[FOM] model theory and nonstandard analysis

[Jose Iovino]

This is my first post on FOM, and it follows up on a recent thread in
which is was argued that in applications of nonstandard analysis to
mathematics, generally little model theory is used beyond the compactness
theorem. John Baldwin replied citing some recent work in which we have
been involved which has exposed deep connections between Shelah's
classification apparatus and one of the currently most active areas of
Banach space theory, namely, asymptotic structure. This post is in
response to an informal invitation from Baldwin to say a bit about these
connections.

As stated above, the study of asymptotic structure is one of the most
active areas of Banach space theory. It has proved to be a powerful
instrument for the solution of long-standing problems (e.g., the
distortion problem, solved by E. Odell and Th. Schlumprecht in the mid
1990's), and it has been shown that it is closely connected with classical
problems that remain open (e.g., the famous separable quotient problem).

The kinds of questions that arise in this field are as follows: one has a
bounded sequence (e_n) in a Banach space X, and a geometric property P;
can we find a subsequence (f_n) of (e_n) such that every subsequence of
(f_n) has a subsequence satisfying P "at infinity"?

An important example is when P is the property of being equivalent to the
standard unit basis of one of the classical sequence spaces (ell_p or
c_0).

>From the point of view of nonstandard analysis, what is studied is the
geometric properties of spaces spanned by sequences of the form (e*_n),
where (e*_n) is a continuation of (e_n) within some nonstarndard hull of
X. An well-known application of Ramsey's Theorem shows that, by replacing
(e_n) by a subsequence if necessary, it can be assumed that the sequence
(e*_n) is indiscernible. In this case (e*_n) is a called a _spreading
model_ of the sequence (e_n). Sometimes the term "spreading model" is also
used to refer to the subspace of the nonstandard hull spanned by (e*_n).

The definition of spreading model that appears in analysis textbooks is of
course different from that outlined above and does not include mention of
nonstanard hulls; it involves instead iterated limits. However, the two
definitions are equivalent in a way that is rather evident to someone
familiar with nonstandard hulls or Banach space ultrapowers.

The concept of spreading model was developed by analysts. However, given
the close connection with model theoretic ideas, it does not come as a
surprise that terms such as "ultraproduct", "space of types", and
"stability" have become part of the Banach space theory vernacular. It
should be noted, however, that the way that some of these concepts are
defined in analysis does not copy literally from their first-order
homonyms. In fact, the analogy with first-order often occurs well below
the surface; for instance, it is not apparent that what analysts call
"type" is closely related to what model theorists call "type".

The origin of many of these developments can be traced back to the
fundamental work of J.-L. Krivine in Banach space theory, in which Banach
space ultrapowers played a crucial role. Today, thirty years later, the
power of these ideas in functional analysis has transcended Banach space
geometry; for instance, ultraproducts are now widely regarded as an
important tool in the area of operator spaces.

For a survey of the role played by asymptotic structure in recent progress
in Banach space theory see [O], especially Chapters 5 and 6.

There are profound connections between asymptotic structure and parts of
Shelah's classification apparatus. This is thanks to a smooth way to
translate between bounded sequences in a Banach space X and model
theoretic types over X. The translation is based on Henson's concept of
approximate truth (for the definition of this concept, see [HI]). Under
this translation, spreading models of a sequence (e_n) correspond
precisely to what Shelah calls "averages" (see [Sh, Chapter VII]; in
Shelah's terminology, the indiscernible sequence (e*_n) satisfies the
property that the type of each term of the sequence is semidefinable over
the previous terms.

The connections between geometric properties (on the Banach space theory
side of the translation) and model-theoretic properties (on the model
theory side) are striking. For example, a fundamental question in Banach
space theory is under what conditions a sequence has a spreading model
equivalent to one of the classical sequence spaces (l_p or c_0); it turns
out that this holds precisely when, in an asymptotic sense, the sequence
approximates a type that is stable in the sense of model theory; in this
case, a spreading model for the sequence is precisely a Morley sequence
for this type. Then, when information given by stability theory (e.g.,
number of nonisomorphic extensions) is translated into geometric
information on the Banach space theory side, new insight is obtained on
the structure of the spreading models of the sequence (e.g., the number of
nonequivalent spreading models).


REFERENCES:

[HI] Henson, C. Ward; Iovino, Jose. Ultraproducts in analysis. Analysis
and Logic, 1--110, London Math. Soc. Lecture Note Ser., 262, Cambridge
Univ. Press, Cambridge, 2002.

[O] Odell, Edward. On subspaces, asymptotic structure, and distortion of
Banach spaces; connections with logic. Analysis and Logic, 189--267,
London Math. Soc. Lecture Note Ser., 262, Cambridge Univ. Press,
Cambridge, 2002.

[Sh] Shelah, S. Classification theory and the number of nonisomorphic
models. Second edition. Studies in Logic and the Foundations of
Mathematics, 92. North-Holland Publishing Co., Amsterdam, 1990.