Date: Nov 4, 2010 1:28 AM
Author: Bill Taylor
Subject: Re: Mathematics as a language

On Nov 4, 3:12 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:

> >What logically could exist -- that is, what is not inherently self-
> >contradictory -- has mathematical existence.

>
> The problem with this is that there could be two different
> mathematical objects, A and B, such that neither is inherently
> self-contradictory, but the existence of A contradicts the
> existence of B. They can't, therefore, both exist.


Exactly so!

My favorite example being,
(as someone just now pointed out half of),
it is consistent with ZF that

1) there exists a set of cardinality strictly between N & R;

2) there exists a function on P(R) whose values
are bijections between the argument and elements of N u {N,R}.

1 & 2 are individually consistent with but jointly inconsistent
with ZFC. (1) means CH is false, (2) means CH is true.

I once asked about this of Thomas Forster, an admirable logician,
who often claims informally that "consistency is existence".

He refused to answer, saying only that "ontology is poisonous".

I took that as meaning I'd won the argument! :)

-- Browbeating Bill