Hey Michael,

Initially, the set *T* is comprised of components s0, s1, s2, and so on.

Set *T* can be put on a one-to-one correspondence with the natural numbers (0, 1, 2, and so on).

s0 -> 0

s1 -> 1

s2 -> 2

…

Each element of *T* is accounted for in the natural numbers, hence you can claim that the two have the same cardinality, or length.

However, Cantor’s diagonal argument shows that *T* is continually growing.

s0 -> 0

s1 -> 1

s2 -> 2

…

n -> ?? (There is no element of natural numbers to correspond to this new sequence)

Because all the natural numbers are preoccupied being accompanied with an element of *T*, *n* cannot be put on a one-to-one correspondence with the natural numbers. Hence, even though both sets are infinite, *T* is longer than the set of natural numbers.

This article wasn’t very mathematically rigorous, so assumptions or steps taken may seem willy-nilly and flexible. You can find a mathematically rigorous version with definitions and theorems here.