A. J. Wilkie

Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function

Recall that a subset of R is called semi-algebraic if it can be represented as a (finite) boolean combination of sets of the form {~ α ∈ R : p(~ α) = 0}, {~ α ∈ R : q(~ α) > 0} where p(~x), q(~x) are n-variable polynomials with real coefficients. A map from R to R is called semi-algebraic if its graph, considered as a subset of R, is so. The geometry of such sets and maps (“semi-algebraic geometry”) is now a widely studied and flourishing subject that owes much to the foundational work in the 1930s of the logician Alfred Tarski. He proved ([11]) that the image of a semi-algebraic set under a semi-algebraic map is semi-algebraic. (A familiar simple instance: the image of {〈a, b, c, x〉 ∈ R : a 6= 0 and ax +bx+c = 0} under the projection map R×R→ R is {〈a, b, c〉 ∈ R : a 6= 0 and b−4ac ≥ 0}.) Tarski’s result implies that the class of semi-algebraic sets is closed under firstorder logical definability (where, as well as boolean operations, the quantifiers “∃x ∈ R . . . ” and “∀x ∈ R . . . ” are allowed) and for this reason it is known to logicians as “quantifier elimination for the ordered ring structure on R”. Immediate consequences are the facts that the closure, interior and boundary of a semialgebraic set are semi-algebraic. It is also the basis for many inductive arguments in semi-algebraic geometry where a desired property of a given semi-algebraic set is inferred from the same property of projections of the set into lower dimensions. For example, the fact (due to Hironaka) that any bounded semi-algebraic set can be triangulated is proved this way. In the 1960s the analytic geometer Lojasiewicz extended the above theory to the analytic context ([8]). The definition of a semi-analytic subset of R is the same as above except that for the basic sets the p(~x)’s and q(~x)’s are allowed to be analytic functions and we only insist that the boolean representations work locally around each point of R (allowing different representations around different points). It is also necessary to restrict the maps to be proper (with semi-analytic graph). With this restriction it is true that the image of a semi-analytic set, known as a sub-analytic set, is semi-analytic provided that the target space is either R or R. Counterexamples have been known since the beginning of this century for maps to R for m ≥ 3. (They are due to Osgood, see [8].) However, the situation was clarified in 1968 by Gabrielov ([5]) who showed that the class of sub-analytic sets

Gromov's theorem on groups of polynomial growth and elementary logic

In the fall of 1980 the authors attended Professor Tits’ course at Yale University in which he gave an account of Gromov’s beautiful proof that every finitely generated group of polynomial growth has a nilpotent subgroup of finite index. An essential part of Gromov’s argument consists of constructing for each group of polynomial growth a locally compact metric space and an action of a subgroup of finite index on that space. The intuitive motivation underlying this construction is fairly clear but it required an elaborate theory of “limits” of metric spaces to be carried out. It occurred to us to give a simple nonstandard definition of a space which has all the nice properties needed in the rest of Gromov’s argument. Besides shortening proofs our construction works for arbitrary finitely generated groups, not only for those of polynomial growth, and it has functorial properties. This enables us to state some of Gromov’s lemmas without the restriction of polynomial growth, e.g., (4.2) and (5.5). We also found a new proof of local compactness of the space, see Section 6, under an a priori weaker hypothesis than polynomial growth, and this led to a slight extension of Gromov’s theorem: If‘ the group r with finite generating set X has growth function G, with G,(n) < c . nd for infinitely many n and positive constants c, d, then r has a nilpotent subgroup of jinite index. (Gromov’s hypothesis is that G,(n) < c . nd for afl n > 0.)

Quasianalytic Denjoy-Carleman classes and o-minimality

We show that the expansion of the real field generated by the functions of a quasianalytic Denjoy-Carleman class is model complete and o-minimal, provided that the class satisfies certain closure conditions. Some of these structures do not admit analytic cell decomposition, and they show that there is no largest o-minimal expansion of the real field.

Provability of the pigeonhole principle and the existence of infinitely many primes

In this note we shall be interested in the following problems. Problem 1. Can I Δ 0 ⊢ ∀ x ∃ y > x ( y is prime)? Here I Δ 0 is Peano arithmetic with the induction axiom restricted to bounded (i.e. Δ 0 ) formulae. Problem 2. Can I Δ 0 ⊢ Δ 0 PHP? Here Δ 0 PHP (Δ 0 pigeonhole principle) is the schema for θ ∈ Δ 0 , or equivalently in I Δ 0 , for a Δ 0 formula F ( x,y ) written . By obtaining partial solutions to Problem 2 we shall show that Problem 1 has a positive solution if I Δ 0 is replaced by I Δ 0 + ∀ xx log( x ) exists. Our notation will be entirely standard (see for example [3] and [4]). In particular all logarithms will be to the base 2 and in expressions like log( x ), (1 + e ) x , etc. we shall always mean the integer part of these quantities. Concerning Problem 2 we remark that it is shown in [5] that for k ∈ N and F ∈ Δ 0 , As far as we know this is the best result of this form, in that we do not know how to replace log( z ) k by anything larger. However, as we shall show in Theorem 1, we can do much better if we increase the difference between the sizes of the domain and range of F . In what follows let M be a countable nonstandard model of I Δ 0 , and let be those subsets of M defined by Δ 0 formulae with parameters from M . Theorem 1. For k ∈ N and F ∈ Δ 0 , Here log 0 ( x ) = x , log k + 1 ( x ) = log(log k ( x )). Proof. To simplify matters, consider first the case k = 1. So assume M ⊨ a log(a) exists and with and a > 1. The idea of the proof is the following.

The rational points of a definable set

Let

Model theory with applications to algebra and analysis

X\R^n

On the Existence of end Extensions of Models of Bounded Induction

be a set that is definable in an o-minimal structure over

Locally polynomially bounded structures

R

A Schanuel property for exponentially transcendental powers

. This article shows that in a suitable sense, there are very few rational points of

A survey of asymptotic classes and measurable structures

X