The classification of algebraic structures which can be embedded into ℛ, the uppersemilattice of recursively enumerable degrees, is the key to answering certain questions about Th(ℛ), the elementary theory of ℛ. In particular, these classification problems are important for answering decidability questions about fragments of Th(ℛ). Thus the solutions of Fried berg [F] and Mučnik [M] to Post's problem were easily extended to show that all finite partially ordered sets are embeddable into ℛ, and hence that ∃1 ∩ Th(ℛ), the existential theory of ℛ, is decidable. (The language used is ℒ′, the pure predicate calculus together with a binary relation symbol ≤ to be interpreted as the ordering of ℛ) The problem of determining which finite lattices are embeddable into ℛ has been a long-standing open problem, and is one of the major obstacles to determining whether ∀2 ∩ Th(ℛ), the universal-existential theory of ℛ, is decidable. Shore has obtained some nice partial results in this direction. Embeddings also played a central role in showing that Th(ℛ) is not ℵ0-categorical (Lerman, Shore and Soare [LeShSo]), thus resolving a problem posed by Jockusch. Harrington and Shelah [HS] embedded all 0′-presentable partially ordered sets into ℛ in such a way that the partially ordered sets can be uniformly recovered from four parameters. They used these embeddings to show that Th(ℛ) is undecidable.

The first nontrivial extension of the embeddings of Friedberg and Mučnik to lattice embeddings was obtained independently by Lachlan [La1] and Yates [Y] who showed that the four-element Boolean algebra can be embedded into ℛ. Thomason [T] and Lerman independently extended this result to include all finite distributive lattices. The nondistributive case, however, was much more difficult. Lachlan [La2] embedded the two five-element nondistributive lattices M 5 and N 5 (see Figures 1 and 2) into ℛ, and his proof could easily have been extended to include a larger class of lattices.

We give a new characterization of the hyperarithmetic sets: a set X of integers is recursive in e α if and only if there is a Turing machine which computes X and “halts” in less than or equal to the ordinal number ωα of steps. This result represents a generalization of the well-known “limit lemma” due to J. R. Shoenfield [Sho-1] and later independently by H. Putnam [Pu] and independently by E. M. Gold [Go]. As an application of this result, we give a recursion theoretic analysis of clopen determinacy: there is a correlation given between the height α of a well-founded tree corresponding to a clopen game A ⊆ ωω and the Turing degree of a winning strategy ƒ for one of the players—roughly, ƒ can be chosen to be recursive in 0 α and this is the best possible (see §4 for precise results).

Given an admissible indexing φ of the countable atomless Boolean algebra ℬ, an automorphism F of ℬ is said to be recursively presented (relative to φ) if there exists a recursive function p ϵ Sym(ω) such that F ∘ φ = φ ∘ p. Our key result on recursiveness: Both the subset of Aut(ℬ) consisting of all those automorphisms which are recursively presented relative to some indexing, and its complement, the set of all “totally nonrecursive” automorphisms, are uncountable.

This arises as a consequence of the following combinatorial investigations: (1) A comparison of the cycle structures of ƒ and , where ƒ is a permutation of some free basis of ℬ and is the automorphism of ℬ induced by ƒ.(2) An explicit description of the permutations of ω whose conjugacy classes in Sym(ω) are (a) uncountable, (b) countably infinite, and (c) finite.

Let κ and λ be infinite cardinals such that λ ≤ λ (we have new information for the case when κ ≤ λ). Let T be a theory in Lκ +, ω of cardinality at most κ, let . Now define

Our main concept in this paper is is a theory in Lκ +, ω of cardinality κ at most, and φ(x, y) ϵ Lκ +, ω }. This concept is interesting because of

Theorem 1. Let T ⊆ Lκ +, ω of cardinality ≤ κ, and . If

then (∀χ > κ)I(χ, T) = 2 χ (where I(χ, T) stands for the number of isomorphism types of models of T of cardinality χ).

Many years ago the second author proved that . Here we continue that work by proving

Theorem 2. .

Theorem 3. For every κ ≤ λ we have .

For some κ or λ we have better bounds than in Theorem 3, and this is proved via a new two cardinal theorem.

Theorem 4. For every T ⊆ Lκ +, ω, and any set of formulas ⊆ Lκ +, ω such that T ⊇ Lκ +, ω, if T is (, μ)-unstable for μ satisfying μμ*(λ,κ) = μ then T is -unstable (i.e. for every χ ≥ λ, T is (, χ)-unstable). Moreover, T is Lκ +, ω-unstable.

In the second part of the paper, we show that always in the applications it is possible to replace the function I(χ, T) by the function IE(χ, T), and we give an application of the theorems to Boolean powers.

Suppose T is superstable. Let ≤ denote the fundamental order on complete types, [p] the class of the bound of p, and U(—) Lascar's foundation rank (see [LP]). We prove

Theorem 1. If q < p and there is no r such that q < r < p, then U(q) + 1 = U(p).

Theorem 2. Suppose U(p) < ω and ξ1 < … < ξk is a maximal descending chain in the fundamental order with ξk = [p]. Then k = U(p).

