A pair of r.e. sets A1, A2 are said to split an r.e. set A (written A1 Δ A2 = A) if A1 ∩ A2 = ∅ and A1 ∪ A2 = A. In the literature there are various results asserting certain splitting properties hold for all r.e. sets. For example Sacks' splitting theorem (cf. [So]) asserts that an r.e. nonrecursive set A may be split into a pair of Turing incomparable r.e. sets A1, A2, and Lachlan's splitting theorem [La5] asserts that A may be split into a pair of r.e. sets A1, A2 for which there exists an r.e. set B with B ⊕ A1, B ⊕ A2 <TA and with the infinum of the Turing degrees of B ⊕ A1 and B ⊕ A2 existing in the upper semilattice of r.e. degrees.

Two of the earliest observations establishing that splitting properties possessed by some r.e. sets are not possessed by others, are Lachlan's nondiamond theorem [La1] (and so, in particular, no complete r.e. set can be split nontrivially with degree theoretic inf 0) and the Yates-Lachlan construction of a minimal pair (a pair of nonzero r.e. degrees with inf 0). Other examples of this phenomenon, which are not obtained by interpreting degree theoretic results in the r.e. sets, are Lachlan's construction of nonmitotic r.e. sets [La2] (sets which cannot be split into a pair of r.e. sets of the same degree) and, later, Ladner's [Ld1, 2] and Ingrassia's [In] analysis of their degrees.

If L is a first order language and n is a natural number, then Ln is the set of formulas which only make use of the variables x 1,…,xn . While every finite structure is determined up to isomorphism by its theory in L, the same is no longer true in Ln . This simple observation is the source of a number of intriguing questions. For example, Poizat [2] has asked whether a complete theory in Ln which has at least two nonisomorphic finite models must necessarily also have an infinite one. The purpose of this paper is to present some counterexamples to this conjecture.

Theorem. For each n ≤ 3 there are complete theories in L 2n−2 and L 2n−1 having exactly n + 1 models.

In our notation and definitions, we follow Poizat [2]. To test structures for elementary equivalence in Ln , we shall use the modified Ehrenfeucht-Fraïssé games of Immerman [1]. For convenience, we repeat his definition here.

Suppose that L is a purely relational language, each of the relations having arity at most n. Let and ℬ be two structures for L. Define the Ln game on and ℬ as follows. There are two players, I and II, and there are n pairs of counters a 1, b 1, …, an , bn . On each move, player I picks up any of the counters and places it on an element of the appropriate structure.

Namba [N] proved that the nonstationary ideal over a measurable (NSκ) cannot be κ+-saturated. Baumgartner, Taylor and Wagon [BTW] asked if it is possible for NSκ to be precipitous over a measurable κ. A model with this property was constructed by the author, and shortly after Foreman, Magidor and Shelah [FMS] proved a general theorem that after collapsing of a supercompact or even a superstrong to the successor of κ, NSκ became precipitous. This theorem implies that it is possible to have the nonstationary ideal precipitous over even a supercompact cardinal. Just start with a supercompact κ and a superstrong λ > κ. Make supercompactness of κ indistractible as in [L] and then collapse λ to be κ+.

The aim of our paper is to show that the existence of a supercompact cardinal alone already implies the consistency of the nonstationary ideal precipitous over a supercompact. The proof gives also the following: if κ is a λ-supercompact for λ ≥ (2κ)+, then there exists a generic extension in which κ is λ-supercompact and NSκ is precipitous. Thus, for a model with NSκ precipitous over a measurable we need a (2κ)+-supercompact cardinal κ. Jech [J] proved that the precipitous of NSκ over a measurable κ implies the existence of an inner model with o(κ) = κ+ + 1. In §3 we improve this result a little by showing that the above assumption implies an inner model with a repeat point.

The paper is organized as follows. In §1 some preliminary facts are proved. The model with NSκ precipitous over a supercompact is constructed in §2.

