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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.logic,sci.math
Subject: Re: Replacement of Cardinality
Date: Thu, 15 Aug 2024 23:48:27 -0700
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On 8/15/2024 10:57 PM, Moebius wrote:
> Am 16.08.2024 um 07:50 schrieb Moebius:
> 
>> Of course, _if_ we already have introduced the real numbers (i.e. IR) 
>> we may define
>>
>>        √2 = the real number x such that x^2 = 2 ,    (*)
>>
>> _after_ we have shown that that there is exactly one x e IR such that 
>> x^2 = 2.
>>
>> From (*) we get immediately: (√2)^2 = 2
> 
> Now we may assume that there are natural numbers n,m such that √2 = n/m.
> 
> (...)

The infinite convergents of continuous fractions can describe sqrt(2) up 
to any desired precision?