[FOM] A Defence of Set Theory as Foundations

[Andrej Bauer]

Dmytro Taranovsky wrote:
> 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.

I should have anticipated this comment.

Yes, it might be true that every mathematical statement is expressible
set-theoretically. This is beside the point, as it is a
meta-mathematical statement, while my criticism is of a sociological
nature (and people are not machines, yet). It is based on the
observation that classical set-theory imposes a certain kind of thinking
on those who use it (as does any foundation). They are thus inclined to
reject fresh ideas that come from alternative foundations. This is a
sociological observation (based on scant evidence).

If we express ideas from alternative foundations set-theoretically, they
become complicated and unnatural, as we are forced to add a lot of
machinery to them. For example, to explain how there can be nilpotent
infinitesimals, we need to first build sheaf models over certain
C^\infty rings. This is a lot of overhead which obscures matters. But if
you work directly in inuitionistic foundation, the infinitesimals are
immediate. It takes some time to get used to them, but after you do it
is a lot of fun. It's like having a new toy.

Similarly, while it is technically possible to explain constructive
mathematics by means of set-theoretic models that involve computability,
such explanations do not convey well the intuitions that constructive
mathematicans possess.

> 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.

The interoperability between various foundations does not preserve
_mathematical intuitions_ that one gets when working in various
foundations. Working in only one foundation makes it easier to learn
just one mathematics. But what about all the others? Classical set
theory does not make it easier to learn constructive mathematics. In
fact, it makes it harder.

Andrej Bauer