We show that the following results in the classical theory of unbounded linear operators on Hilbert spaces can be proved within the framework of Bishop's constructive mathematics: the Kato-Rellich theorem, the spectral theorem. Stone's theorem, and the self-adjointness of the most common quantum mechanical operators, including the Hamiltonians of electro-magnetic fields with some general forms of potentials.

We provide a general framework for studying the expansion of strongly minimal sets by adding additional relations in the style of Hrushovski. We introduce a notion of separation of quantifiers which is a condition on the class of expansions of finitely generated models for the expanded theory to have a countable ω-saturated model. We apply these results to construct for each sufficiently fast growing finite-to-one function μ from ‘primitive extensions’ to the natural numbers a theory Tμ of an expansion of an algebraically closed field which has Morley rank 2. Finally, we show that if μ is not finite-to-one the theory may not be ω-stable.

The study of continuous group actions is ubiquitous in mathematics, and perhaps the most general kinds of actions for which we can hope to prove theorems in just ZFC are those where a Polish group acts on a Polish space.

For this general class we can find works such as [29] that build on ideas from ergodic theory and examine actions of locally compact groups in both the measure theoretic and topological contexts. On the other hand a text in model theory, such as [12], will typically consider issues bearing on the actions by the symmetric group of all permutations of the integers. More generally [1] shows that the orbit equivalence relations induced by closed subgroups of the infinite symmetric group can be reduced to the isomorphism relation on corresponding classes of countable models.

This paper considers a third category formed by the continuous actions of separable Banach spaces on Polish spaces. These examples cannot be subsumed under the two earlier headings, and it is known from [10] that the Borel cardinalities of the quotient spaces that arise from such actions are incomparable with the equivalence relations induced by the symmetric group or any locally compact Polish group action.

One of the first things to be addressed concerns the complexity of these equivalence relations. This question for appears in [1].

We discuss ways in which a typical one-dimensional Brownian motion can be approximated by oscillations which are encoded by finite binary strings of high descriptive complexity. We study the recursive properties of Brownian motions that can be thus obtained.

In this paper we construct a non-CM -trivial stable theory in which no infinite field is interpretable. In fact our theory will also be trivial and ω-stable, but of infinite Morley rank. A long term aim would be to find a non CM-trivial theory which has finite Morley rank (or is even strongly minimal) and does not interpret a field. The construction in this paper is direct, and is a “3-dimensional” version of the free pseudoplane. In a sense we are cheating: the original point of the notion of CM-triviality was to describe the geometry of a strongly minimal set, or even of a regular type. In our example, non-CM-triviality will come from the behaviour of three orthogonal regular types.

A stable theory is said to be CM-trivial if whenever A ⊆ B and acl(Ac) ∩ acl(B) = acl(A) in Teq, then Cb(stp(c/A)) ⊆ Cb(stp(c/B)). ( An infinite stable field will not be CM-trivial.) The notion is due to Hrushovski [3], where he gave several equivalent definitions, as well as showing that his new strongly minimal sets constructed “ab ovo” were CM-trivial. The notion was studied further in [6] where it was shown that CM-trivial groups of finite Morley rank are nilpotent-by-finite. These results were generalized in various ways to the superstable case in [8].

In this paper, we study partition properties of the set of real numbers. The meaning of “set of real numbers” will vary, referring at times to the collection of sequences of natural numbers, ωω; the collection of infinite sets of natural numbers [ω]ω; the collection of infinite sequences of zeroes and ones, 2ω; or (ω), the power set of ω.

The archetype for the relations is the property: “all sets of reals are Ramsey,” in the notation of Erdős and Hajnal, ω → (ω)ω. This states that for every partition F : [ω]ω → 2, there is an infinite set H ∈ [ω]ω such that F is constant on [H]ω. Like virtually all of the properties we will discuss, it contradicts the Axiom of Choice but is compatible with the principle of dependent choices (DC). DC will be used throughout the paper wihtout further mention.

The properties discussed in this paper will vary in two respects. Some, like ω → (ω)ω, will be incompatible with the existence of an ultrafilter on ω (UF) and some will not. Some are known to be consistent relative to ZF alone, and for some, such as ω → (ω)ω, the question is still open. All properties, however, are true in Solovay's model and hence are consistent relative to Con(ZF + “there exists an inaccessible cardinal”).

CM-triviality of a stable theory is a notion introduced by Hrushovski [1]. The importance of this property is first that it holds of Hrushovski's new non 1-based strongly minimal sets, and second that it is still quite a restrictive property, and forbids the existence of definable fields or simple groups (see [2]). In [5], Frank Wagner posed some questions about CM-triviality, asking in particular whether a structure of finite rank, which is “coordinatized” by CM-trivial types of rank 1, is itself CM-trivial. (Actually Wagner worked in a slightly more general context, adapting the definitions to a certain “local” framework, in which algebraic closure is replaced by P-closure, for P some family of types. We will, however, remain in the standard context, and will just remark here that it is routine to translate our results into Wagner's framework, as well as to generalise to the superstable theory/regular type context.) In any case we answer Wagner's question positively. Also in an attempt to put forward some concrete conjectures about the possible geometries of strongly minimal sets (or stable theories) we tentatively suggest a hierarchy of geometric properties of forking, the first two levels of which correspond to 1-basedness and CM-triviality respectively. We do not know whether this is a strict hierarchy (or even whether these are the “right” notions), but we conjecture that it is, and moreover that a counterexample to Cherlin's conjecture can be found at level three in the hierarchy.