Positive Model Theory and Compact Abstract Theories

Abstract

We develop positive model theory, which is a non first order analogue of classical model theory where compactness is kept at the expense of negation. The analogue of a first order theory in this framework is a compact abstract theory: several equivalent yet conceptually different presentations of this notion are given. We prove in particular that Banach and Hilbert spaces are compact abstract theories, and in fact very well-behaved as such.

Group configurations and germs in simple theories

“…Although we assume that we work in a first order theory, we hardly use first order logic's strength: negation is never used, since the negation of a partial type is not, in general, a partial type (and the need for negation never arises); although universal quantification does make sense for a partial type (even for one in a hyperimaginary sort, as long as the ambient theory is first order), we never need to use it either; and the only primitive building blocks we use are complete types and partial types that define indiscernibility. It follows that the natural context for this paper is not a first order theory, but rather the much more general one of a thick compact abstract theory (see [Ben03a,Ben03b]): in a compact abstract theory there is no essential distinction between real and hyperimaginary sorts, and all the manipulations of partial types we propose to use make sense; thickness means that indiscernibility is type-definable, so all the primitives are available; finally, first order simplicity generalises fully to a simple thick compact abstract theory. Once one understand this context, it is a mere observation that all that we say in this paper holds without modification, and we will discuss it no further.…”

Imaginaries in Hilbert spaces

“…He also used some definable equivalence relations and their classes to generalise the Finite Equivalence Relation Theorem and to define strong types. Another approach for this subject comes from the work of the first author on compact abstract theories (see [Ben03]). We approach Hilbert spaces from this second point of view:…”

“…The picture changes slightly when we also consider imaginaries. It is well understood how to add imaginary (or hyperimaginary) sorts to a cat, and such a sort enjoys pretty much the same status as the real sort ( [Ben03]). One could also treat imaginary sorts with positive bounded formulas in the same way that hyperimaginaries are treated in first-order theories.…”

“…We assume familiarity with [Ben03] for the general framework of cats. Results related to stability in Hilbert spaces will be quoted from [BB02].…”

Abstract elementary classes: some answers, more questions

“…Various other attempts to formalize analytic structures (notably Banach spaces [Hen72,HI02]) provide examples of 'homogeneous model theory' ( [She70,BL03] and many more); Banach spaces are also an example of CATS [BY03a]. Strictly speaking, the class of Banach spaces is not closed under unions of chains so doesn't form an AEC.…”