Cantor's Infinities

What just happened?

The sets $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$ could be arranged in an enumerable sequence — every number eventually gets its ordinal position. This is surprising especially for $\mathbb{Q}$: between any two rational numbers there are infinitely many of them, and yet they can be numbered by listing the fractions along the diagonals of a grid. For $\mathbb{R}$, however, no list is ever complete — you can always construct a number that is not on it.

Hierarchy of number sets:

Despite this inclusion, the cardinality (the size of infinity) of the first three sets is identical — we call it $\aleph_0$ (“aleph-null”, the smallest infinite cardinal number):

Cantor's diagonal argument (1891):

Suppose (for contradiction) that all the real numbers in the interval (0, 1) could be numbered as a list $x_1, x_2, x_3, \dots$. We build a new number $y$, taking the i-th decimal digit of the number $x_i$ and changing it:

The number $y$ differs from $x_1$ in the 1st digit, from $x_2$ in the 2nd digit, from $x_3$ in the 3rd digit — and so on. So $y$ differs from EVERY number on the list, meaning it is not on the list. But $y \in (0,1)$, while the list was supposed to contain all such numbers — a contradiction. Therefore no list can be complete: $\mathbb{R}$ is uncountable.

Where did the resistance to this discovery come from?

When Georg Cantor published these results (1874 and 1891), some mathematicians of the time — led by Leopold Kronecker — criticized them sharply, regarding different “sizes” of infinity as philosophically disturbing, even unacceptable. Today Cantor’s set theory is the foundation of all of modern mathematics.

Why this matters for your exam:

Distinguishing countable from uncountable sets is the basis for understanding how rational numbers differ from irrational and real ones — useful in proofs of irrationality and in problems about the density of number sets.

Remember: $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ can always be arranged in an enumerable sequence (cardinality $\aleph_0$) — $\mathbb{R}$ never can (the cardinality of the continuum, a larger infinity).