On 04 Jun 2009, at 15:40, Brian Tenneson wrote:

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> This is a denial of the axiom of infinity. I think a foundational
> set theorist might agree that it is impossible to -construct- an
> infinite set from scratch which is why they use the axiom of infinity.
> People are free to deny axioms, of course, though the result will
> not be like ZFC set theory. The denial of axiom of foundation is
> one I've come across; I've never met anyone who denies the axiom of
> infinity.
Among mathematicians nobody denies the axiom of infinity, but many
philosopher of mathematics are attracted by finitism.
But Torgny is ultrafinitist. That is much rare. he denies the
existence of natural numbers above some rather putative biggest
natural number.
>
>
> For me it is strange that the following statement is false: every
> natural number has a natural number successor.
I thought he would have said this, and accepted that the successor of
its N is equal to N+1. Nut in a reply he says that N+1 exists but is
not a natural number, which I think should not be consistent.
> To me it seems quite arbitrary for the ultrafinitist's statement:
> every natural number has a natural number successor UNTIL we reach
> some natural number which does not have a natural number successor.
> I'm left wondering what the largest ultrafinist's number is.
It cannot be a constructive object. It is a number which is so big
that if you add 1 to it, the "everything" explodes!
I dunno. I still suspect that ultrafinitism in math cannot be
consistent, unlike the many variate form of finitism. Comp is arguably
a form of finitism at the ontological level, yet an ultra-infinitism,
if I can say, at the epistemological level.
Bruno
http://iridia.ulb.ac.be/~marchal/
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