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From: Moebius <invalid@example.invalid>
Newsgroups: sci.logic,sci.math
Subject: Re: Replacement of Cardinality
Date: Fri, 16 Aug 2024 09:22:55 +0200
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Am 16.08.2024 um 08:48 schrieb Chris M. Thomasson:
> 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 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?

Sure.