FOM: trivial amalgamation; Hilbertism; consistency equals existence; applications to analysis; practical completeness
Stephen G Simpson
simpson at math.psu.edu
Wed Feb 16 21:24:26 EST 2000
Here are some quick and essentially random comments on just a few
aspects of John Steel's long posting:
Tue Jan 25 20:18:57 2000 - John Steel - FOM: generic absoluteness
> One ambition of set theory is to be a useful universal framework in
> which all our mathematical theories can be interpreted. (Of
> course, one can always amalgamate theories by interpreting them as
> speaking of disjoint universes; this is the paradigm for a useless
> framework.)
Of course such trivial amalgamations are useless. But then, what
additional property should a universal framework have, in order to be
useful? Some sort of coherence condition? Later in the same posting,
Steel mentions the desirability of natural interpretations. Could
somebody make this notion of ``natural interpretation'' a little more
precise?
> the most interesting and useful thing about Hilbertism was the
> definite way in which it was false.
I don't remember hearing the term ``Hilbertism'' before. Recently
Borzacchini and Ketland have been discussing here on FOM whether Plato
was a Platonist. Was Hilbert a ``Hilbertist''?
If ``Hilbertism'' refers to what has been called Hilbert's Program
(i.e., the program of Hilbert's paper On the Infinite), then I have to
dispute what Steel has said, because it seems to me there are a lot of
good things that came out of Hilbert's Program, beyond its definitive
falsifiability. In particular, the original false version of
Hilbert's Program immediately suggests at least one viable alternative
program that has a lot of mathematical substance and scientific
coherence. I refer to the program of ``Partial Realizations of
Hilbert's Program'' a la my paper of that title, JSL 53 (1988) pp
349-363 (on-line at http://www.math.psu.edu/simpson/papers/).
Steel asks rhetorically:
> Why not adopt Peano (or primitive recursive) Arithmetic plus
> Con(there is a measurable), Con(there is a supercompact),
> Con(...),... as our official theory?
....
> Isn't this progress?
Well, maybe it *is* progress, from a certain viewpoint. We may want
to posit that only Pi^0_1 statements are ``verifiable'' and as
scientists we want to focus on verifiable aspects of our theories.
Then the ``official theory'' mentioned above seems like a good
approach. PRA + Con(measurable) would seem to have all the verifiable
consequences of a measurable, without getting involved in stuff that
is otherwise unverifiable.
This is not at all similar to the idea that the world was created in
1998 complete with fossils, memories, etc. The big difference is that
Con(measurable) is in an important sense *equivalent* to the existence
of a measurable. As Steel says:
> we have just found a very strange way of saying that there are
> measurable cardinals.
But, what's so ``strange'' about it?
Let me try to make it a little more plausible.
There is an old dictum (Hilbert's?) that existence (of mathematical
objects) is the same as consistency.
[ Here is the argument. Let S be a system of mathematical objects
whose existence is in question. Let T be a set of axioms describing
all of the properties that S is to have. Since mathematical objects
are completely described and axiomatized by their properties, T ``is''
S. But according to the G"odel Completeness Theorem, T is consistent
if and only if S ``exists'', in the sense that a model of T exists.
So, existence (of S) equals consistency (of T). ]
Now, this dictum has a lot of appeal. In fact, it is a rigorous
theorem, the G"odel Completeness Theorem. Why does Steel apparently
think it is ``strange''?
> The theory of projective sets one gets from PD generalizes the
> theory of Sigma^1_2 sets one gets from open determinacy in a
> natural way. ( Although, contrary to a remark Steve Simpson made on
> FOM some time ago, the generalization is often far from routine,
> and entirely new phenomena do show up.)
Did I say that? Where? I know that there are new phenomena at the
higher levels, and I also know that some of the details of the
generalization are not routine, e.g., the correct formulation of the
scale property.
Still, the exposition by Moschovakis in his book seems to suggest that
there is a lot of similarity with the Pi^1_1 and Sigma^1_2 cases.
Anyway, the real question for Steel is, what do higher projective sets
have to do with mathematical practice? Borel sets, yes. Especially
low-level Borel sets. But what about high-level projective sets?
The most effective way to convince core mathematicians that they need
axioms asserting the existence of Woodin cardinals would be to use
such axioms to prove some theorems that they find interesting from
their own standpoint as mathematicians. Has this been done? I don't
think so.
Suppose for instance that our test case is a classical analyst:
somebody like E. Stein, C. Fefferman, Bourgain, etc. Now, I know that
Howard Becker (and collaborators?) proved some very nice results
showing that the *definitions* of some standard concepts of classical
analysis are at fairly high levels in the projective hierarchy (Pi^1_3
or Pi^1_4, perhaps). But, did this work ever pay off in the direction
of a proof that Woodin cardinals have some ineliminable application to
prove a *theorem* that fits in with classical analysis?
Or, would Steel claim that uniformization properties of projective
sets *are* classical analysis? Hmmmmm .... Try to convince Stein of
that!
> It also appears that the theory we get from PD is complete in a
> practical sense, insofar as natural non-Godel-like statements in
> LSA go.
(LSA = the language of second order arithmetic.)
This fact, that PD is ``complete in a practical sense'' for the
language of second order arithmetic, has always struck me as a very
remarkable and interesting phenomenon. It is similar to the fact that
PA ``appears to be complete in a practical sense'' for the language of
first order arithmetic. This phenomenon of ``practical completeness''
has been discussed before here on FOM.
So, we are now looking at two examples of pairs (L,T), such that L is
a language, and T is a theory which ``appears to be complete for L in
a practical sense'': (L_1, PA), and (L_2, Z_2 + DC + PD), where L_n =
the language of nth order arithmetic.
Can anyone point to any other examples of pairs (L,T) as above?
-- Steve
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