Need example from number theory
JOSEPH SHIPMAN
joeshipman at aol.com
Sat Aug 28 16:50:33 EDT 2021
I don’t want a necessary use of AC. I want an unnecessary use, that isn’t obviously unnecessary!
— JS
Sent from my iPhone
> On Aug 28, 2021, at 4:40 PM, Gyorgy Sereny <sereny at math.bme.hu> wrote:
>
>
> 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
>>
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