Different concepts of infinities and their comparisons, have been
introduced in mathematics. The most important ones are the
ordinals, the cardinals and the nonstandard numbers.
We already presented the nonstandard numbers in the previous text. This was due to the coexistence of different models of arithmetic. Now we shall present the ordinals and cardinals, which are "pure» set theoretic concepts (describing objects and staying in a fixed model of set theory) not involving any consideration of other theories and models.
Like with most other mathematical concepts, the notions of "size" and "size comparison" for sets, however intuitive they may seem, do not have any a priori meaning, but any meaning such words can have must come from somewhere, namely from a definition that must be given. So, the notion of cardinal is a concept defined in set theory to "compare the size" of any sets, including the infinite sets, generalizing the counting of elements by natural numbers.
Philosophical note : Some marginal philosophers would
believe in the preexistence of a meaning of the words "size»,
"smaller» and "larger» for sets,
purely based on their intuitive feeling that these words should
make a priori sense, prior to any formal definition, and might
even want to argue that this meaning differs from the concept of
cardinal as it is commonly accepted; but such a discussion in
fact has nothing to do with mathematics, since there is no
logical way to express or discuss what some words of human
language such as "size» and "comparison of size»
should "really mean» as suggested by the intuition
rather than from any formal definition. Only a few words and
concepts from natural language can and need to be accepted as
making a priori clear (unambiguous) mathematical sense without
further explanations, in order for mathematics to have a start
somewhere. The essential thing is to care only admitting this
way the concepts that are indeed mathematically clear without
trap. Such distinctions are not always obvious for beginners,
but may require some experience to be done correctly. But well,
nowadays this work has been completed.
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 equipotence 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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