# [FOM] A Defence of Set Theory as Foundations

Dmytro Taranovsky dmytro at MIT.EDU
Wed Oct 5 17:36:24 EDT 2005

On Wed, 2005-10-05 at 16:31 +0200, Andrej Bauer wrote:
> I would like to offer a criticism not of set-theory itself, but of the
> idea that having a single powerful foundation of mathematics is a good
> thing.
>
> A reasonably powerful foundation of mathematics imposes a certain way of
> thinking onto those who have been trained to use it. When such a
> foundation is as wide-spread as classical set theory is among today's
> mathematicians, this hinders development and acceptance of those new
> mathematical ideas that can only be expressed in alternative foundations.

A key thesis is that every mathematical statement is expressible
set-theoretically.  For example, non-well founded sets can be treated as
graphs.  Urelements can be treated as sets.  Higher order set theory can
be expressed through an additional natural predicate.

In intuitionism, statements are treated not literally but
"constructively".  For example, "for all x there is y such that P(x, y)"
would mean that there is a procedure that when given x outputs y such
that P(x, y).   This way intuitionism can be interpreted in a classical
framework.

The law of the excluded middle is a logical truth.  Therefore, literal
treatment of statements in intuitionism would be incomplete or (if the
law is denied as opposed to just not asserted) contradictory.  I should
note, however, that in certain areas, intuitionism is a useful way of
managing complexity.

While other universal foundations are conceivable (like a large universe
with second order logic but without special predicates), they are
interpretable through sets.  For reasons of efficiency and truth, the
set theoretical foundation is chosen.   Having a single universal
foundation makes it easier to learn mathematics.

Dmytro Taranovsky