An explanation is given of why, after adding to a model M of ZFC first a Solovay real r and next a Cohen real c, in M[r] [c] a Cohen real over M[c] is produced. It is also shown that a Solovay algebra iterated with a Cohen algebra can be embedded into a Cohen algebra iterated with a Solovay algebra.

It is often asserted in the literature that any theorem which can be proved using nonstandard analysis can also be proved without it. The purpose of this paper is to show that this assertion is wrong, and in fact there are theorems which can be proved with nonstandard analysis but cannot be proved without it. There is currently a great deal of confusion among mathematicians because the above assertion can be interpreted in two different ways. First, there is the following correct statement: any theorem which can be proved using nonstandard analysis can be proved in Zermelo-Fraenkel set theory with choice, ZFC, and thus is acceptable by contemporary standards as a theorem in mathematics. Second, there is the erroneous conclusion drawn by skeptics: any theorem which can be proved using nonstandard analysis can be proved without it, and thus there is no need for nonstandard analysis.

The reason for this confusion is that the set of principles which are accepted by current mathematics, namely ZFC, is much stronger than the set of principles which are actually used in mathematical practice. It has been observed (see [F] and [S]) that almost all results in classical mathematics use methods available in second order arithmetic with appropriate comprehension and choice axiom schemes.

Let X be any infinite, coinfinite r.e. set. We show that the index set {e: We ≡1X) is -complete, answering a question posed by Odifreddi in [2].

The complexity of the set of truths of arithmetic is determined for various theories of truth deriving from Kripke and from Gupta and Herzberger.

The following is a contribution to the abstract study of the model theory of modal logics. Historically, individual modal logics have been specified deductively; and, as a result, it has seemed natural to view modal logics as sets of sentences provable in some deductive system. This proof theoretic view has influenced the abstract study of modal logics. For example, Fine [1975] defines a modal logic to be any set of sentences in the modal language L□ which contains all tautologies, all instances of the schema (□(ϕ ⊃ Ψ) ⊃ (□ϕ ⊃ □Ψ)), and which is closed under modus ponens, necessitation and substitution.

Here, however, modal logics are equated with classes of “possible world” interpretations. “Worlds” are taken to be ordered pairs (a, λ), where a is a sentential interpretation and λ is an ordinal. Properties of the accessibility relation are identified with those classes of binary relational systems closed under isomorphisms. The origin of this approach is the study of alternative Kripke semantics for the normal modal logics (cf. Weaver [1973]). It is motivated by the desire that modal logics provide accounts of both logical truth and logical consequence (cf. Corcoran and Weaver [1969]) and the realization that there are alternative Kripke semantics for S4, B and M which give identical accounts of logical truth, but different accounts of logical consequence (cf. Weaver [1973]). It is shown that the Craig interpolation theorem holds for any modal logic which has characteristic modal axioms and whose associated class of binary relational systems is closed under subsystems and finite direct products. The argument uses a back and forth construction to establish a modal analogue of Robinson's theorem. The argument for the interpolation theorem from Robinson's theorem employs modal analogues of the Ehrenfeucht-Fraïssé characterization of elementary equivalence and Hintikka's distributive normal form, and is itself a modal analogue of a first order argument (cf. Weaver [1982]).

Let PA be Peano arithmetic with function symbols′, + and ·. The length of a proof P, denoted by lh(P), is the maximum length of threads of P (for the term ‘thread’, see [T, p. 14]). For a formula A and a natural number m, PA ⊢mA denotes the fact that there is a proof in PA of A whose length is less than or equal to m. PA ⊢ A denotes the fact that there is a proof in PA of A.

G. Kreisel conjectured that the following proposition holds.

“Let m be a natural number and A(a) be a formula. If for each natural number n, then PA ⊢ ∀xA(x)”.

