However in fact we do not necessarily need the ZFC axiomatic theory for considering cardinals as a legitimate notion : instead they can be formalized without being uniquely determined objects, by letting sets play their role : the role of a variable with value a cardinal can be played by a variable with value any set in the cardinal (as a class of sets in bijection with each other), and formalizing formulas on cardinals by formulas whose value remains unchanged by replacing this set by any other set in bijection with it.

The main properties of cardinals are the following:

Except if X=∅, the relation "There is a surjection from Y to X" is weaker than (implied by) "there is an injection from X to Y" (section 2.5), and is equivalent if the Axiom of Choice is accepted.

Cantor's theorem (2.5) says there cannot be any surjection from X to P(X). Thus there cannot be any injection from P(X) to X either (independently of the axiom of choice).

For any set of cardinals, that is concretely, a set of sets
representing different cardinals, their union is larger than each
(as there is an obvious injection); then its powerset is even
larger. Thus no set of sets can ever represent all cardinals.

Indeed we can easily find a bijection between ℕ and ℕ×ℕ. A
set with the same cardinality as ℕ is said *countable*.
Generally, many infinite sets are countable. Namely, those that
can defined using no more than a finite sequence of numbers. For
example the set of (positive or negative) integers, the set of
rational numbers, the set of algebraic numbers (solutions of
algebraic equations with rational coefficients)

Without the axiom of choice, from a surjection from X to Y we can
define an injection from P(Y) to P(X), and more generally from Z^{Y}
to Z^{X} for any set Z (see 2.5).

Indeed we can find an injection from ℝ to [0,1], then from [0,1]
to P(ℕ) given by the binary expansion; then we can find an
injection from [0,1] to ℝ using the development in base 3 (or any
other number than 2).

We can also find an injection from ℕ^{ℕ} to ℝ using
continuous fractions. But the equinumerosity between ℕ^{ℕ}and
P(ℕ) can as well be deduced from P(ℕ)≈P(ℕ×ℕ)≈P(ℕ)^{ℕ}≈P(ℕ)

The continuum hypothesis is the claim "There is no cardinal
between those of ℕ and P(ℕ)". A very hard result of mathematical
logic is that the number of intermediate cardinals between ℕ and
P(ℕ) can roughly be any number between zero and many infinities,
depending on the model of set theory. However it would be naive to
say "There may exist an intermediate cardinal but we cannot find
it". In fact the problem (cause of undecidability) is not to find
it, but it is whether the explicitly constructed set (that is the
candidate set for having an intermediate cardinal), is in
bijection with P(ℕ) or not. We shall explain this below.

This can be contrasted with the case of finite sets : if X is a
finite set with n elements then P(X) has 2^{n} elements.
Thus there are manier and manier intermediate cardinals when n
increases.

If the axiom of choice is accepted then cardinals form a total order (for any sets X and Y there either exists an injection from X to Y or from Y to X). And more precisely, a well-ordering, as cardinals can be represented by ordinals, that is, essentially, well-ordered sets.