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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Fri, 17 Jan 2025 09:52:22 -0500
Organization: i2pn2 (i2pn.org)
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On 1/17/25 5:39 AM, WM wrote:
> On 17.01.2025 01:37, Richard Damon wrote:
>> On 1/16/25 8:18 AM, WM wrote:
>>> On 15.01.2025 20:54, joes wrote:
>>>> Am Wed, 15 Jan 2025 18:58:40 +0100 schrieb WM:
>>>>> On 14.01.2025 13:37, Richard Damon wrote:
>>>>>
>>>>>> EVERY Natural Number is "Definable",
>>>>> Then remove the set ℕ by application of only definable numbers:
>>>>> ℕ \ {1} \ {1, 2} \ {1, 2, 3} \ ...
>>>> Yes, N \ {1, 2, 3, ...} = {}.
>>>
>>> Of course. But what FISON can be dropped?
>>>
>> Of course no FINITE (the F of FISON) will be the match for the 
>> INFINITE set of Natural Numbers.
>>
>> That doesn't mean the whole set of them fail, but of course, you need 
>> to be using logic that CAN use *ALL* of them
> 
> Logic says that every FISON which is smaller than another FISON can be 
> dropped. Logic says even more: Every FISON which is smaller than ℕ can 
> be dropped. Therefore all can be dropped.

ANY can be dropped, but not ALL.

The problem is that "ALL" FISONs is an infinite set of them, and your 
operation can be done on an infinite set.nn

> 
> Do you know a FISON that is smaller than ℕ but cannot be dropped?n

No, and all that shows is that there isn't a "largest" FISON that has 
all the Natural Numbers.

Or, is simpler words there isn't a FINITE set that is INFINITE. This is 
the distinction between "Any" and "All" when talking about infinite 
sets. You can not do a finite sequential operation on "All" of an 
infinite set.

> 
> Regards, WM
> 
> Regards, WM