Trouble with definition of countable, denumerable

Question

I found the following definition:

Definition. A set is countable iff its cardinality is either finite or equal to $\aleph_0$. A set is denumerable iff its cardinality is exactly $\aleph_0$. A set is uncountable iff its cardinality is greater than $\aleph_0$.

The null set is countable. The finite set, {A, B, C}, is countable. The infinite set, $\mathbb{N}$, is countable and denumerable. Sets with a larger cardinality than $\mathbb{N}$ are uncountable.

I have trouble with seeing the difference between countable and denumerable, apart from the part that the cardinality is finite. Isn't "A set is countable iff its cardinality equal to $\aleph_0$" and "A set is denumerable iff its cardinality is exactly $\aleph_0$" the same?

Answer 1

Every square is a rectangle, but not every rectangle is a square. Similarly, every denumerable set is countable, but not every countable set is denumerable. If you want, think of "denumerable" as an abbreviation for "countable and infinite" (or think of "countable" as an abbreviation for "denumerable or finite").

Answer 2

Basically, the issue is a terminological one, and thus subject to change.

There are book were "countable" is not used at all.

See :

• Patrick Suppes, Axiomatic set theory (1960 - Dover reprint) :

page 100 : Definition 5. [A set] $A$ is finite if and only if ...

page 150 : Definition 23. A set is infinite if and only if it is not finite.

page 151 : Theorem 41. The set $\omega$ of natural numbers is infinite.

Definition 24. A set is denumerable if and only if it is equipollent to the set $\omega$ of all natural numbers.

Theorem 43. Every denumerable set is infinite.

page 191 : Theorem 59. The set of real numbers is not denumerable.

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Other authors "suppress" denumerable; see

• Kenneth Kunen, The Foundations of Mathematics (2009), page 52 :

Definition I.11.14 $A$ is countable iff $A \preccurlyeq \omega$. $A$ is finite iff $A \preccurlyeq n$ for some $n \in \omega$. "infinite" means "not finite". "uncountable" means "not countable". $A$ is countably infinite iff $A$ is countable and infinite.

Answer 3

A denumerable set has a bijection in $\mathbb N$. A countable set is either finite or denumerable. Either A or B is a nice way of saying A xor B, isn't English a beautiful language.