FOM: trivial amalgamation; Hilbertism; consistency equals existence; applications to analysis; practical completeness
John Steel
steel at math.berkeley.edu
Fri Feb 25 13:04:13 EST 2000
Here are some replies to Steve Simpson's 2/16/00 comments on my generic
absoluteness posting.
> Steel mentions the desirability of natural interpretations. Could
> somebody make this notion of ``natural interpretation'' a little more
> precise?
>
Probably not me. The number of times "natural" occurs in that posting
is quite embarrassing. I meant to distinguish the interpretation
phi-->(phi)^L of ZFC+V=L into ZFC from the interpretation of ZFC+"there is
a measurable" into PRA + Con(measurable) one gets by e.g. choosing the
leftmost branch in the tree of attempts to build a model with universe
omega. Perhaps you could say that the latter interpretation doesn't really
capture the meaning of the language of set theory as used by the advocate
of ZFC + measurables, since that person actually (ought to) believe
certain statements about that tree of attempts which don't come out true
under this interpretation. ( This is clearer if we talk about
interpreting ZFC + "there is a measurable" as the theory of the L-least
wellfounded model of ZFC + "there is a measurable". The statement
"there is a wellfounded model of..." comes out false.) In contrast, if
we have a set theorist who insists V=L, but is otherwise rational, we
can understand everything he says now or ever will say by relativising
his quantifiers to L. Our interpretation really captures the full meaning
of the language of set theory, as he is using it.
> > 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''?
>
I meant by "Hilbertism" Hilbert's philosophical case for the
importance of his program, and his belief that program would succeed.
> 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/).
I didn't mean to denigrate this work. My point was that Hilbert's
philosophy lead him to a definite mathematical program ( maybe the
program led to the philosophy, I don't really know, this is Oliver Stone
style history). The fact that successful programs spun off his failed
one is due to its definiteness.
>
> 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.
What sense is this? The way we use the language of set theory
today is such that "there is a measurable" and "Con(measurables)" make
quite different assertions.
As Steel says:
>
> > we have just found a very strange way of saying that there are
> > measurable cardinals.
"We" referred here to the instrumentalists (not me). The
instrumentalist distorts the meaning of the language of set theory. He
makes the assertion we make by uttering "there are measurables" by
uttering "Con(measurables)", or more likely perhaps "ZFC + 'there are
measurables" is part of our best mathematical theory". It is really quite
parallel to using " the world behaves exactly as if there were dinosaurs"
to assert that there were dinosaurs.
>
> But, what's so ``strange'' about it?
What's strange is that the instrumentalist distorts our language.
We have a meaning for "T is a our best theory" which is different from
that of T. (The former makes an assertion about humans, the latter may
not.) The instrumentalist uses "T is a good theory" to assert T. It
would seem that his non-standard language is actually poorer than ours.
>
> 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''?
>
I think the dictum pre-dates Hilbert; I seem to remember being told
that Cantor had said something similar.
If the dictum is just an early intuition that Godel's Completeness
theorem is true, then there is nothing in it to dispute. However, I
believe it was intended to be more. It was intended to give some advice to
mathematicians as to what basic existence assumptions they should accept.
In this connection, the Godel Completeness Theorem interpretation of the
dictum would suggest reading it as: use any consistent theory. This is
pretty poor advice. It is no advice at all really, a sort of anything-goes
approach that gets you nowhere. It amounts to abandoning the search for a
universal framework theory.
I'm not sure what Steve himself is getting at. We agree that all
consistent theories have models. How does that fact support
instrumentalism?
> 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?
It depends on what you mean by mathematical practice. Borel, Baire,
Lebesgue, et al., were analysts interested in the foundations of their
subject. They found the problem of the Lebesgue measurability of
projective sets compelling.
> 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!
>
I'm not sure how the "real question" got to be "what does E. Stein
think?". Why don't we go all the way, and demand some engineering
applications?
I take the real question in this area to be: what are the proper
axioms for mathematics? The evidence is pretty good that these
include ZFC + large cardinal hypotheses. One strong piece of evidence is
the "complete" theory of projective sets one gets.
Whether these axioms get used in their full strength right now in
more applied areas is not crucial. They are meant as a foundation for the
long run, and in this respect they meet important theoretical tests. In
fact, all roads toward a stronger foundation seem to lead to these axioms.
There is an extensive mathematical theory around them, and we should
continue to develop it. This can only make broader applications more
likely.
John Steel
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