FOM: "collection" as a basic mathematical concept

[Stephen G Simpson]

I wrote:

 > I proposed to define f.o.m. (= foundations of mathematics) as "the
 > systematic study of the most basic mathematical concepts and the
 > logical structure of mathematics, with an eye to the unity of human
 > knowledge."
 > ...
 > I presented a tentative list of the most basic mathematical
 > concepts: number, shape, set, function, algorithm, mathematical
 > axiom, mathematical proof, mathematical definition.

Charles Silver responded:

 > I don't mean to quibble, but isn't the *basic* or *foundational*
 > concept that of a "collection" rather than of a "set"?  ....

My list was tentative.  My purpose in presenting it was to delimit
f.o.m. in what seems like a reasonable way, so that the discussion on
the FOM list wouldn't randomly wander all over the mathematical and
intellectual landscape.

I would have no objection if you were to modify my list and replace
"set" by "collection".  I said "set" rather than "collection", because
I wanted to acknowledge that set theory is the current orthodoxy.  But
I'm not particularly content or dogmatic about this orthodoxy, and we
could certainly discuss alternative foundational schemes based on a
notion of "collection" other than sets.  What did you have in mind?
How about foundational schemes based on "predicate"?

-- Steve