[FOM] A Defence of Set Theory as Foundations

[Patrick Caldon]

> 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.

I would suggest the opposite. In terms of pedagogical value, set theory
seems to make things harder.  We teach young children to count 1,2,3,
... and not {{}}, {{{}},{}}, ... .  The former is an easier
representation for young minds to grasp, and its lack of generality
does not seem to trouble them too much.  Similarly we teach (for
instance) measure theory directly targeted to R^n or C^n, whereas it
would be possible to teach abstract measure theory based entirely on
sets directly.  I'm reading at the moment a book on numerical
optimization; every set is a subset of R^n.  A few facts about R^n are
a more than adequate foundation for a great deal of mathematics, and
it's much easier to use a few facts about R^n and its subsets for
learning maths than set theory in its full generality.

If you're looking for truth, I'd suggest that your axioms should be
obvious facts.  Why is replacement necessarily obvious (axiomatic)?
Technically, I think it's brilliant and subtle, but it is not (in my
view) obvious.  The writer of every text who's bothered to include
subset comprehension and pairing as axioms would implicitly agree with
me.  We now know that naive comprehension doesn't work; near the turn
of the last century this was obviously true to the point where it was
made an axiom of set theory by the leading minds of the day!

I think we can all agree that set theory is a convenient basis for
exploring the foundations of mathematics, and further (given the
struggles to find something that works) that it's not a simple matter
to find a good notion for exploring the foundations of mathematics.
Furthermore several attempts to build a foundation have all come out
at this equivalent notion.  This hints that it is quite fundamental
indeed, certainly powerful and convenient, and so worthy of serious
study, but I'm not sure how you can make the stronger statement that
it is a true foundation.

Patrick.