Abstract We study the relative computational power of structures related to the ordered field of reals, specifically using the notion of generic Muchnik reducibility. We show that any expansion of the reals by a continuous function has no more computing power than the reals, answering a question of Igusa, Knight, and Schweber [7]. On the other hand, we show that there is a certain Borel expansion of the reals that is strictly more powerful than the reals and such that any Borel quotient of the reals reduces to it.

Abstract We ask when, for a pair of structures $\mathcal {A}_1,\mathcal {A}_2$ , there is a uniform effective procedure that, given copies of the two structures, unlabeled, always produces a copy of $\mathcal {A}_1$ . We give some conditions guaranteeing that there is such a procedure. The conditions might suggest that for the pair of orderings $\mathcal {A}_1$ of type $\omega _1^{CK}$ and $\mathcal {A}_2$ of Harrison type, there should not be any such procedure, but, in fact, there is one. We construct an example for which there is no such procedure. The construction involves forcing. On the way to constructing our example, we prove a general result on modifying Cohen generics.

Abstract We improve on and generalize a 1960 result of Maltsev. For a field F, we denote by $H(F)$ the Heisenberg group with entries in F. Maltsev showed that there is a copy of F defined in $H(F)$ , using existential formulas with an arbitrary non-commuting pair of elements as parameters. We show that F is interpreted in $H(F)$ using computable $\Sigma _1$ formulas with no parameters. We give two proofs. The first is an existence proof, relying on a result of Harrison-Trainor, Melnikov, R. Miller, and Montalbán. This proof allows the possibility that the elements of F are represented by tuples in $H(F)$ of no fixed arity. The second proof is direct, giving explicit finitary existential formulas that define the interpretation, with elements of F represented by triples in $H(F)$ . Looking at what was used to arrive at this parameter-free interpretation of F in $H(F)$ , we give general conditions sufficient to eliminate parameters from interpretations.

Abstract There is a Turing computable embedding $\Phi $ of directed graphs $\mathcal {A}$ in undirected graphs (see [15]). Moreover, there is a fixed tuple of formulas that give a uniform effective interpretation; i.e., for all directed graphs $\mathcal {A}$ , these formulas interpret $\mathcal {A}$ in $\Phi (\mathcal {A})$ . It follows that $\mathcal {A}$ is Medvedev reducible to $\Phi (\mathcal {A})$ uniformly; i.e., $\mathcal {A}\leq _s\Phi (\mathcal {A})$ with a fixed Turing operator that serves for all $\mathcal {A}$ . We observe that there is a graph G that is not Medvedev reducible to any linear ordering. Hence, G is not effectively interpreted in any linear ordering. Similarly, there is a graph that is not interpreted in any linear ordering using computable $\Sigma _2$ formulas. Any graph can be interpreted in a linear ordering using computable $\Sigma _3$ formulas. Friedman and Stanley [4] gave a Turing computable embedding L of directed graphs in linear orderings. We show that there is no fixed tuple of $L_{\omega _1\omega }$ -formulas that, for all G, interpret the input graph G in the output linear ordering $L(G)$ . Harrison-Trainor and Montalbán [7] have also shown this, by a quite different proof.

Abstract This article contributes to the general program of extending techniques and ideas of effective algebra to computable metric space theory. It is well-known that relative computable categoricity (to be defined) of a computable algebraic structure is equivalent to having a c.e. Scott family with finitely many parameters (e.g., [1]). The first main result of the article extends this characterisation to computable Polish metric spaces. The second main result illustrates that just a slight change of the definitions will give us a new notion of categoricity unseen in the countable case (to be stated formally). The second result also shows that the characterisation of computably categorical closed subspaces of ${\Cal R}^n $ contained in [17] cannot be improved. The third main result extends the characterisation to not necessarily separable structures of cardinality κ using κ-computability.

We say that a structure $\mathcal{A}$ admits \emph{strong jump inversion} provided that for every oracle $X$, if $X'$ computes $D(\mathcal{C})'$ for some $\mathcal{C}\cong\mathcal{A}$, then $X$ computes $D(\mathcal{B})$ for some $\mathcal{B}\cong\mathcal{A}$. Jockusch and Soare \cite{JS} showed that there are low linear orderings without computable copies, but Downey and Jockusch \cite{DJ} showed that every Boolean algebra admits strong jump inversion. More recently, D.\ Marker and R.\ Miller \cite{MM} have shown that all countable models of $DCF_0$ (the theory of differentially closed fields of characteristic $0$) admit strong jump inversion. We establish a general result with sufficient conditions for a structure $\mathcal{A}$ to admit strong jump inversion. Our conditions involve an enumeration of $B_1$-types, where these are made up of formulas that are Boolean combinations of existential formulas. Our general result applies to some familiar kinds of structures, including some classes of linear orderings and trees. We do not get the result of Downey and Jockusch for arbitrary Boolean algebras, but we do get a result for Boolean algebras with no $1$-atom, with some extra information on the complexity of the isomorphism. Our general result gives the result of Marker and Miller. In order to apply our general result, we produce a computable enumeration of the types realized in models of $DCF_0$. This also yields the fact that the saturated model of $DCF_0$ has a decidable copy.

