Every infinite set which can be enumerated by natural numbers is the same size (cardinality) as N, and is said to be countable.
Then a cardinal number is, by definition, a class consisting of all sets of the same size.
Then an ordinal number is, by definition, a class consisting of all well-ordered sets of the same order type.
The converse is not true in general for infinite sets: it is possible to impose different well-orderings on the set of natural numbers that give rise to different ordinal numbers.
By defining the notion of the size of a set in terms of its cardinality, the issue can be settled.
The evidence strongly suggests that Cantor was quite confident in the result itself and that his comment to Dedekind refers instead to his then-still-lingering concerns about the validity of his proof of it.
[1] Nevertheless, Cantor's remark would also serve nicely to express the surprise that so many mathematicians after him have experienced on first encountering a result that is so counter-intuitive.
In 1904 Ernst Zermelo proved by means of the axiom of choice (which was introduced for this reason) that every set can be well-ordered.
In 1963 Paul J. Cohen showed that in Zermelo–Fraenkel set theory without the axiom of choice it is not possible to prove the existence of a well-ordering of the real numbers.
However, the ability to well order any set allows certain constructions to be performed that have been called paradoxical.
It states that it is possible to decompose a ball of a fixed radius into a finite number of pieces and then move and reassemble those pieces by ordinary translations and rotations (with no scaling) to obtain two copies from the one original copy.
But it is still a natural philosophical question to contemplate some physical action that actually completes after an infinite number of discrete steps; and the interpretation of this question using set theory gives rise to the paradoxes of the supertask.
Tristram Shandy, the hero of a novel by Laurence Sterne, writes his autobiography so conscientiously that it takes him one year to lay down the events of one day.
An increased version of this type of paradox shifts the infinitely remote finish to a finite time.
The 20th century saw a resolution to these paradoxes in the development of the various axiomatizations of set theories such as ZFC and NBG in common use today.
However, the gap between the very formalized and symbolic language of these theories and our typical informal use of mathematical language results in various paradoxical situations, as well as the philosophical question of exactly what it is that such formal systems actually propose to be talking about.
In 1897 the Italian mathematician Cesare Burali-Forti discovered that there is no set containing all ordinal numbers.
In letters to David Hilbert and Richard Dedekind he wrote about inconsistent sets, the elements of which cannot be thought of as being all together, and he used this result to prove that every consistent set has a cardinal number.
Russell himself explained this abstract idea by means of some very concrete pictures.
In 1905, the Hungarian mathematician Julius König published a paradox based on the fact that there are only countably many finite definitions.
This result is known as Tarski's indefinability theorem; it applies to a wide class of formal systems including all commonly studied axiomatizations of set theory.
In the same year the French mathematician Jules Richard used a variant of Cantor's diagonal method to obtain another contradiction in naive set theory.
Based upon work of the German mathematician Leopold Löwenheim (1915) the Norwegian logician Thoralf Skolem showed in 1922 that every consistent theory of first-order predicate calculus, such as set theory, has an at most countable model.
The root of this seeming paradox is that the countability or noncountability of a set is not always absolute, but can depend on the model in which the cardinality is measured.