# Need example from number theory

Gyorgy Sereny sereny at math.bme.hu
Sat Aug 28 16:40:13 EDT 2021

The most elementary example of the necessary use of AC I know
(of course, it is not number theory):

(5.3) The union of a countable family of countable sets is countable.
(By the way, this often used fact cannot be proved in ZF alone.)

To prove (5.3) let A_n be a countable set for each n \in N.
For each n, let us _choose_ an enumeration (a_nk: k \in N) of A_n. [...]

(Thomas Jech: Set Theory
Second Corrected Edition, p.39)

Gyorgy Sereny

On Sat, 28 Aug 2021, JOSEPH SHIPMAN wrote:

> What is an example of a theorem of number theory which has a
> well-known proof *suitable for undergraduates*,
> that uses the Axiom of Choice in a way that is not obviously unnecessary?
>
> We know that AC can be eliminated from the proof of any arithmetical
> statement, but I’d like an example I can easily explain.
>
> — JS
>
>