Abstract In [8], the third author defined a reducibility $\le _w^{\rm{*}}$ that lets us compare the computing power of structures of any cardinality. In [6], the first two authors showed that the ordered field of reals ${\cal R}$ lies strictly above certain related structures. In the present paper, we show that $\left( {{\cal R},exp} \right) \equiv _w^{\rm{*}}{\cal R}$ . More generally, for the weak-looking structure ${\cal R}$ ℚconsisting of the real numbers with just the ordering and constants naming the rationals, allo-minimal expansions of ${\cal R}$ ℚare equivalent to ${\cal R}$ . Using this, we show that for any analytic functionf, $\left( {{\cal R},f} \right) \equiv _w^{\rm{*}}{\cal R}$ . (This is so even if $\left( {{\cal R},f} \right)$ is noto-minimal.)

Abstract Schweber [10] defined a reducibility that allows us to compare the computing power of structures of arbitrary cardinality. Here we focus on the ordered field ${\cal R}$ of real numbers and a structure ${\cal W}$ that just codes the subsets of ω. In [10], it was observed that ${\cal W}$ is reducible to ${\cal R}$ . We prove that ${\cal R}$ is not reducible to ${\cal W}$ . As part of the proof, we show that for a countable recursively saturated real closed field ${\cal P}$ with residue field k, some copy of ${\cal P}$ does not compute a copy of k.

Abstract In this paper, we investigate connections between structures present in every generic extension of the universe V and computability theory. We introduce the notion of generic Muchnik reducibility that can be used to compare the complexity of uncountable structures; we establish basic properties of this reducibility, and study it in the context of generic presentability, the existence of a copy of the structure in every extension by a given forcing. We show that every forcing notion making ω2 countable generically presents some countable structure with no copy in the ground model; and that every structure generically presentable by a forcing notion that does not make ω2 countable has a copy in the ground model. We also show that any countable structure ${\cal A}$ that is generically presentable by a forcing notion not collapsing ω1 has a countable copy in V, as does any structure ${\cal B}$ generically Muchnik reducible to a structure ${\cal A}$ of cardinality ℵ1. The former positive result yields a new proof of Harrington’s result that counterexamples to Vaught’s conjecture have models of power ℵ1 with Scott rank arbitrarily high below ω2. Finally, we show that a rigid structure with copies in all generic extensions by a given forcing has a copy already in the ground model.

Abstract For a countable structure , the spectrum is the set of Turing degrees of isomorphic copies of . For a complete elementary first order theory T, the spectrum is the set of Turing degrees of models of T. We answer a question from [1] by showing that there is an atomic theory T whose spectrum does not match the spectrum of any structure.

We continue recent work on computable structure theory in the setting of ω1. We prove the analogue of a result from Fokina et al. (2012 J. Symbolic Logic, 77, 122–132) saying that isomorphism of computable structures lies ‘on top’ among Σ11 equivalence relations on ω. Our equivalence relations are on ω1. In the standard setting, Σ11 sets are characterized in terms of paths through trees. In the setting of ω1, we use a new characterization of Σ11 sets that involves clubs in ω1. Finally, we present some new results about ω1-computable categoricity for fields.

Abstract We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of FF-reducibility introduced in [9] to show completeness of the isomorphism relation on many familiar classes in the context of all equivalence relations on hyperarithmetical subsets of ω.

Abstract The first main result isolates some conditions which fail for the class of graphs and hold for the class of Abelianp-groups, the class of Abelian torsion groups, and the special class of “rank-homogeneous” trees. We consider these conditions as a possible definition of what it means for a class of structures to have “Ulm type”. The result says that there can be no Turing computable embedding of a class not of Ulm type into one of Ulm type. We apply this result to show that there is no Turing computable embedding of the class of graphs into the class of “rank-homogeneous” trees. The second main result says that there is a Turing computable embedding of the class of rank-homogeneous trees into the class of torsion-free Abelian groups. The third main result says that there is a “rank-preserving” Turing computable embedding of the class of rank-homogeneous trees into the class of Boolean algebras. Using this result, we show that there is a computable Boolean algebra of Scott rank .