Let PA1 be the corresponding system with + and · replaced by ternary predicates and axioms saying that these predicates represent functions. Parikh [P] proved the following proposition which is obtained from Kreisel's conjecture by replacing PA by PA1.

Proposition. Let A(a) be a formula in PA1and m be a natural number. Assume thatfor each natural number n. Then PA1 ⊢ ∀xA(x).

The reason why Parikh's method succeeds is the fact that the only function symbol ′ in PA1 is unary. So his method fails for PA.

To solve this conjecture for PA, we must make syntactical investigation into proofs in PA of formulas of the form A() with length ≤ m. Even if lengths of proofs are less than or equal to m, depths of occurrences of bound variables in induction axiom schemata or equality axiom schemata in proofs are not always bounded.

wtt-reducibility has become of some importance in the last years through the works of Ladner and Sasso [1975], Stob [1983] and Ambos-Spies [1984]. It differs from Turing reducibility by a recursive bound on the use of the reduction. This makes some proofs easier in the wtt degrees than in the Turing degrees. Certain proofs carry over directly from the Turing to the wtt degrees, especially those based on permitting. But the converse is also possible. There are some r.e. Turing degrees which consist of a single r.e. wtt degree (the so-called contiguous degrees; see Ladner and Sasso [1975]). Thus it suffices to prove a result about contiguous wtt degrees using an easier construction, and it carries over to the corresponding Turing degrees.

In this work we prove some results on pairs of r.e. wtt degrees which have no infimum. The existence of such a pair has been shown by Ladner and Sasso. Here we use a different technique of Jockusch [1981] to prove this result (Theorem 1) together with some stronger ones. We show that a pair without infimum exists above a given incomplete wtt degree (Theorem 5) and below any promptly simple wtt degree (Theorem 12). In Theorem 17 we prove, however, that there are r.e. wtt degrees such that any pair below them has an infimum. This shows that certain initial segments of the wtt degrees are lattices. Thus there is a structural difference between the wtt and Turing degrees where the pairs without infimum are dense (Ambos-Spies [1984]).

Epistemic arithmetic—that is, first-order arithmetic with S4 as the underlying logic—was introduced by Shapiro in [7] and independently by Reinhardt in [6]. Shapiro showed that intuitionistic first-order arithmetic HA can be embedded in epistemic arithmetic EA . Moreover he showed that some of the basic proof-theoretic facts about HA , such as the existence and disjunction properties, can be extended to EA . In [3] we showed that the interpretation of HA in EA is faithful. (G.E. Mine has independently also proved this theorem.) Finally, in [2], Flagg showed that a suitable form of Church's thesis is consistent with EA . (Carlson [1] has announced another proof of this result.) Flagg's argument involves an ingenious realizability notion for EA which, as it stands, is not very perspicuous. The purpose of the present paper is to give a more transparent treatment of Flagg realizability. We obtain a new version of Flagg's proof of the consistency of Church's thesis with EA . Our main new result is that, in a sense to be made precise below, Flagg realizability coincides on HA embedded in EA with Kleene's 1945 realizability (e.g. see [5, pp. 501–516]). Thus it turns out once more that methods and results proved for EA can be viewed as extensions or generalizations of well-known methods and results for HA .

In his famous paper [1] on the elementary theory of finite fields Ax considered fields K with the property that every absolutely irreducible variety defined over K has K-rational points. These fields have been called pseudo algebraically closed (pac) and also regularly closed, and extensively studied by Jarden, Éršov, Fried, Wheeler and others, culminating with the basic works [8] and [11].

The above algebraic-geometric definition of pac fields can be put into the following equivalent model-theoretic version: K is existentially complete (ec) relative to the first order language of fields into each regular field extension of K. It has been this characterization of pac fields which the author extended in [2] to ordered fields. An ordered field (K, <) is called in [2] pseudo real closed (prc) if (K, <) is ec in every ordered field extension (L, <) with L regular over K. The concept of pre ordered field has also been introduced by McKenna in his thesis [15] by analogy with the original algebraic-geometric definition of pac fields.

