FOM: Pin-pointing the cause of dispute ...
Andrian-Richard-David Mathias
Andrian-Richard-David.Mathias at univ-reunion.fr
Tue Feb 22 01:54:09 EST 2000
... between set-theorists and category-theorists ?
A. Some history
1. Let me distinguish between a comprehension axiom,
asserting that there is a set whose members are all
those x with some property;
and a separation axiom, asserting that there is a set whose members are
all those x in the set a with some property.
Distinguish too between typed variables, which range over all the objects of
a particular type, and untyped variables, which range over all the
objects of the universe.
2. Russell's paradox results from attempting to have both all comprehension
axioms and untyped variables.
The two lines of response were the type-theoretical one, of
introducing typed variables, and banning untyped variables,
but retaining the comprehension axioms for all properties expressible
in this typed language; and the set-theoretical one of keeping the
untyped variables and allowing the separation axioms for properties
expressible in the untyped language, but banning the comprehension axioms
for that language; and requiring the universe not to be a set.
3. The category-theoretic presentation of mathematics
descends from the type-theoretic response.
4. Mac Lane, in his book "Mathematics: form and function" has given a
set-theoretic system which captures the capacity for object-formation
that suffices for mathematics as he sees it. Mac Lane's
sytem is much weaker than Zermelo's system and is equiconsistent with the
simple theory of types.
B. The point
5. The point of psychological divergence of the two schools seems to be
Mostowski's isomorphism theorem, (MIT for short),
that every extensional well-founded relation
is isomorphic to a transitive set, with its corollary that every well-ordering
is isomorphic to a von Neumann ordinal.
MIT is essential to modern set theory, but rejected with passion by
Mac Lane and his school.
6. MIT is not provable in Mac Lane' system Mac (nor in Zermelo's),
its proof requiring instances of the axiom scheme of collection
which are not available in those systems; but,
ironically, there is a natural interpretation of Mac + MIT in Mac, so that
the two systems are equiconsistent. Similar comments apply to
the systems Zermelo + MIT and Zermelo.
[See my paper "The Strength of Mac Lane Set Theory", to appear in the
Annals of Pure and Applied Logic, for the proofs of these and other
remarks.]
7. MIT offers a way of selecting a canonical representative of each
isomorphism class of well-orderings, or indeed of each isomorphism class of
well-founded extensional relations.
To set-theorists, the existence of such canonical representatives
greatly increases the clarity of Goedel's constructible universe and his
Condensation Lemma, and leads to such things as Jensen's construction
principles Diamond and Square, and many things besides.
To (extreme) category-theorists, the very idea of having canonical
representatives of isomorphism classes is a violation of their fundamental
doctrine that isomorphism is the only important thing, indeed the only
meaningful thing.
C. Comment
8. Thus there is an enormous psychological gap between
the set-theorists' exploration of Cantor's Absolute and
the dogma so clearly stated by Mac Lane that "mathematics
is protean and therefore does not have an ontology".
9. For a possible historical parallel, let me quote from
page 193 of Juergen Neukirch's book, "Algebraic Number Theory"
as Englished by Norbert Schappacher (Springer, 1999):
"Throughout the historical development of algebraic number
theory, a controversy persisted between the followers of Dedekind's
ideal-theoretic approach and the divisor-theoretic method of building
up the theory from the valuation-theoretic notion of primes. Both theories
are equivalent in the sense of [certain] isomorphism results, but they are
also fundamentally different in nature. The controversy has finally given way
to the realisation that neither approach is dominant, that each one instead
emanates from its own proper world, and that the relation between these
worlds is expressed by an important mathematical principle."
Something of Neukirch's sage portrait holds true of the sets-categories
controversy; but, alas, not yet all.
D. A third way ...
(10.) ... is to consider set theories (such as ones proposed by Church and
by Quine) with a universal set; that is, where the
universe is considered to be a set. A book by Thomas Forster discusses
those.
11. Some years ago I saw a preprint of a paper by Feferman investigating the
possibility of a category of all categories within the framework of
Quine's system of New Foundations, which admits a set of all sets.
Can anyone tell me more about that ?
A. R. D. Mathias
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