Shepherdson [14] showed that for a discrete ordered ring I, I is a model of I Open iff I is an integer part of a real closed ordered field. In this paper, we consider integer parts satisfying PA. We show that if a real closed ordered field R has an integer part I that is a nonstandard model of PA (or even IΣ4), then R must be recursively saturated. In particular, the real closure of I, RC (I), is recursively saturated. We also show that if R is a countable recursively saturated real closed ordered field, then there is an integer part I such that R = RC(I) and I is a nonstandard model of PA.

Abstract We show that for every computable limit ordinal α, there is a computable structure that is categorical, but not relatively categorical (equivalently, it does not have a formally Scott family). We also show that for every computable limit ordinal α, there is a computable structure with an additional relation R that is intrinsically on , but not relatively intrinsically on (equivalently, it is not definable by a computable Σα formula with finitely many parameters). Earlier results in [7], [10], and [8] establish the same facts for computable successor ordinals α.

Abstract This paper calculates, in a precise way. the complexity of the index sets for three classes of computable structures: the class of structures of Scott rank , the class , of structures of Scott rank , and the class K of all structures of non-computable Scott rank. We show that I(K) is m-complete is m-complete relative to Kleene's and is m-complete relative to .

Abstract Here we prove that if T and T′ are strongly minimal theories, where T′ satisfies a certain property related to triviality and T does not, and T′ is model complete, then there is no computable embedding of Mod(T) into Mod(T′). Using this, we answer a question from [4], showing that there is no computable embedding of VS into ZS, where VS is the class of infinite vector spaces over ℚ, and ZS is the class of models of Th(ℤ, S). Similarly, we show that there is no computable embedding of ACF into ZS, where ACF is the class of algebraically closed fields of characteristic 0.

Abstract In [3]. two different effective versions of Borel embedding are defined. The first, called computable embedding, is based on uniform enumeration reducibility. while the second, called Turing computable embedding, is based on uniform Turing reducibility. While [3] focused mainly on computable embeddings, the present paper considers Turing computable embeddings. Although the two notions are not equivalent, we can show that they behave alike on the mathematically interesting classes chosen for investigation in [3]. We give a “Pull-back Theorem”, saying that if Ф is a Turing computable embedding of K into K′, then for any computable infinitary sentence φ in the language of K′, we can find a computable infinitary sentence φ* in the language of K such that for all A ∈ K A ⊨ φ* iff Φ (A) ⊨ φ and φ* has the same “complexity” as φ (i.e., if φ is computable Σα or computable Πα, for α ≥ 1, then so is φ*). The Pull-back Theorem is useful in proving non-embeddability, and it has other applications as well.

Abstract Makkai [10] produced an arithmetical structure of Scott rank ω1CK. In [9], Makkai's example is made computable. Here we show that there are computable trees of Scott rank ω1CK. We introduce a notion of “rank homogeneity”. In rank homogeneous trees, orbits of tuples can be understood relatively easily. By using these trees, we avoid the need to pass to the more complicated “group trees” of [10] and [9], Using the same kind of trees, we obtain one of rank ω1CK that is “strongly computably approximable”. We also develop some technology that may yield further results of this kind.

Abstract. A set X is prime bounding if for every complete atomic decidable (CAD) theory T there is a prime model of T decidable in X. It is easy to see that X = 0′ is prime bounding. Denisov claimed that every X

When bounds on complexity of some aspect of a structure are preserved under isomorphism, we refer to them as intrinsic. Here, building on work of Soskov [34], [33], we give syntactical conditions necessary and sufficient for a relation to be intrinsically on a structure. We consider some examples of computable structures and intrinsically relations R. We also consider a general family of examples of intrinsically relations arising in computable structures of maximum Scott rank. For three of the examples, the maximal well-ordered initial segment in a Harrison ordering, the superatomic part of a Harrison Boolean algebra, and the height-possessing part of a Harrison p-group, we show that the Turing degrees of images of the relation in computable copies of the structure are the same as the Turing degrees of paths through Kleene's . With this as motivation, we investigate the possible degrees of these paths. We show that there is a path in which ∅′ is not computable. In fact, there is one in which no noncomputable hyperarithmetical set is computable. There are paths that are Turing incomparable, or Turing incomparable over a given hyperarithmetical set. There is a pair of paths whose degrees form a minimal pair. However, there is no path of minimal degree.

