FOM: Orthogonal roles
Andrian-Richard-David Mathias
Andrian-Richard-David.Mathias at univ-reunion.fr
Fri Mar 3 10:38:02 EST 2000
1. Todd Wilson draws attention to two roles of set theory, which he summarises
in these phrases:
W1 Set theory provides a very abundant ontology for mathematics;
W2 Set theory provides an axiomatization of our pre-theoretic notion
of collection (or at least iteratively conceived collections);
and remarks that
W3 I think that the foundational roles of
ontology and axiomatization are basically orthogonal.
I am intrigued since I too have perceived an orthogonality, and I see
a possible link between Wilson's dichotomy and mine.
As antithetical labels I have at different times since 1986 used "geometrical"
versus "arithmetical", "spatial" versus "temporal", and "type-theorist"
versus "set-theorist", in an attempt to identify those parts of mathematical
thought where the one mindset is more effective than the other.
2. Start from Wilson's remark that
W4 topos theory can provide an ontology that is just as
abundant as the one set theory provides.
To a set-theorist that is plainly untrue: taking topos theory to be a
decorated version of Mac Lane set theory, it cannot, for example, prove the
existence of an infinite set of infinite cardinals.
[Mac Lane, be it noted, does not see the issue between topos theory and set
theory as an ontological competition.]
3. But let us turn Wilson's remark around and take it to mean that the objects
supplied by a system of the strength of topos theory meet Wilson's
ontological hunger --- as he says,
W5 What matters is that there are enough objects available, and enough
relations between these objects representable, that we can map whatever
we wish to speak about onto what's provided.
[Aside: that smacks slightly of the idea of a once-for-all foundational job
being done, an idea I distrust as I believe foundational questions permeate
mathematics. I think of particular superstructures of abstract ideas being
called into being to solve particular problems; different superstructures
being invoked at different times.]
Now let us ask what is it that Wilson senses that set theory provides
that goes beyond that supply of objects.
4. In my terms, the objects that satisfy Wilson are those generated by the
spatial, geometric or type-theoretic part of mathematics; the objects
missing from that world are those generated by "recursive" constructions
which build into the unknown, something that type-theoretic constructions
cannot do.
A difficulty in the debate is that the objects that only the set-theoretic,
arithmetical or temporal side of mathematics can produce are prima facie
not objects for which the type-theoretic, geometric or spatial school see any
need. A formal expression of that at one level is offered by the result
established in "The Strength of Mac Lane Set Theory", that any stratified
formula provable in Mac + Kripke-Platek is provable in Mac. When schemes of
separation and collection are admitted for richer classes of formulae, that
conservative extension phenomenon breaks down: ZF proves with ease the
stratifiable statement that there is an infinite set of infinite cardinals,
which Z cannot; but the topos theorists question the relevance of these
richer classes.
5. Personally I define set theory as the study of well-foundedness, and regard
its foundational successes as occurring when it meets a need for a new
framework for a "recursive" construction (in a suitably abstract sense).
I don't think it succeeds at all in accounting for geometric intuition.
[In line with Wilson's final point, that failure should not be allowed to
obscure its successes; but nor should its successes be judged a reason for
sweeping its failures under the foundational carpet.]
6. To sum up, it seems to me plausible that where Wilson speaks of the success
of set theory for exploring new axiomatisations he is responding to the
capacity of set theory to build new frameworks, whereas when he speaks of its
ontological success he might merely be recognising that set theory is adequate
to produce all the objects that the type-theoretic view of mathematics can
generate; for if he thinks that type theory and set theory are equally
abundant he is not taking into account the numerous counter-instances
supplied, for example, by the Mostowski Isomorphism Theorem.
***************
Some assorted remarks:
7. Here is a portion of Wilson's fifth question:
W6 If forging a representation is a one-time job, why should we discount
a foundational scheme because it's a little harder to set up the
basic representations, since once this is done, mathematics can
(for the most part) proceed as usual ?
I am dubious that forging a representation is a one-time job: this belief was
a fundamental error of the Bourbaki group; its effect is to paralyse
foundational thought. That point apart, I believe that Bourbaki's foundations
for mathematics come under this heading, as shown in my posting of a few
months back computing the length of Bourbaki's definition of the number 1
(over 4 times (10 exp 12) symbols if ordered pair is taken as primitive and
over 2 times (10 exp 54) if the Wiener--Kuratowski definition is used). I
think theirs is a scheme which *should* be discounted because of the
psychological impairment it causes the reader who tries to take it seriously.
8. There is an error related to the Mostowski Isomophism Theorem in the book
"Locally Presentable and Accessible Categories" by Adamek and Rosicky,
mentioned by Wilson in an earlier posting. On page 293, line -2 the authors say
``every model of ZC is isomorphic to a transitive model". Under the axiom of
foundation that is false, since it would hold only for well-founded models,
and by the compactness theorem there can easily be ill-founded models of any
reasonable set theory. [Scott years ago proved the relative consistency of the
set theory obtained from, say, ZFC by dropping the axiom of foundation and
adding in its place the statement that every extensional relation is isomorphic
to a transitive set; and Aczel in his book on ill-founded sets takes that a
stage further.]
The error is not important in its context, since the authors are concerned
only with elementary submodels of transitive sets, which (by the axiom
of foundation) are indeed well-founded.
9. Wilson writes:
W7 These axioms can be seen as an attempt to capture everything about the
notion of collection that can be subjected to rigorous examination.
Everything ? surely incompleteness means that there is no hope of doing that.
Ideas slowly evolve.
10. He also writes
W8 The drive to ever larger cardinals is fueled by a desire to understand
the limits of our notion of collection and to determine whether there is
hidden there any secrets that will shed light on the striking
incompletenesses of our current axiomatic setup.
A point I made in my posting "Strong Statements of Analysis" is that it
can be misleading to speak of large cardinals; one should speak instead of
large-cardinal properties, which certain ordinals might possess in certain
inner models. Such properties have a way of revealing themselves to be
embedded in apparently innocent questions of ordinary mathematics; so
that it is sometimes these other questions that fuels the drive to explore
the strength of such properties.
A.R.D.Mathias
professeur associ'e
Universit'e de la R'eunion.
ardm at univ-reunion.fr
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