FOM: Pin-pointing the cause of dispute ...

[Andrian-Richard-David Mathias]

... 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