We consider only computable languages, and countable structures, with universe a subset of ω, which we think of as a set of constants. We identify sentences with their Gödel numbers. Thus, for a structure , the complete (elementary) diagram, Dc( ), and the atomic diagram, D( ), are subsets of ω. We classify formulas as usual. A formula is both Σ0 and Π0 if it is open. For n > 0, a formula, in prenex normal form, is Σn, or Πn, if it has n blocks of like quantifiers, beginning with ∃, or ∀. For a formula θ, in prenex normal form, we let neg(θ) denote the dual formula that is logically equivalent to ¬θ—if θ is Σn, then neg(θ) is Πn, and vice versa.

The results in this paper say that natural upper bounds for sets of degrees associated with theories and models of arithmetic cannot be minimal. The basic new result says that for any completion T of PA, there is another completion S such that S

Feiner [F] showed that a Boolean algebra need not have a computable copy (see also [T2]). Downey and Jockusch [D-J] showed that every low Boolean algebra does have a computable copy. Thurber [T3], showed that every low2 Boolean algebra has a computable copy. Here we show that every Boolean algebra which is low3, or even low4, has a computable copy. The results of [D-J] and [T3] were obtained by passing to linear orderings. In [D-J], there is an embedding theorem saying that any linear ordering which is with the successor relation as an added predicate can be embedded in a slightly larger linear ordering which is computable. An isomorphism theorem of Remmel [R] is used to show that the interval algebras of the two linear orderings are isomorphic (except in a trivial case). In [T3], there is an embedding theorem saying that any linear ordering which is with certain added predicates can be embedded in one which is with successor. Again the isomorphism theorem of Remmel is used to show that the interval algebras are isomorphic (except in a trivial case). Here, instead of passing to linear orderings, we work directly with Boolean algebras. We begin with a review of the known results. We re-formulate the embedding theorems of Downey-Jockusch and Thurber in terms of Boolean algebras. We extract from Remmel's isomorphism theorem some information on complexity. In this way, we show that a low Boolean algebra is isomorphic to a computable one by an isomorphism which is , at worst, and the same is true for a low2 Boolean algebra.

In this paper, we state a metatheorem for constructions involving coding. Using the metatheorem, we obtain results on coding a family of sets into a family of relations, or into a single relation. For a concrete example, we show that the set of limit points in a recursive ordering of type ω2 can have arbitrary 2-REA degree.

We extend results of Harizanov and Barker. For a relation R on a recursive structure /oA, we give conditions guaranteeing that the image of R in a recursive copy of /oA can be made to have arbitrary ∑α0 degree over Δα0. We give stronger conditions under which the image of R can be made ∑α0 degree as well. The degrees over Δα0 can be replaced by certain more general classes. We also generalize the Friedberg-Muchnik Theorem, giving conditions on a pair of relations R and S under which the images of R and S can be made ∑α0 and independent over Δα0 in a recursive copy of /oA.

Let A be a recursive structure, and let R be a recursive relation on A. Harizanov (1991) isolated a syntactical condition which (with additional effectiveness conditions) is necessary and sufficient for A to have recursive copies in which the image of R is r.e. of arbitrary r.e. degree. We had conjectured that a certain extension of Harizanov's syntactical condition would (with some effectiveness conditions) be necessary and sufficient for A to have recursive copies in which the image of R is ∑α0 of arbitrary ∑α0 degree, but this is not the case. Here we give examples illustrating some restrictions on the possible ∑α0 degrees. In these examples, the image of R cannot be ∑α0 of degree d unless d possesses an “α-table” (a sequence of sets in which each one is r.e. in and above the earlier ones).

Methods for carrying out transfinitely nested priority constructions have been developed by Harrington [7] and by Ash [2, 1, 3, 4]. Ash's method has different versions, with later ones becoming simpler. Lemmp and Lerman [11] have also developed a method, for finitely nested constructions. Ash formulated abstractly the object of a nested priority construction, and he proved a metatheorem for what he called “α-systems”, listing conditions which guarantee the success of the construction. Harrington's method of “workers”, at least in its original, informal state, seems more flexible than Ash's α-systems. In [10, 9], there are finite and transfinite versions of a metatheorem for workers. The statements are complicated, and these metatheorems have not proved to be very useful. The present paper gives a new transfinite metatheorem. The statement is considerably simpler than the one in [9], although not so simple as that in [3]. The new metatheorem grew, in part, out of an effort to find a new proof of Ash's metatheorem. The new metatheorem yields the one in [3], and it seems more flexible. A different generalization of Ash's metatheorem will be given in [5]. Ash's metatheorem is easier to use than the one in the present paper, and the result in [3] is certainly the one to use wherever it applies. Here we give one application of the new metatheorem which does not seem to follow from the result in [3]. This is a theorem on models “representing” a given Scott set, which implies one half of a recent result of Solovay [18], on Turing degrees of models of particular completions of Peano arithmetic (PA).

