For any countable transitive complete theory T with infinite models and the finite model property, we construct a minimal structure M such that the theory of M is small if and only if T is small, and is λ-stable if and only if T is λ-stable. This gives a series of new examples of minimal structures.

If X and Y are orthogonal hyperdefinable sets such that X is simple, then any group G interpretable in X ∪ Y has a normal hyperdefinable X-internal subgroup N such that G/N is Y-internal; N is unique up to commensurability. In order to make sense of this statement, local simplicity theory for hyperdefinable sets is developed. Moreover, a version of Schlichting’s Theorem for hyperdefinable families of commensurable subgroups is shown.

A Skolem class is a class of formulas of pure quantification theory in Skolem normal form: closed, prenex formulas with prefixes ∀…∀∃…∃. (Pure quantification theory contains quantifiers, truth-functions, and predicate letters, but neither the identity sign nor function letters.) The Gödel Class, in which the number of universal quantifiers is limited to two, was shown effectively solvable (for satisfiability) sixty years ago [G1]. Various solvable Skolem classes that extend the Gödel Class can be obtained by allowing more universal quantifiers but restricting the combinations of variables that may occur together in atomic subformulas [DG, Chapter 2]. The Gödel Class and these extensions are also finitely controllable, that is, every satisfiable formula in them has a finite model. In this paper we isolate a model-theoretic property that connects the usual solvability proofs for these classes and their finite controllability. For formulas in the solvable Skolem classes, the property is necessary and sufficient for satisfiability. The solvability proofs implicitly relied on this fact. Moreover, for any formula in Skolem normal form, the property implies the existence of a finite model.

The proof of the latter implication uses the random models technique introduced in [GS] for the Gödel Class and modified and applied in [Go] to the Maslov Class. The proof thus substantiates the claim made in [Go] that random models can be adapted to the Skolem classes of [DG, Chapter 2]. As a whole, the results of this paper provide a more general, systematic approach to finite controllability than previous methods.

It is proved consistent that there be a proper σ-ideal on ω1 and an ℵ1-preserving poset ℙ such that ℙ ⊩ the σ-ideal generated by is not proper.

The notion of the next admissible set has proved to be a very useful notion in definability theory and generalized recursion theory, a unifying notion that has produced further interesting results in its own right. The basic treatment of the next admissible set above a structure ℳ of urelements is to be found in Barwise's [75] book Admissible sets and structures. Also to be found there are many of the interesting characterizations of the next admissible set. For further justification of the interest of the next admissible set the reader is referred to Moschovakis [74], Nadel and Stavi [76] and Schlipf [78a, b, c].

One of the most interesting single properties of is its ordinal (ℳ). It coincides, for example, with Moschovakis' inductive closure ordinal over structures ℳ with pairing functions—and over some, such as algebraically closed fields of characteristic 0, without pairing functions (by recent work of Arthur Rubin) (although a locally famous counterexample of Kunen, a theorem of Barwise [77], and some recent results of Rubin and the author, show that the inductive closure ordinal may also be strictly smaller in suitably pathological structures). Further justification for looking at (ℳ) alone may be found in the above-listed references. Loosely, we can consider the size of to be a useful measure of the complexity of ℳ. One of the simplest measures of the size of —and yet a very useful measure—is its ordinal, (ℳ). Keisler has suggested thinking of (ℳ) as the information content of a model—the supremum of lengths of wellfounded relations characterizable in the model.

The purpose of this paper is to extend the coding method (see Beller, Jensen and Welch [82]) into the context of large cardinals.

Theorem. Suppose μ is a normal measure on κ in V and 〈 V, A〉 ⊨ ZFC. Then there is a 〈V, A〉-definable forcing for producing a real R such that:

(a) V[R] ⊨ ZFC and A is V[R]-definable with parameter R.

(b) V[R] = L[μ*, R], where μ* is a normal measure on κ in V[R] extending μ.

(c) V ⊨ GCH → is cardinal and cofinality preserving.

Corollary. It is consistent that μ is a normal measure, R ⊆ ω is not set-generic over L[μ] and 0+ ∉ L[μ, R].

Some other corollaries will be discussed in §4 of the paper.

The main difficulty in L[μ]-coding lies in the problem of “stationary restraint”.

As in all coding constructions, conditions will be of the form belonging to an initial segment of the cardinals, where p(γ) is a condition for almost disjoint coding into a subset of γ +. In addition for limit cardinals γ in Domain(p), 〈p γ′∣γ′ < γ〉 serves to code pγ .

An important restriction in coding arguments is that for inaccessible for only a nonstationary set of γ′ < γ. The reason is that otherwise there are conflicts between the restraint imposed by the different and the need to code extensions of pγ below γ.

Hereditarily finite sets are sets which are finite, whose members are finite, the members of whose members are finite, and so on. In ZF there are but countably many such sets; they constitute Vω. Were ZF to lose its axiom of regularity, however, one could not guarantee that the number of hereditarily finite sets would remain countable.

In Mostowski set theory, in which atomic sets are permissible, each atom, in isolation, would form a hereditarily finite collection. The number of hereditarily finite sets could be anything one should choose.

Even in a world that did not permit the free adjunction of arbitrary, meaningless atoms, the number of hereditarily finite sets could remain large. In Finsler set theory, it is shown as Theorem 22, below, that there are uncountably many hereditarily finite sets.

The reader who is interested in this paradoxical sounding fact can turn directly to §4 after grasping these introductory concepts. §3 is an exhaustive list of the smallest Finsler sets; it is hoped that this list will prove useful in checking future attempts to classify the finite Finsler sets.

Finsler set theory is not a firmly axiomatized theory. It is, at its present stage, a family of theories undergoing evolution. It permits the usual mathematical operations with sets. One can employ ordinal numbers, cardinal numbers, and the usual methods of Cantorian set theory freely. But there is a somewhat different interpretation attached to the concept “set” than one is used to in Zermelo-Fraenkel set theory, ZF.