Cardinal Numbers and Arithmetic
Ordinals measure
arrangement; cardinals measure raw size. Two sets have the same
cardinal when there is a bijection between them, and cardinals are the answer to the plain question
"how many?" — extended to infinite sets. For finite sets the cardinals are just
0, 1, 2, 3, \dots. Past that lies a tower of infinite sizes, each named by
an aleph:
\aleph_0 \;<\; \aleph_1 \;<\; \aleph_2 \;<\; \cdots \;<\; \aleph_\omega \;<\; \cdots
\aleph_0 ("aleph-null") is the size of
the countable sets —
\mathbb{N}, \mathbb{Z}, \mathbb{Q}. Then \aleph_1
is the very next infinite size, \aleph_2 the one after, and the alephs march
on indexed by the ordinals — one aleph \aleph_\alpha for
every ordinal \alpha, defined by
transfinite recursion.
The power set always jumps
Cardinals wouldn't be interesting if there were only one infinity.
Cantor's theorem guarantees
otherwise: for every set, its power set is strictly larger,
|A| < |\mathcal{P}(A)| = 2^{|A|}. For finite sets that jump from
n to 2^n is already explosive — and it never stops
being explosive:
Applied to \mathbb{N}, this gives
2^{\aleph_0} = |\mathcal{P}(\mathbb{N})| = |\mathbb{R}| — the size of
the continuum, strictly bigger than
\aleph_0. Iterating gives an endless ladder of ever-larger infinities with
no top.
Infinite arithmetic collapses
Here is the surprise that makes cardinal arithmetic so different from ordinal arithmetic. For infinite
cardinals, addition and multiplication are trivial — they just take the bigger of the
two:
- For infinite \kappa, \lambda (at least one infinite):
\kappa + \lambda = \kappa \cdot \lambda = \max(\kappa, \lambda).
- So \aleph_0 + \aleph_0 = \aleph_0 and
\aleph_0 \cdot \aleph_0 = \aleph_0 — the reason
\mathbb{Z} and \mathbb{Q} stay countable.
- Exponentiation is the exception — it genuinely grows:
2^{\kappa} > \kappa always, by Cantor.
Addition and multiplication cannot escape the size you start with; only exponentiation — the
power-set operation — breaks out to a strictly larger cardinal. That single asymmetry is the whole
drama of infinite arithmetic.
// Cardinal arithmetic on infinite sizes just takes the max — a demo with symbolic "ranks".
// Represent aleph_k by its index k; addition & multiplication = max, exponentiation 2^k jumps up.
const add = (k: number, m: number): number => Math.max(k, m); // aleph_k + aleph_m
const mul = (k: number, m: number): number => Math.max(k, m); // aleph_k * aleph_m
console.log("aleph_0 + aleph_0 -> aleph_" + add(0, 0)); // aleph_0
console.log("aleph_0 * aleph_1 -> aleph_" + mul(0, 1)); // aleph_1
console.log("aleph_2 + aleph_5 -> aleph_" + add(2, 5)); // aleph_5
console.log("but 2^aleph_0 is STRICTLY bigger than aleph_0 (Cantor) — exponentiation escapes.");
This is the Continuum Hypothesis (CH): is the size of the reals,
2^{\aleph_0}, the very next cardinal
\aleph_1 — with nothing in between? Cantor believed yes and couldn't
prove it; it was Hilbert's very first problem in 1900. The astonishing resolution: Gödel (1940) and
Cohen (1963) showed CH can be neither proved nor disproved from
ZFC.
It is independent — there are consistent mathematical universes where
2^{\aleph_0} = \aleph_1 and equally consistent ones where it is
\aleph_2, or larger.
-
Cardinals are not ordinals. \omega, \omega+1, \omega\cdot 2
are different ordinals but all have the same cardinal
\aleph_0. Ordinal \omega + 1 > \omega, yet
cardinal \aleph_0 + 1 = \aleph_0. Don't mix the two arithmetics.
-
$2^{\aleph_0}$ is not "obviously" $\aleph_1$. It is some aleph bigger
than \aleph_0, but exactly which one is undecidable in ZFC — that's the
Continuum Hypothesis. Writing 2^{\aleph_0} = \aleph_1 as a fact is a
classic slip.