In [A1]–[A4] there is an abstract description, in terms of “α-systems”, of the object of a nested priority construction, and there is a metatheorem listing conditions which guarantee the success of the construction. The metatheorem has different versions, with later versions becoming simpler and more general. The information needed to meet the requirements is enumerated by a function, where α is an arbitrary recursive ordinal. All other objects associated with the α-system are r.e. In the basic metatheorem, the requirements must all be at level α, although there is a special result, for limit α, which allows one requirement at each level in an increasing sequence with limit α. There are other abstract descriptions of nested priority arguments in [K1], [K2], [K3], and [L-L]. A typical use of α-systems is that in [B], where it is shown that, for a relation R on a recursive structure , under suitable assumptions, if R does not have a “recursive Σα” definition in , then there is a recursive isomorphic copy of in which the image of R is not . There is no difficulty in generalizing this to make each of infinitely many relations not , when the assumptions are uniformly satisfied and each relation has no recursive Σα definition. However, the metatheorems for α-systems do not apply to the situation where we have several (even two) relations Ri, which do not have recursive Σβi definitions and whose images we wish to make not for distinct βi. An even more basic problem is to construct a recursive copy of a recursive structure via an isomorphism whose restriction to various sets Ui, is not If βi, = α for all i, then we can use an (α + 1)-system; otherwise, the old metatheorems do not apply.

In what follows, L is a recursive language. The structures to be considered are L-structures with universe named by constants from ω. A structure is recursive A if the open diagram D( ) is recursive. Lerman and Schmerl [L-S] proved the following result. Let T be an ℵ0-categorical elementary first-order theory. Suppose that for all n, , and T is arithmetical. Then T has a recursive model. The aim of this paper is to extend Theorem 0.1. Stating the extension requires some terminology. Consider finitary formulas with symbols from L and sometimes extra constants from ω. For each n ∈ ω, the Σn and Πn formulas are as usual. Then Bnformulas are Boolean combinations of Σn formulas. For an L-structure , Dn( ) denotes the set of Bn sentences in the complete diagram Dc( ). A complete Σn theory is a maximal consistent set of ΣnL-sentences. We may write φ(x), or Γ(x), to indicate that the free variables of the formula φ, or the set Γ, are among those in x. A complete Bn type for x is a maximal consistent set Γ(x) of Bn formulas with just the free variables x. If T is ℵ0-categorical, then for each x only finitely many complete types Γ(x) are consistent with T. While Lerman and Schmerl stated their result just for ℵ0-categorical theories, essentially the same proof yields the following. Theorem 0.2. Let T be a consistent, complete theory such that for all n andx, only finitely many complete Bn types Γ(x) are consistent with T.

In one of the author's earlier articles the main result is a metatheorem for constructions by finitely many workers. In an article of Ash. Jokusch and Knight, there were two constructions which had, for an arbitrary recursive ordinal α, one worker for each β≤α. The present paper gives a metatheorem for constructions like these ones. The object of the construction is to produce: 1) a sequence of instructions with «labels» attached, following a prescribed «instruction function» that is recursive in 0 (α) , and 2) a recursive sequence of neighborhoods in a metric space, determining a point that «adheres» to the sequence of labels. The metatheorem is applied to pairs of structures

The aim of the present paper is to give some general conditions for constructions by finitely many workers. Constructions using infinitely many workers will not be considered here, although there are examples of such constructions. The original construction using the method of workers, due to Harrington [H], has a worker n for each n ∈ ω, as do the constructions in [K1] and [K2]. Marker [M] obtains a result using three workers. In [AJK], there are two constructions that use three workers. There are also two constructions that have, for an arbitrary recursive ordinal α, one worker for each β < α. The main result here is a metatheorem, which is patterned after Proposition 1 of Ash [A]. As in [A], the object of the construction is to attach “labels” to the nodes in a highly nonrecursive path through a tree, while recursively enumerating neighborhoods of an “adherent” point in a metric space. There is a family of relations associated with the labels, and the metatheorem here and the one in [A] both say that the construction will succeed if these relations satisfy a list of properties. There are significant differences between the result here and that in [A]. One difference is that certain relations which in [A] were required to be r.e. need not be r.e. here. Another difference is that there are extra relations here, and as a result, the list of properties to be satisfied is longer and more horrible than that in [A].

