Re: Why is there something rather than nothing?

From: Bruno Marchal <marchal.domain.name.hidden>
Date: Thu, 20 Nov 2003 12:57:55 +0100

At 18:30 19/11/03 -0500, Jesse Mazer wrote:
>Does anyone know, are there versions of philosophy-of-mathematics that
>would allow no distinctions in infinities beyond countable and
>uncountable? I know intuitionism is more restrictive about infinities than
>traditional mathematics, but it's way *too* restrictive for my tastes, I
>wouldn't want to throw out the law of the excluded middle.


I don't know. In general people who accept
the uncountable accept most big cardinals.
This follows from the fact that in most set
theories you can prove Cantor theorem
which say that card(power of A) is always
strictly bigger than card(A).

In some "toposes" you can certainly build
ad hoc models of set theory with a modified
power set axioms making possible to collapse
the uncountable sets (but that's whisfull
ad hoc mathematics).

Note that by "Skolem Paradox" the notion
of cardinality is shown to be relative. (I have
use this to refute a set-theoretic analog of Lucas'
godelian refutation of mechanism).
That relativity is a reason for me (like you it
seems) to be unplatonic with the notion of sets
(unlike the notion numbers).

One way out is to have some constructive
axiomatic where you modelize "uncountable"
by "non recursively countable" ...

But for machine, you can guess, the big
whole is not even nameable, and we
(would-be computationalist) must learn to
accept the plausibility of it. Set theories,
by their failure, render that more palpable.

Bruno
Received on Thu Nov 20 2003 - 06:52:56 PST