Given a positive integer e, a system K = (K; P1, …, Pe), where K is a field and P1, …, Pe are orders of K (identified with the corresponding positive cones), is called an e-fold ordered field (e-field). In his thesis [9] van den Dries developed a model theory for e-fields. The main result proved in [9, Chapter II] states that the theory e-OF of e-fields is model con. panionable, and the models of the model companion e-OF are explicitly described.

Suppose κ is a supercompact cardinal. It is known that for every λ ≥ κ, many normal ultrafilters on Pκ(λ) have the partition property. It is also known that certain large cardinal assumptions imply the existence of normal ultrafilters without the partition property. In [1], we introduced the tree T of normal ultrafilters associated with κ. We investigate the distribution throughout T of normal ultrafilters with and normal ultrafilters without the partition property.

The comprehension principle of set theory asserts that a set can be formed from the objects satisfying any given property. The principle leads to immediate contradictions if it is formalized as an axiom scheme within classical first order logic. A resolution of the set paradoxes results if the principle is formalized instead as two rules of deduction in a natural deduction presentation of logic. This presentation of the comprehension principle for sets as semantic rules, instead of as a comprehension axiom scheme, can be viewed as an extension of classical logic, in contrast to the assertion of extra-logical axioms expressing truths about a pre-existing or constructed universe of sets. The paradoxes are disarmed in the extended classical semantics because truth values are only assigned to those sentences that can be grounded in atomic sentences.

We investigate the set S2 of “quickly sharped” reals:

in the manner of [K] defining a natural hierarchy and quasi-hierarchy of constructibility degrees and identifying their termination points.

The axioms for Zermelo-Fraenkel (ZF) set theory are an appealing but somewhat arbitrary-seeming assortment. A survey of the axioms does not suffice to reveal the source of their attraction. Accordingly, attempts have been made to ground ZF in principles whose appeal can be felt immediately. These attempts can be classified as follows. First, some of them propose to rest the ZF axioms directly on informal doctrine. The others propose to ground the ZF axioms in other formal axioms that can be regarded as more basic. When the latter approach is taken, ZF continues to draw on informal support, but the draft is made at a more basic level.

The same research can be classified in another way, according to the informal diagnosis offered for the paradoxes of set theory. In some cases, the diagnosis is that the paradoxical sets (such as the Russell set) fail to exist only because they would have to be too large; a set that would be sufficiently small must always exist. This is the doctrine of limitation of size. In other cases, the diagnosis is that the paradoxical sets fail to exist only because they would have to lie too high in a certain hierarchy of sets; a set that would lie sufficiently low in the hierarchy must always exist. This is the doctrine of the hierarchy. The present paper will investigate the latter doctrine, with the doctrine of size making a brief appearance at the end.

Let M be an O-minimal structure. We use our understanding, acquired in [KPS], of the structure of definable sets of n-tuples in M, to study definable (in M) equivalence relations on Mn. In particular, we show that if E is an A-definable equivalence relation on Mn (A ⊂ M) then E has only finitely many classes with nonempty interior in Mn, each such class being moreover also A-definable. As a consequence, we are able to give some conditions under which an O-minimal theory T eliminates imaginaries (in the sense of Poizat [P]).

If L is a first order language and M an L-structure, then by a definable set in M, we mean something of the form X ⊂ Mn, n ≥ 1, where X = {(a1…,an) ∈ Mn: M ⊨ϕ(ā)} for some formula ∈ L(M). (Here L(M) means L together with names for the elements of M.) If the parameters from come from a subset A of M, we say that X is A-definable.