The complete diagram of a structure , denoted by Dc( ), is the set of all sentences true in the structure ( , a)a∈ . A structure is said to be resplendent if for every sentence θ involving a new relation symbol R in addition to symbols occurring in Dc( ), if θ is consistent with Dc( ), then there is a relation P on such that (see[1]). Baldwin asked whether a homogeneous recursively saturated structure is necessarily resplendent. Here it is shown that this need not be the case. It is shown that if is an uncountable homogeneous resplendent model of an unstable theory, then must be saturated. The proof is related to the proof in [5] that an uncountable homogeneous recursively saturated model of first order Peano arithmetic must be saturated. The example for Baldwin's question is an uncountable homogeneous model for a particular unstable theory, such that is recursively saturated and omits some type. (The continuum hypothesis is needed to show the existence of such a model in power ℵ1.) The proof of the main result requires two lemmas.

All structures to be considered here have universe ω, and all languages come equipped with Gödel numberings. If is a structure, then D( ), the open diagram of , can be thought of as a subset of ω, and it makes sense to talk about the Turing degree deg(D( )). This depends on the presentation as well as the isomorphism type of . For example, consider the ordering = (ω, <). For any B ⊆ ω, it is possible to code B in a copy of as follows: Let π be the permutation of ω such that for each n ∈ ω, π leaves 2n and 2n + 1 fixed if n ∈ B and switches 2n with 2n + 1 if n ∉ B. Let be the copy of such that ≃π . Then n ∈ B iff the sentence 2n < 2n + 1 is in D( ). In §4, this idea will be used to show that for any structure that is not completely trivial, {deg(D( )): ≃ } is closed upwards. It would be satisfying to have a way of assigning Turing degrees to structures such that the degree assigned to a given structure measured the recursion-theoretic complexity of the isomorphism type and was independent of the presentation. Jockusch suggested the following.

Let PA be the theory of first order Peano arithmetic, in the language L with binary operation symbols + and ·. Let N be the theory of the standard model of PA. We consider countable models M of PA such that the universe ∣M∣ is ω. The degree of such a model M, denoted by deg(M), is the (Turing) degree of the atomic diagram of M. The results of this paper concern the degrees of models of N, but here in the Introduction, we shall give a brief survey of results about degrees of models of PA. Let D0 denote the set of degrees d such that there is a nonstandard model of M of PA with deg(M) = d. Here are some of the more easily stated results about D0. (1) There is no recursive nonstandard model of PA; i.e., 0 ∈ D0. This is a result of Tennenbaum [T]. (2) There existsd ∈ D0such thatd ≤ 0′. This follows from the standard Henkin argument. (3) There existsd ∈ D0such thatd < 0′. Shoenfield [Sh1] proved this, using the Kreisel-Shoenfield basis theorem. (4) There existsd ∈ D0such thatd′ = 0′. Jockusch and Soare [JS] improved the Kreisel-Shoenfield basis theorem and obtained (4). (5) D0 = Dc = De, where Dc denotes the set of degrees of completions of PA and De the set of degrees d such that d separates a pair of effectively inseparable r.e. sets. Solovay noted (5) in a letter to Soare in which in answer to a question posed in [JS] he showed that Dc is upward closed.

In [6], Nadel showed that if is a recursively saturated model of Pr = Th(ω, +) of power at most ℵ1, then there is a model such that ≡ ∞ω and can be expanded to a recursively saturated model of P. For a fixed completion T of P, can be chosen to have a recursively saturated expansion to a model of T just in case is recursive in T-saturated. (“Recursive in T-saturation” is defined just like recursive saturation except that the sets of formulas considered are those that are recursive in T.) Nadel also showed in [6] that for a fixed completion T of P, a countable nonstandard model of Pr can be expanded to a model of T (not necessarily recursively saturated) iff satisfies a condition called “exp(T)-saturation.” This condition is stronger than recursive saturation but weaker than recursive in T-saturation. Nadel left open the problem of characterizing the models of Pr of power ℵ1 such that for some , ≣ ∞ω and can be expanded to a model of T. The present paper gives such a characterization. The condition on is that it is recursively saturated, and for each n ∈ ω, the set Tn of Πn-sentences of T is recursive in some type realized in . This result can be interpreted in various ways, just as the results from [6] were interpreted in various ways in [4]. Friedman [2] introduced the notion of a “standard system.”

