Let T1 be the complete first-order theory of the additive group of the integers with 1 as distinguished element (in symbols, T1 = Th(Z, +, 1)). In this paper we prove that all models of T1 are ℵ0-homogeneous (§2), classify them (and lists of elements in them) up to isomorphism or L∞κ-equivalence (§§3 and 4) and show that they may be as complex as arbitrary sets of real numbers from the point of view of admissible set theory (§5). The results of §§2 and 5 together show that while the Scott heights of all models of T1 are ≤ ω (by ℵ0-homogeneity) their HYP-heights form an unbounded subset of the cardinal . In addition to providing this unusual example of the relation between Scott heights and HYP-heights, the theory T1 has served (using the homogeneity results of §2) as an example for certain combinations of properties that people had looked for in stability theory (see end of §4). In §6 it is shown that not all models of T = Th(Z, +) are ℵ0-homogeneous, so that the availability of the constant for 1 is essential for the result of §2. The two main results of this paper (2.2 and essentially Theorem 5.3) were obtained in the summer of 1979. Later we learnt from Victor Harnik and Julia Knight that T1 is of some interest for stability theory, and were encouraged to write up our proofs. During 1982/3 we improved the proofs and added some results.

Abstract We characterize the ordinals α of uncountable cofinality such that α is the standard part of a nonstandard model of ZFC (or equivalently KP).

The borderline between decidable and undecidable Propositional Dynamic Logic (PDL) is sought when iterative programs represented by regular expressions are augmented with increasingly more complex recursive programs represented by context-free languages. The results in this paper and its companion [HPS] indicate that this line is extremely close to the original regular PDL. The main result of the present paper is: The validity problem for PDL with additional programs αΔ(β)γΔ for regular α, β and γ, defined as Uiαi; β; γi, is Π11-complete. One of the results of [HPS] shows that the single program AΔ(B) AΔ for atomic A and B is actually sufficient for obtaining Π11- completeness. However, the proofs of this paper use different techniques which seem to be worthwhile in their own right.

The use of the temporal logic formalism for program reasoning is reviewed. Several aspects of responsiveness and fairness are analyzed, leading to the need for an additional temporal operator: the 'until' operator -U. Some general questions involving the 'until' operator are then discussed. It is shown that with the addition of this operator the temporal language becomes expressively complete. Then, two deductive systems DX and DUX are proved to be complete for the languages without and with the new operator respectively.

Determining the truth value of self-referential sentences is an interesting and often tricky problem. The Gödel sentence, asserting its own unprovability in P (Peano arithmetic), is clearly true in N(the standard model of P), and Löb showed that a sentence asserting its own provability in P is also true in N (see Smorynski [Sm, 4.1.1]). The problem is more difficult, and still unsolved, for sentences of the kind constructed by Kreisel [K1], which assert their own falsity in some model N* of P whose complete diagram is arithmetically defined. Such a sentence χ has the property that N ⊨ iff N* ⊭ χ (note that ¬χ has the same property). We show in §1 that the truth value in N of such a sentence χ, after a certain normalization that breaks the symmetry between it and its negation, is determined by the parity of a natural number, called the rank of N, for the particular construction of N* used. The rank is the number of times the construction can be iterated starting from N and is finite for all the usual constructions. We also show that modifications of, e.g., Henkin's construction (in his completeness proof of predicate calculus) allow arbitrary finite values for the rank of N. Thus, on the one hand the truth value of χ in N, for a given “nice” construction of N*, is independent of the particular (normalized) choice of χ, and we shall see that χ is unique up to (provable) equivalence in P. On the other hand, the truth value in question is sensitive to minor changes in the definition of N* and its determination seems to be largely a combinatorial problem.

Denoting a version of Hoare's system for proving partial correctness of recursive programs by H, we present an extension D which may be thought of as H υ {@@@@,@@@@,@@@@,@@@@} υ H-1, including the rules of H, four special purpose rules and inverse rules to those of Hoare. D is shown to be a complete system (in Cook's sense) for proving deductions of the form σ1,....σn @@@@ σ over a language, the wff's of which are assertions in some assertion language L and partial correctness specifications of the form p(α)q. All valid formulae of L are taken as axioms of D. It is shown that D is sufficient for proving partial correctness, total correctness and program equivalence as well as other important properties of programs, the proofs of which are impossible in H. The entire presentation is worked out in the framework of nondeterministic programs employing iteration and mutually recursive procedures.

Abstract Let ℳ be a structure for a language ℒ on a set M of urelements. HYP(ℳ) is the least admissible set above ℳ. In §1 we show that pp(HYP(ℳ)) [= the collection of pure sets in HYP(ℳ)] is determined in a simple way by the ordinal α = ° (HYP(ℳ)) and the ℒxω theory of ℳ up to quantifier rank α. In §2 we consider the question of which pure countable admissible sets are of the form pp(HYP(ℳ)) for some ℳ and show that all sets Lα (α admissible) are of this form. Other positive and negative results on this question are obtained.

In this paper a converse of Barwise's completeness theorem is proved by cut-elimination considerations applied to inductive definitions. We show that among the transitive sets T satisfying some weak closure conditions (closure under primitive-recursive set-functions is more than enough), only the unions of admissible sets satisfy Barwise's completeness theorem in the form stating that if φ ∊ T is a sentence which has a derivation (in the universe) then φ has a derivation in T. See §1 for the origin of the problem in Barwise's paper [Ba]. Stated quite briefly the proof is as follows (a step-by-step account including relevant definitions is given in the body of the paper): Let T be a transitive prim.-rec. closed set, and let is nonempty, transitive and closed under pairs}. For each let κ(A) be the supremum of closure ordinals of first-order positive operators on subsets of A (first-order with respect to By Theorem 1 of [BGM], it is enough to prove that rank(T) in order to obtain that T is a union of admissible sets. (The rank of a set is defined by rank(x) = supy ∊ x (rank(y) + 1); since T is prim.-rec. closed, rank(T) = smallest ordinal not in T.) Let We show how to find in T (in fact, in Lω(A)) a derivable sentence τ that has no derivation D such that rank(D) ≤ α. Thus, if τ is to have a derivation in T, rank(T) > α. α is arbitrary (< κ(A)), so rank(T) ≥ κ(A). Q.E.D.