M is said to be O-minimal if M = (M, <,…), where < is a dense linear order with no first or last element, and every definable set X ⊂ M is a finite union of points, and intervals (a, b) (where a, b ∈ M ∪ {± ∞}). (This notion is as in [PS] except here we demand the underlying order be dense.) The complete theory T is said to be O-minimal if every model of T is O-minimal. (Note that in [KPS] it is proved that if M is O-minimal, then T = Th(M) is O-minimal.) In the remainder of this section and in §2, M will denote a fixed but arbitrary O-minimal structure. A,B,C,… will denote subsets of M.

This paper introduces and investigates a notion that approximates decidability with respect to countable structures. The paper demonstrates that there exists a decidable first order theory with a prime model that is not almost decidable. On the other hand it is proved that if a decidable complete first order theory has only countably many complete types, then it has a prime model that is almost decidable. It is not true that every decidable complete theory with only countably many complete types has a decidable prime model. It is not known whether a complete decidable theory with only countably many countable models up to isomorphism must have a decidable prime model. In [1] a weaker result was proven—if every complete extension, in finitely many additional constant symbols, of a theory T fails to have a decidable prime model, then T has 2 ω nonisomorphic countable models. The corresponding statement for saturated models is false, even if all the complete types are recursive, as was shown in [2]. This paper investigates a variation of the open question via a different notion of effectiveness—almost decidable.

A tree Tr will be a subset of ω<ω that is closed under predecessor. For elements f, g in ω<ω ∪ ωω , ƒ ⊲ g iffdf ∀i < lh(ƒ)[ƒ(i) = g(i)].

This paper is a continuation of the authors' paper [7]; in particular, we give a sharper and more useful criterion for the approximate elementary equivalence of Cσ(K) spaces, where K is a totally disconnected compact Hausdorff space. (See Theorem 2 below.) As an application, we obtain a complete description of the Banach spaces X which are approximately equivalent to c0. Namely, X ≡ Ac0 iff X = Cσ(K) where K is a totally disconnected compact Hausdorff space which has a dense set of isolated points and σ is an involutory homeomorphism on K which has a unique fixed point t, and t is not an isolated point. (See Theorem 7.)

These results are derived from an analysis which we give of the elementary theories of structures (B, σ), where B is a Boolean algebra and σ is an involutory automorphism of B which leaves at most one nontrivial ultrafilter invariant. Let U(σ) denote this ultrafilter, if it exists; let U(σ) = B in case σ leaves no nontrivial ultrafilter invariant. We show that the elementary theory of (B, σ) is completely determined by the theory of (B, U(σ)) (and conversely, because U(σ) is definable in (B, σ)). This makes possible the use of the explicit invariants given by Éršov [4] for structures (B, U) where U is an ultrafilter on B. (These generalize the Tarski invariants for Boolean algebras [15].) We also use the Éršov invariants in the proof of our main result.

An attempt to answer the following question gave rise to the results of the present paper. Let be an arbitrary model of set theory. Does there exist an elementary extension of satisfying the two requirements: (1) contains an ordinal exceeding all the ordinals of ; (2) does not enlarge any (hyper) integer of ? Note that a trivial application of the ordinary compactness theorem produces a model satisfying condition (1); and an internal ultrapower modulo an internal ultrafilter produces a model satisfying condition (2) (but not (1), because of the axiom of replacement). Also, such a satisfying both conditions (1) and (2) exists if the external cofinality of the ordinals of is countable, since by [KM], would then have an elementary end extension.

Using a class of models constructed by M. Rubin using in [RS], and already employed in [E1], we prove that our question in general has a negative answer (see Theorem 2.3). This result generalizes the results of M. Kaufmann and the author (appearing respectively in [Ka] and [E1]) concerning models of set theory with no elementary end extensions.

In the course of the proof it was necessary to establish that all conservative extensions (see Definition 2.1) of models of ZF must be cofinal. This is in direct contrast with the case of Peano arithmetic where all conservative extensions are end extensional (as observed by Phillips in [Ph1]). This led the author to introduce two useful weakenings of the notion of a conservative end extension which, as shown by the “completeness” theorems in §3, can exist.