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NNTP-Posting-Date: Mon, 19 Aug 2024 22:02:33 +0000
Subject: Re: Replacement of Cardinality (infinite middle)
Newsgroups: sci.logic,sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Mon, 19 Aug 2024 15:02:37 -0700
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On 08/19/2024 02:43 PM, Jim Burns wrote:
> On 8/19/2024 3:27 PM, Ross Finlayson wrote:
>> On 08/18/2024 09:56 PM, Jim Burns wrote:
>
>>> [...]
>>
>> I mean it's a great definition that finite has that
>> there exists a normal ordering that's a well-ordering
>
> ...in both directions...
>
>> and that all the orderings of the set are well-orderings.
>
> ...in both directions...
>
>> That's a great definition of finite and now it stands
>> for itself in enduring mathematical definition in defense.
>
> ...for comfortably more than a century.
>
>> Why is it you think that Stackel's definition of finite
>> and "not Dedekind's definition of countably infinite"
>> don't agree?
>
> They mostly agree.
>
> Given the Axiom of Choice
> (let us say, if an inaccessible cardinal exists),
> they completely agree.
>
> My impression from somewhere is that,
> if they disagree,
> they disagree on some very weird sets.
>
> https://en.wikipedia.org/wiki/Finite_set
>
>> [...] is another little fact of mathematics
>> missing from your neat little hedgerow.
>
> I mark my neat little hedgerow, and
> I describe what's true everywhere inside the hedgerow.
> That allows me to learn about
> what's inside the hedgerow,
> even though it's infinite and I am finite.
>
> I like doing that.
> It isn't wrong for me to do that.
> I will continue doing that.
>
>

No, not just that its normal ordering is ...,
that _all_ its orderings are ....

Finite, ....