Defining and comparing infinities : ordinals and cardinals


The concept of cardinals consists in saying that two sets E and F "have the same cardinal", to mean that they are equinumerous (there exists a bijection between them). In the early times of set theory, people had a problem dealing with this name of "cardinals" that behave as sorts of "objects" of set theory which they could not easily formalize as effective objects among those admitted in the basic formalism. This problem was later resolved by taking the ZFC axiomatic system where ordinals appear as specific objects, and all cardinals are represented by ordinals (any set is in bijection with an ordinal) and there is a standard way to pick up a specific ordinal representing each cardinal.
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:

The class of cardinals is ordered by the relation of existence of injections between sets

It is easy to see that the relation between X and Y defined by "There is an injection from X to Y" is a preorder, that only depends on the cardinals of X and Y. It is an order (the existence of injections from X to Y and from Y to X implies the existence of a bijection, section 3.5).

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.

The powerset of X is stricly bigger than X

We have the obvious injection from X to P(X) defined by x {x}.
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).

There is no set of all cardinals

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.

Infinite cardinalities cannot easily increase without the powerset

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)

The powerset preserves inequalities of cardinals

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).

The set ℝ of real numbers has the same cardinality as P(ℕ)

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 to ℝ using continuous fractions. But the equinumerosity between ℕand P(ℕ) can as well be deduced from P(ℕ)≈P(ℕ×ℕ)≈P(ℕ)≈P(ℕ)

The continuum hypothesis is undecidable

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 increases.

More properties use the axiom of choice

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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