That the finiteness of U(p) in Theorem 2 is necessary follows from

Theorem 3. There is an ω-stable theory with a type p ϵ S 1(ϕ) such that

(1) U(p) = ω + 1, and

(2) there is a maximal descending chain of proper extensions of [p] which has order type ω.

If a uniform ultrafilter U over an uncountable cardinal κ is not outright countably complete, probably the next best thing is that it have a finest partition: a master function f:κ → ω with ƒ −({n}) ∉ U for each n ϵ ω such that for any g: κ → κ, either (a) it is one-to-one on a set in U, or (b) it factors through ƒ (mod U), i.e. for some function h, {α < κ ∣ h(f(α)) = g(α)} ϵ U. In this paper, it is shown that recent contructions of irregular ultrafilters over ω 1 can be amplified to incorporate a finest partition.

Henceforth, let us assume that all ultrafilters are uniform.

There has been an extensive study of substantial hypotheses, which are nonetheless weaker than countable completeness, on ultrafilters over uncountable cardinals. To survey some results and to establish a context, let us first recall the Rudin-Keisler (RK) ordering on ultrafilters: If U i is an ultrafilter over I i i for i = 1, 2, then U 1 ≤RK U 2 iff there is a projecting function Ψ:I 2 → I 1 such that U 1 = Ψ *(U 2) = {X ⊆, I 1∣ Ψ −1(X) ϵ U 2}· U 1, =RK U 2 iff U 1, ≤RK and U 2 and U 2≤RK U 1; and U 1<RK U 2 iff U 1≤RK U 2 yet U 1 ≠RK U 2. In terms of this ordering, if an ultrafilter U has a finest partition ƒ, then ƒ *(U) over ω is maximum amongst all RK predecessors of U: for any g:κ → κ, if g *(U) <RK U, then g is not one-to-one on a set in U, so since g factors through ƒ with some h,g *(U) = h *(ƒ *(U)). Say now that an ultrafilter U over κ > ω is indecomposable iff whenever ω < λ < κ, there is no V ≤RK U such that V is a (uniform) ultrafilter over λ.

C. U. Jensen suggested the following construction, starting from a field K:

and asked when two fields Kα and Kβ are equivalent. We give a complete answer in the case of a field K of characteristic 0.

Let A and B be subsets of the reals. Say that A K ≥ B, if there is a real a such that the relation “x ∈ B” is uniformly ⊿1 (a, A) in . This reducibility induces an equivalence relation ≡ K on the sets of reals; the ≡ K -equivalence class of a set is called its Kleene degree. Let be the structure that consists of the Kleene degrees and the induced partial order ≥. A substructure of that is of interest is , the Kleene degrees of the sets of reals. If sharps exist, then there is not much to , as Steel [9] has shown that the existence of sharps implies that has only two elements: the degree of the empty set and the degree of the complete set. Legrand [4] used the hypothesis that there is a real whose sharp does not exist to show that there are incomparable elements in ; in the context of V = L, Hrbáček has shown that is dense and has no minimal pairs. The Hrbáček results led Simpson [6] to make the following conjecture: if V = L, then forms a universal homogeneous upper semilattice with 0 and 1. Simpson's conjecture is shown to be false by showing that if V = L, then Gödel's maximal thin set is the infimum of two strictly larger elements of .

The second main result deals with the notion of jump in . Let A′ be the complete Kleene enumerable set relative to A. Say that A is low-n if A (n) has the same degree as ⊘(n), and A is high-n if A(n) has the same degree as ⊘(n+1). Simpson and Weitkamp [7] have shown that there is a high (high-1) incomplete set in L. They have also shown that various other sets are neither high nor low in L. Legrand [5] extended their results by showing that, if there is a real x such that x# does not exist, then there is an element of that, for all n, is neither low-n nor high-n. In §2, ZFC is used to show that, for all n, if A is and low-n then A is Borel. The proof uses a strengthened version of Jensen's theorem on sequences of admissible ordinals that appears in [7, Simpson-Weitkamp].

We give a new elementary proof of the comparison theorem relating and ; the proof does not use Skolem theories.

By the same method we prove:

a) , for suitable classes of sentences;

b) proves the consistency of , for finite k, and hence is stronger than .

a) and b) answer a question of Feferman and Sieg.

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.

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.

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 .

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.

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)].

Let Pr be Presburger's arithmetic, i.e., the complete theory of the structure (ω, +). Lipshitz and Nadel showed in [4] that a countable model of Pr is recursively saturated iff it can be expanded to a model of Peano arithmetic, PA. As a starting point for an introductory discussion, let us mention one more fact about countable recursively saturated models of Pr. If is such a model then the following is readily seen (as explained also in §1):

Any two countable recursively saturated elementary endextensions of are isomorphic.

