In this paper we analyze some aspects of the question of using methods from model theory to study structures of functional analysis.

By a well known result of P. Lindström, one cannot extend the expressive power of first order logic and yet preserve its most outstanding model theoretic characteristics (e.g., compactness and the Löwenheim-Skolem theorem). However, one may consider extending the scope of first order in a different sense, specifically, by expanding the class of structures that are regarded as models (e.g., including Banach algebras or other structures of functional analysis), and ask whether the resulting extensions of first order model theory preserve some of its desirable characteristics.

A formal framework for the study of structures based on Banach spaces from the perspective of model theory was first introduced by C. W. Henson in [8] and [6]. Notions of syntax and semantics for these structures were defined, and it was shown that using them one obtains a model theoretic apparatus that satisfies many of the fundamental properties of first order model theory. For instance, one has compactness, Löwenheim-Skolem, and omitting types theorems. Further aspects of the theory, namely, the fundamentals of stability and forking, were first introduced in [10] and [9].

The classes of mathematical structures formally encompassed by this framework are normed linear spaces, possibly expanded with additional structure, e.g., operations, real-valued relations, and constants. This notion subsumes wide classes of structures from functional analysis. However, the restriction that the universe of a structure be a normed space is not necessary. (This restriction has a historical, rather than technical origin; specifically, the development of the theory was originally motivated by questions in Banach space geometry.) Analogous techniques can be applied if the universe is a metric space. Now, when the underlying metric topology is discrete, the resulting model theory coincides with first order model theory, so this logic extends first order in the sense described above. Furthermore, without any cost in the mathematical complexity, one can also work in multi-sorted contexts, so, for instance, one sort could be an operator algebra while another is. say, a metric space.

The paper considers the set of first-order polymodal formulas the modal operators in which can be applied to subformulas of at most one free variable. Using a mosaic technique, we prove a general satisfiability criterion for formulas in , which reduces the modal satisfiability to the classical one. The criterion is then used to single out a number of new, in a sense optimal, decidable fragments of various modal predicate logics.

We show how the forcing of [5] can be iterated so as to get a model containing supercompact cardinals in which every measurable cardinal δ is δ+ supercompact. We then apply this iteration to prove three additional theorems concerning the structure of the class of supercompact cardinals.

We show that under Martin's Axiom, the cofinality cf(Aut(Γ)) of the automorphism group of the random graph Γ is 2ω.

It is shown that for each computably enumerable set of n-element subsets of ω there is an infinite set A ⊆ ω such that either all n-element subsets of A are in or no n-element subsets of A are in . An analogous result is obtained with the requirement that A be replaced by the requirement that the jump of A be computable from 0(n). These results are best possible in various senses.

A number of specific cardinal numbers have been defined in terms of /fin or ωω. Some have been generalized to higher cardinals, and some even to arbitrary Boolean algebras. Here we study eight of these cardinals, defining their generalizations to higher cardinals, and then defining them for Boolean algebras. We then attempt to completely describe their relationships within each of several important classes of Boolean algebras.

The generalizations to higher cardinals might involve several cardinals instead of just one as in the case of ω, For example, the number a associated with maximal almost disjoint families of infinite sets of integers can be generalized to talk about maximal subsets of [κ]μ subject to the pairwise intersections having size less than ν. (For this multiple generalization of . see Monk [2001].) For brevity we do not consider such generalizations, restricting ourselves to just one cardinal. The set-theoretic generalizations then associate with each infinite cardinal κ some other cardinal λ, defined as the minimum of cardinals with a certain property.

The generalizations to Boolean algebras assign to each Boolean algebra some cardinal λ, also defined as the minimum of cardinals with a certain property.

For the theory of the original “continuum” cardinal numbers, see Douwen [1984]. Balcar and Simon [1989]. and Vaughan [1990].

I am grateful to Mati Rubin for some conversations concerning these functions for superatomic algebras, and to Bohuslav Balcar for information concerning the function h.

The notation for set theory is standard. For Boolean algebras we follow Koppelberg [1989], but recall at the appropriate place any somewhat unusual notation.

The results in this paper say that natural upper bounds for sets of degrees associated with theories and models of arithmetic cannot be minimal. The basic new result says that for any completion T of PA, there is another completion S such that S<TT and Rep(S) = Rep(T). This immediately implies that deg(T) is not minimal over {deg(X): X ∈ Rep(T)}. As an application of the basic result, we obtain the fact that if is a non-standard model of TA (true arithmetic), then deg () cannot be minimal over {deg(X): X is arithmetical}. More generally, if is a non-standard model of an arbitrary completion T of PA, then deg() cannot be minimal over {deg(): X ∈ Rep(T)}. We vary the basic result, making S′ ≡TT′. As an application of the variant, we obtain the fact that if is a non-standard model of PA, then {deg(): ≅ } has no minimal element.

