is incomplete. The diagonal number differs from every row at its own position.
Sizes of infinity: a strict hierarchy
N: aleph-0 Z (integers) same size as N Q (rationals) same size as N R (reals): strictly larger uncountable: cannot be listed countable |P(N)| = |R| = 2^(aleph-0) (the continuum)

The natural numbers, integers, and rationals are all countably infinite: they can all be put in a one-to-one correspondence with each other. The real numbers are uncountably infinite: a strictly larger infinity. Between these two sizes, the Continuum Hypothesis asks whether there is anything in between.

Hilbert's Hotel: a hotel with infinitely many rooms, all full, always has room
HILBERT'S HOTEL (fully occupied) {[1,2,3,4,5,6,7].map((n, i) => `${n}`).join('')} ... New guest Solution: move guest n to room n+1. Room 1 is now free. infinity + 1 = infinity.
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