The theories considered here are countable and complete, and the types are all complete too. Let T be an L-theory. A sequence σ = (σn(ν))n∈ω of L-formulas is said to be independent (with respect to T) if for each α ∈ 2<ω, the sentence is in T. As an example, let T = Th(Z, +), and let σ be the sequence of formulas saying (in the language of groups) ν is divisible by the nth prime, for n ∈ ω. A theory T has an independent sequence of formulas just in case it has types. If T has one independent sequence σ, then it has other independent sequences of arbitrarily high degree. (These can be obtained by taking conjunctions of the formulas from σ. If T has an independent sequence that is recursive, or one that is recursive in some type, then T will have types of arbitrarily high degree. (This follows from the fact that the independent sequence can be used to encode any set in a type.) Nadel and the author had wondered whether a theory with types must have an independent sequence of formulas that is recursive in one of the types. The main result of the present paper is an example of a recursive theory for which this is not the case.

Abstract An example is given of a complete theory with minimal models of arbitrarily large minimality rank.

If is a countable recursively saturated structure and T is a recursively axiomatizable theory that is consistent with Th( ), then it is well known that can be expanded to a recursively saturated model of T [7, p. 186]. This is what has made recursively saturated models useful in model theory. Recursive saturation is the weakest notion of saturation for which this expandability result holds. In fact, if is a countable model of Pr = Th(ω, +), then can be expanded to a model of first order Peano arithmetic P just in case is recursively saturated (see [3]). In this paper we investigate two natural sets of Turing degrees that tell a good deal about the expandability of a given structure. If is a recursively saturated structure, I( ) consists of the degrees of sets that are recursive in complete types realized in . The second set of degrees, D( ), consists of the degrees of sets S such that is recursive in S-saturated. In general, I( ) ⊆ D( ). Moreover, I( ) is obviously an “ideal” of degrees. For countable structures , D( ) is “closed” in the following sense: For any class C ⊆ 2ω, if C is co-r.e. in S for some set S such that , then there is some σ ∈ C such that . For uncountable structures , we do not know whether D( ) must be closed.

A set J of Turing degrees is called an ideal if (1) J ≠ ∅, (2) for any pair of degrees ã, , if ã, ϵ J, then ã ⋃ ϵJ, and (3) for any ⋃ ϵ J and any , if < ⋃, then ϵ J. A set J of degrees is said to be closed if for any theory T with a set of axioms of degree in J, T has a completion of degree in J. Closed ideals of degrees arise naturally in the following way. If is a recursively saturated structure, let I( ) = { for some ā ϵ }. Let D( ) = { : is recursive in d-saturated}. (Recursive in d-saturation is defined like recursive saturation except that the sets of formulas considered are recursive in d.) These two sets of degrees were investigated in [2]. It was shown that if is a recursively saturated model of P, Pr = Th(ω, +), or Pr′ = Th(Z, +, 1), then I( ) = D( ), and this set is a closed ideal. Any closed ideal J can be represented as I( ) = D( ) for some recursively saturated model of Pr′. For sets J of power at most ℵ1, Pr′ can be replaced by P. Assuming CH, all closed ideals have power at most ℵ1, but if CH fails, there are closed ideals of power greater than ℵ1, and it is not known whether these can be represented as I( ) = D( ) for a recursively saturated model of P. In the present paper, it will first be shown that information about representation of closed ideals provides new information about an old problem of MacDowell and Specker [6] and extends an old result of Scott [8] in a natural way. It will also be shown that the representation results from [2] answer a problem of Friedman [1]. This part of the paper is aimed at convincing the reader that representation problems are worth investigating.

This paper is concerned with algebraic independence in structures that are relatively simple for their size. It is shown that for κ a limit cardinal, if a structure of power at least κ is ∞ω-equivalent to a structure of power less than κ, then must contain an infinite set of algebraically independent elements. The same method of proof yields the fact that if σ is an Lω1ω-sentence (not necessarily complete) and σ has a model of power ℵω then some model of σ contains an infinite algebraically independent set. All structures are assumed to be of countable similarity type. Letters , etc. will be used to denote either a structure or the universe of the structure. If X ⊆ , the algebraic closure of X (in ), denoted by Cl(X), is the union of all finite sets that are weakly definable (in ) by Lωω-formulas with parameters from X. A set S is algebraically independent if for each a in S, a ∉ Cl(S – {a}). An algebraically independent set is sometimes called a “free” set (in [3] and [4], for example). It is known (see [5]) that any structure of power ℵn must have a set of n algebraically independent elements, and there are structures of power ℵn with no independent set of size n + 1. In power ℵω every structure will have arbitrarily large finite algebraically independent sets. However, it is consistent with ZFC that some models of power ℵω do not have any infinite algebraically independent set. Devlin [4] showed that if V = L, then for any cardinal κ, if every structure of power κ has an infinite algebraically independent set, then κ has a certain large cardinal property that ℵω can never possess.

