# Re: The seven step-Mathematical preliminaries

```Torgny sais he is an ultrafinitist, although in some precedent post,
he shows he can be open to usual finitism.```
```
Usual finitism is not in conflict with comp, except that it makes the
math more difficult, and the theology a bit too formal for my taste.

But ultrafinitism is incompatible with comp and its classical Church
thesis. For an ultrafinitist everything is computable, and by some
miracle even the basic program "10 GOTO 10" does stop.

I know only two ultrafinitists on this planet. Torgny, and a Russian
guy who wrote papers that nobody understands (as far as I know).

Bruno

On 02 Jun 2009, at 20:29, Jesse Mazer wrote:

>
>
> > Date: Tue, 2 Jun 2009 19:43:59 +0200
> > From: tor...@dsv.su.se
> > 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?
>
> >

http://iridia.ulb.ac.be/~marchal/

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