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 ZY
to ZX 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 ℕℕ to ℝ using continuous fractions. But the equinumerosity between ℕℕand 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 2n elements.
Thus there are manier and manier intermediate cardinals when n
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.
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