If ωn has the tree property for all 2 ≤ n < ω and , then for all and n < ω. Mnt(X) exists.

We define a novel interpretation of second order arithmetic into Explicit Mathematics. As a difference from standard -interpretation, which was used before and was shown to interpret only subsystems proof-theoretically weaker than T0. our interpretation can reach the full strength of T0. The -interpretation is an adaptation of Kleene's recursive readability, and is applicable only to intuitionistic theories.

One of the main goals in the theory of forcing iteration is to formulate preservation theorems for not collapsing ω1 which are as general as possible. This line leads from c.c.c. forcings using finite support iterations to Axiom A forcings and proper forcings using countable support iterations to semi-proper forcings using revised countable support iterations, and more recently, in work of Shelah, to yet more general classes of posets. In this paper we concentrate on a special case of the very general iteration theorem of Shelah from [5, chapter XV]. The class of posets handled by this theorem includes all semi-proper posets and also includes, among others, Namba forcing.

In [5, chapter XV] Shelah shows that, roughly, revised countable support forcing iterations in which the constituent posets are either semi-proper or Namba forcing or P[W] (the forcing for collapsing a stationary co-stationary subset of

with countable conditions) do not collapse ℵ1. The iteration must contain sufficiently many cardinal collapses, for example, Levy collapses. The most easily quotable combinatorial application is the consistency (relative to a Mahlo cardinal) of ZFC + CH fails + whenever A ∪ B = ω2 then one of A or B contains an uncountable sequentially closed subset. The iteration Shelah uses to construct this model is built using P[W] to “attack” potential counterexamples, Levy collapses to ensure that the cardinals collapsed by the various P[W]'s are sufficiently well separated, and Cohen forcings to ensure the failure of CH in the final model.

In this paper we give details of the iteration theorem, but we do not address the combinatorial applications such as the one quoted above.

These theorems from [5, chapter XV] are closely related to earlier work of Shelah [5, chapter XI], which dealt with iterated Namba and P[W] without allowing arbitrary semi-proper forcings to be included in the iteration. By allowing the inclusion of semi-proper forcings, [5, chapter XV] generalizes the conjunction of [5, Theorem XI.3.6] with [5, Conclusion XI.6.7].

Certain basic concepts of geometrical stability theory are generalized to a class of closure operators containing algebraic closure. A specific case of a generalized closure operator is developed which is relevant to Vaught's conjecture. As an application of the methods, we prove

Theorem A. Let G be a superstate group of U-rank ω such that the generics of G are locally modular and Th(G) has few countable models. Let G− be the group of nongeneric elements of G. G+ = Go + G−. Let Π = {q ∈ S(∅): U(q) < ω}. For any countable model M of Th(G) there is a finite A ⊂ M such thai M is almost atomic over A ∪ (G+ ∩ M) ∪ ⋃p∈Πp(M).

This paper discusses two combinatorial problems in stability theory. First we prove a partition result for subsets of stable models: for any A and B, we can partition A into ∣B∣<κ(T) pieces. 〈Ai ∣ i < ∣B∣<κ(T)〉. such that for each Ai there is a Bi ⊆ B where ∣Bi∣ < κ(T) and , Second, if A and B are as above and ∣A∣ > ∣B∣, then we try to find A′ ⊂ A and B′ ⊂ B such that ∣A′∣ is as large as possible. ∣B′∣ is as small as possible, and . We prove some positive results in this direction, and we discuss the optimality of these results under ZFC + GCH.

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.

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.

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 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.