The remainder of the present section gives a brief account of the background needed for the basic new result and the variant. These two results are proved in Section 2. The applications are given in Section 3, along with further background needed for the applications. One important source of ideas used in the present paper is a paper of Scott [9]. In addition, there are ideas taken from Tennenbaum [12], Feferman [3], Marker [6], [7], and Solovay. Chapter 19 of [1] gathers together most of this material. In fact, it contains all that is really essential. In one application, we appeal to Solovay's result on degrees of models of an arbitrary completion of PA, a result which is not completely proved in [1]. However, for the best application, which implies all the others, we use only some ideas from the proof of Solovay's theorem. These are given in lemmas that are proved in Section 3, where they are needed, or taken from [1], While the proof of Solovay's result requires an infinitely nested priority construction, our best application rests on nothing more than finite-injury constructions.

We show there is a non-recursive r.e. set A such that if W is any low r.e. set. then the join W ⊕ A is also low. That is. A is “almost deep”. This answers a question of Joekusch. The almost deep degrees form an definable ideal in the r.e. degrees (with jump.)

Strengthening known instances of Vaught Conjecture, we prove the Glimm-Effros dichotomy theorems for countable linear orderings and for simple trees. Corollaries of the theorems answer some open questions of Friedman and Stanley in an Lω1ω-interpretability theory. We also give a survey of this theory.

The relationship between ideals I of Turing degrees and groups of I-recursive automorphisms of the ordering on rationals is studied. We discuss the differences between such groups and the group of all automorphisms, prove that the isomorphism type of such a group completely defines the ideal I, and outline a general correspondence between principal ideals of Turing degrees and the first-order properties of such groups.

The uniform reflection principle for the theory of uniform T-sentences is added to PA. The resulting system is justified on the basis of a disquotationalist theory of truth where the provability predicate is conceived as a special kind of analyticity. The system is equivalent to the system ACA of arithmetical comprehension. If the truth predicate is also allowed to occur in the sentences that are inserted in the T-sentences. yet not in the scope of negation, the system with the reflection schema for these T-sentences assumes the strength of the Kripke-Feferman theory KF. and thus of ramified analysis up to ε0.

We show that two abelian-by-finite groups are elementarily equivalent if and only if they satisfy the same sentences with two alternations of quantifiers. We also prove that abelian-by-finite groups satisfy a quantifier elimination property. On the other hand, for each integer n, we give some examples of nilpotent groups which satisfy the same sentences with n alternations of quantifiers and do not satisfy the same sentences with n + 1 alternations of quantifiers.

We show that every infinite computable partial ordering has either an infinite chain or an infinite antichain. Our main result is that this cannot be improved: We construct an infinite computable partial ordering that has neither an infinite chain nor an infinite antichain.

In this article we introduce systems for metapredicative Mahlo in explicit mathematics and admissible set theory. The exact upper proof-theoretic bounds of these systems are established.

The current paper proves the results announced in [5].

We isolate a new large cardinal concept, “remarkability.” Consistencywise, remarkable cardinals are between ineffable and ω-Erdös cardinals. They are characterized by the existence of “0#-like” embeddings; however, they relativize down to L. It turns out that the existence of a remarkable cardinal is equiconsistent with L(ℝ) absoluteness for proper forcings. In particular, said absoluteness does not imply determinacy.

Answers are given to two questions concerning the existence of some sparse subsets of ℋ = {0, 1 … ., H – 1} ⊆ *ℕ. where H is a hyperfinite integer. In §1. we answer a question of Kanovei by showing that for a given cut U in ℋ, there exists a countably determined set X ⊆ ℋ which contains exactly one element in each U-monad, if and only if U = a · ℕ for some a Є ℋ ∖ {0}. In §2, we deal with a question of Keisler and Leth in [6]. We show that there is a cut V ⊆ ℋ such that for any cut U, (i) there exists a U-discrete set X ⊆ ℋ with X + X = ℋ (mod H) provided , (ii) there does not exist any U-discrete set X ⊆ ℋ with X + X = ℋ (mod H) provided . We obtain some partial results for the case U = V.

We give an answer to the question as to whether quantifier elimination is possible in some infinite algebraic extensions of ℚp (‘infinite p-adic fields’) using a natural language extension. The present paper deals with those infinite p-adic fields which admit only tamely ramified algebraic extensions (so-called tame fields). In the case of tame fields whose residue fields satisfy Kaplansky's condition of having no extension of p-divisible degree quantifier elimination is possible when the language of valued fields is extended by the power predicates Pn introduced by Macintyre and, for the residue field, further predicates and constants. For tame infinite p-adic fields with algebraically closed residue fields an extension by Pn predicates is sufficient.