> Date: Tue, 2 Jun 2009 19:43:59 +0200
> From: [email protected]
> To: [email protected]
> Subject: Re: The seven step-Mathematical preliminaries
>
>
> Bruno Marchal skrev:
>> 4) The set of all natural numbers. This set is hard to define, yet I
>> hope you agree we can describe it by the infinite quasi exhaustion by
>> {0, 1, 2, 3, ...}.
>>
>
> Let N be the biggest number in the set {0, 1, 2, 3, ...}.
>
> Exercise: does the number N+1 belongs to the set of natural numbers,
> that is does N+1 belongs to {0, 1, 2, 3, ...}?
Not every well-ordered set has a largest member. Every well-ordered set has a
"size" represented by an ordinal (see
http://en.wikipedia.org/wiki/Ordinal_number ) and there is a particular type of
ordinal called a "limit ordinal" which has no largest member, as discussed in
the section of that article at
http://en.wikipedia.org/wiki/Ordinal_number#Successor_and_limit_ordinals
Of course this is just how it works in set theory, I think you have said you
are some type of finitist so unlike a set theorist you may not want to "allow"
sets with no largest member, but in this case you shouldn't even use notation
like {0, 1, 2, 3, ...} that does not specify the largest member. I suppose
instead you could write something like {0, 1, 2, 3, ..., N} but in this case
you should specify what N is supposed to represent...the largest finite number
that any human has conceived of up to the present date? The number of distinct
physical entities in the universe (or the observable universe)? For a finitist
what defines "largest", and can it change over time?
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