FOM: Predicativism, GR, ACA_0 and Zorn's Lemma
Jeffrey Ketland
ketland at ketland.fsnet.co.uk
Fri Feb 18 08:59:04 EST 2000
Dear Matt and Joe
There is a kind of "sociological" reason for studying the foundational
situation wrt GR rather than QFT. Most workers in GR produce very tight
mathematical theorems, using the Definition-Lemma-Theorem approach, whereas
in the textbooks and papers I have read on QFT, the mathematical development
(commutation relations, path integrals, Feynman diagrams, Feynman
propagators, etc.) tends to be developed informally.
Matt's idea that GR might be developed fully within a predicative system
like ACA_0 strikes me as implausible. The prima facie reason is that I can
cite theorems required in the standard development of GR which depend upon
Zorn's Lemma.
If these theorems were transcribed into L_2, the appropriate axioms required
would be certain choice schemes. And we know from Simpson 1999 "Subsystems
of Second-Order Arithmetic" that various choice schemes in L_2 are stronger
than ACA_0 (usually being equivalent to some form of impredicative
comprehension).
For example, take a look at:
[1] Wald, Robert M 1984 "General Relativity" (Chicago), Chapter 10, "The
Initial Value Formulation of GR".
Wald here is discussing the sense in which GR has a well-posed initial value
formulation, the sense in which initial data on a Cauchy surface
"determines" the evolution of 3-geometry on space-like hypersurfaces. The
main theorems developed there are theorems concerning existence and
uniqueness of solutions to various systems of differential equations.
However, look at
Theorem 10.2.2 (page 264): If Sigma is a C^infinity 3-manifold, h_ab a
smooth 3-metric on Sigma and K_ab is a smooth symmetric tensor on Sigma. If
h_ab and K_ab satisfy certain conditions, then THERE EXISTS a unique
C^infinity manifold (M, g_ab), called the Maximal Cauchy Development of
(Sigma, h_ab, K_ab) such that (M, g_ab) is a solution of Einstein's
equations, is globally hyperbolic, etc.
Wald only sketches the proof of this theorem (p. 263), but it depends upon
Zorn's Lemma, applied to the set I of globally hyperbolic manifolds that
solve Einstein's equations and which have Sigma as a Cauchy surface. (There
is a partial order on this set and Zorn's Lemma yields the existence of a
maximal element, which is the required Maximal Cauchy development).
So it looks as if (at least prima facie!) there are theorems in GR which
require Zorn's lemma, which is impredicative and goes beyond ACA_0.
Oddly enough, some of these tight theorems were first given by one Robert
Geroch (who is a colleague of Wald's, at Chicago Physics Dept). I have
Robert Geroch's textbook,
[2] Geroch, R 1985: "Mathematical Physics" (Chicago)
where Geroch is usually explicit about such matters. Geroch notes (p. 49)
that Zorn's Lemma is required in developing some of the basic facts about
vector spaces (in particular, that every vector space has a basis).
However, what Geroch says about the foundational status of Zorn's Lemma is
truly remarkable. Geroch states the lemma (that for any partially ordered
set X, if every totally ordered subset of X is bounded above, then X has a
maximal element) and then gives the proof and comments:
------------------------------
The discussion above is not a proof - it simply gives the flavour of what
the statement asserts. Is there a real proof? The answer is remarkable:
there is neither a proof nor a counterexample. The point is that sets
themselves must be regarded, not as being "handed down from above", but
rather as just symbols which may be related in a certain way (e.g., "is an
element of"). (A set theory is a mathematical structure rather like the
notion of a group).
.... What one normally does is include our statement, called Zorn's Lemma,
as an additional axiom on one's set theory ... Although Zorn's Lemma is
useful in mathematics, I know of no example in which one's stance regarding
Zorn's Lemma has a direct impact on one's mode of description of physical
phenomena. Such an example might be very interesting. [p. 42].
-------------------------------
So: Geroch's position seems very close to formalism. So, what is really
interesting is that the formalist (meta-)position he advocates here seems to
CONTRADICT what he actually DOES in the rest of the book, where he USES
Zorn's Lemma to prove things about various structures. It is hard to see how
one could believe (to be true) such theorems proved using Zorn's Lemma, and
then add that their proof is completely arbitrary! In what sense is a
theorem proved, if its proof is arbitrary? This seems like some kind of
intellectual schizophrenia.
Of course, my sympathies are with the position: "Anything Goes!", rather
than with the various reductionist or anti-realist frameworks (finitism,
constructivism, intuitionism, predicativism, etc.).
Jeff
Dr Jeffrey Ketland
Department of Philosophy C15 Trent Building
University of Nottingham NG7 2RD
Tel: 0115 951 5843
E-mail: Jeffrey.Ketland at nottingham.ac.uk
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