Equivalence of Definitions of Countable Set

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $S$ be a set.


The following definitions of the concept of Countable Set are equivalent:

Definition 1

$S$ is countable if and only if there exists an injection:

$f: S \to \N$

Definition 2

$S$ is countable if and only if it is finite or countably infinite.


Proof

Definition 1 implies Definition 2

Let $S$ be a countable set by Definition 1.

Then there is an injection $f: S \to \N$.

By Law of Excluded Middle, $S$ is finite or infinite.

If $S$ is finite, then it trivially satisfies Definition 2.

If $S$ is infinite, then it is countably infinite by Infinite Set of Natural Numbers is Countably Infinite, and thus satisfies Definition 2.

$\Box$


Definition 2 implies Definition 1

Let $S$ be a countable set by Definition 2.

Then $S$ is finite or countably infinite.

If $S$ is countably infinite, then there is a bijection $f: S \to \N$, which is injective by definition.

If $S$ is finite, then for some natural number $n$ there is a bijection $f: S \to \N_n$, where $\N_n$ is the initial segment of $\N$ determined by $n$.


Let $f': S \to \N$ be the extension of $f$ to $\N$.


Then $f'$ is an injection.

Thus in either case $S$ is countable by Definition 1.

$\blacksquare$


Law of the Excluded Middle

This theorem depends on the Law of the Excluded Middle.

This is one of the axioms of logic that was determined by Aristotle, and forms part of the backbone of classical (Aristotelian) logic.

However, the intuitionist school rejects the Law of the Excluded Middle as a valid logical axiom. This in turn invalidates this theorem from an intuitionistic perspective.



Sources