If we drop “countable” from the assumption of this statement, we can still say that the two models are ∞ω-equivalent. Must they be isomorphic if both have cardinality ℵ1? Certainly not, since one of the models can be ω 1-like while the other is not. Once we realized this much, let us concentrate on ω 1-like structures. We prove in §3:

Theorem. Any countable recursively saturated model of Pr has isomorphism types of ω 1-like recursively saturated elementary endextensions. Only one of these is the isomorphism type of a structure that can be expanded to a model of PA.

The key technical result is proven in §2. It says that an as above has precisely two countable recursively saturated elementary endextensions which are nonisomorphic over .

This paper is concerned with recursive structures and the persistance of an effective notion of categoricity. The terminology and notational conventions are standard. We will devote the rest of this paragraph to a cursory review of some of the assumed conventions. If θ is a formula, then θk denotes θ if k is zero, and ¬θ if k is one. If A is a sequence with domain a subset of ω, then A∣ n denotes the sequence obtained by restricting the domain of A to n. For an effective first order language L, let {ci ∣i<ω} be distinct new constants, and let {θ i ∣i<ω} be an effective enumeration of all sentences in the language L ∪ {ci ∣j<ω}. An infinite L-structure is recursive iff it has universe ω and the set is recursive, where cn is interpreted by n. In general we say that a set of formulas is recursive if the set of its indices with respect to an enumeration such as above is recursive. The ∃-diagram of a structure is recursive if the structure is recursive and the set and θ i is an existential sentence} is also recursive. The definition of “the ∀∃-diagram of is recursive” is similar.

A nonnegative integer is called a number, a collection of numbers a set and a collection of sets a class. We write ε for the set of all numbers, o for the empty set, N(α) for the cardinality of α, ⊂ for inclusion and ⊂+ for proper inclusion. Let α, β 1,…, β k be subsets of some set υ. Then α′ stands for υ−α and β 1 … β k for β 1 ∩ … ∩ β k . For subsets α 1, …, α r of υ we write:

Note that α 0 = υ, hence s 0 = N(υ). If the set υ is finite, the classical inclusion-exclusion principle (abbreviated IEP) states

In this paper we generalize (a) and(b) to the case where α 1, …, α r are subsets of some countable but isolated set υ. Then the role of the cardinality N(α) of the set α is played by the RET (recursive equivalence type) Req α of α. These generalizations of (a) and (b) are proved in §3. Since they involve recursive distinctness, this notion is discussed in §2. In §4 we consider a natural extension of “the sum of the elements of a finite set σ” to the case where σ is countable. §5 deals with valuations, i.e., certain mappings μ from classes of isolated sets into the collection Λ of all isols which permit us to further generalize IEP by substituting μ(α) for Req α.

A many-valued logic version of the Craig-Lyndon interpolation theorem has been given by Gill [1] and Miyama [2]. The former dealt with three-valued logic and the latter generalized it to M-valued logic with M ≥ 3. The purpose of this paper is to improve the form of Miyama's version of the interpolation theorem. The system used in this paper is a many-valued analogue (Takahashi [4], Rousseau [3]) of Gentzen's logical calculus LK. Let T = {1,…, M} be the set of truth values. An M-tuple (Γ 1,…, Γ M ) of sets of formulas is called a sequent, which is regarded as valid if for any valuation (in canonical sense) there is a truth value μ Є T such that the set Γμ contains a formula of the value μ with respect to the valuation. (In the next section, and thereafter, we change the format of the sequent for typographical reasons.) Miyama's result is as follows (in representative form):

• (I) If a sequent ({A}, ∅,…, ∅, {B}) is valid, then there is a formula D such that

• (i) every predicate or propositional variable occurring in D occurs in A and B, and

• (ii) the sequents {{A}, ∅,…, ∅, {D}) and (D}, ∅,…, ∅, {B}) are both valid.

• What shall be proved in this paper is the following (in representative form):

• (II) If a sequent ({A 1}, {A 2}, …, {AM }) is valid, then there is a formula D such that

• (i) every predicate or propositional variable occurring in D occurs in at least two of the formulas A 1,…, AM , and

• (ii) the following M sequents are valid:

• ({A 1},{D},…,{D}),({D},{A 2},…,{D}),…,({D},{D},…,{AM }).

• Clearly the former can be obtained as a corollary of the latter.

Let p be a set. A function Φ is uniformly Σ 1(p) in every admissible set if there is a Σ 1 formula ϕ in the parameter p so that ϕ defines Φ in every Σ 1-admissible set which includes p. A theorem of Van de Wiele states that if Φ is a total function from sets to sets then Φ is uniformly Σ 1 in every admissible set if and only if it is E-recursive. A function is ESp -recursive if it can be generated from the schemes for E-recursion together with a selection scheme over the transitive closure of p. The selection scheme is exactly what is needed to insure that the ESP -recursively enumerable predicates are closed under existential quantification over the transitive closure of p. Two theorems are established: a) If the transitive closure of p is countable then a total function on sets is ESp -recursive if and only if it is uniformly Σ 1(p) in every admissible set. b) For any p, if Φ is a function on the ordinal numbers then Φ is ESP -recursive if and only if it is uniformly Σ 1(p) in every admissible set.