Let L be a countable language including the unary relation symbol U. Let and be L-structures such that is a proper elementary U-extension of ; i.e., , and . Under what conditions will have a proper elementary U-extension? In [2], it was shown that this is not always the case, even if and are countable. However, the examples given are completely artificial, and it still seems that in most cases will have a proper elementary U-extension. Lascar asked whether will necessarily have a proper elementary U-extension whenever it contains an infinite set of indiscernibles over . This paper gives a counterexample for Lascar's question. The example is produced by modifying one of the examples in [2], using an idea of Marcus [5]. Models containing an infinite set of indiscernibles can often be “stretched” to produce larger models that share some desired nonelementary property with the original [1], [6]. However, the mere presence of indiscernibles in a model does not guarantee that it can be used in this way. If the model is not completely determined by the indiscernibles, the nonelementary property may not carry over to larger models. An example of this is given in [3]. The example for Lascar's question is further evidence that models with indiscernibles need not be “elastic”.

This paper gives some simple existence results on prime and atomic models over sets. It also contains an example in which there is no prime model over a certain set even though there is an atomic model over the set. The existence results are “local” in that they deal with just one set rather than all sets contained in models of some theory. For contrast, see the “global” results in [6] or [7, p. 200]. Throughout the paper, L is a countable language, and T is a complete L-theory with infinite models. There is a “large” model of T that contains the set X and any other sets and models to be used in a particular construction of a prime or atomic model over X. A model is said to be prime over X if and every elementary monomorphism on X can be extended to an elementary embedding on all of . This notion is used in a variety of ways in model theory. It aids in distinguishing between models that are not isomorphic, as in Vaught [10]. It also aids in showing that certain models are isomorphic, as in Baldwin and Lachlan [1].

Here an example will be given of a complete Lω1ω-sentence with a model of power ℵ1 but with no model of higher power. The continuum hypothesis is not assumed. The question of whether such an example exists was brought to the author's attention by Professor M. Makkai. An Lω1ω-sentence is said to be complete if its models all satisfy the same Lω1ω-sentences, or, equivalently, if all of the countable L-structures satisfying the sentence are isomorphic. Scott [5] showed that any countable L -structure (where L is countable) must satisfy a complete Lω1ω-sentence. Such a sentence is called a Scott sentence for the structure. An uncountable L-structure need not satisfy any complete Lω1ω -sentence. A complete Lω1ω-sentence σ is said to characterize the infinite cardinal k if σ has a model of power k but not of any higher power. The set of cardinals characterized by complete Lω1ω -sentences will be denoted by CC. By a result of Lopez-Escobar [3], if k ∈ CC, k <⊐ω1. Assuming GCH (so that ⊐α = ℵα, ), Malitz [4] showed that CC = {ℵα: α < ω1}. Without assuming GCH, Baumgartner [1] showed that ⊐α ∈ CC for all α < ω1. Without GCH, it is unknown whether ℵn ∈ CC for n ≥ 2. Now it will be shown that ℵ1, ∈ CC.

Let L be an elementary first order language. Let be an L-structure, and let φ be an L-formula with free variables u1, …, un, and υ. A Skolem function for φ on is an n-ary operation f on such that for all . If is an elementary substructure of , then an n-ary operation f on is said to preserve the elementary embedding of into if f(x)∈ for all x ∈ n, and ( , f ∣ n) ≺ ( , f). Keisler asked the following question: Problem 1. If and are L-structures such that ≺ , and if φ (u, υ) is an L-formula (with appropriate free variables), must there be a Skolem function for φ on which preserves the elementary embedding? Payne [6] gave a counterexample in which the language L is uncountable. In [3], [5], the author announced the existence of an example in which L is countable but the structures and are uncountable. The construction of the example will be given in this paper. Keisler's problem is still open in case both the language and the structures are required to be countable. Positive results for some special cases are given in [4]. The following variant of Keisler's question was brought to the author's attention by Peter Winkler: Problem 2. If L is a countable language, a countable L-structure, and φ(u, υ) an L-formula, must there be a Skolem function f for φ on such that for every countable elementary extension of , there is an extension of f which preserves the elementary embedding of into ?

In [7] it is shown that if Σ is a type omitted in the structure = ω, +, ·, < and complete with respect to Th( ) then Σ is omitted in models of Th( ) of all infinite powers. The proof given there extends readily to other models of P. In this paper the result is extended to models of ZFC. For pre-tidy models of ZFC, the proof is a straightforward combination of the methods in [7] and in Keisler and Morley ([9], [6]). For other models, the proof involves forcing. In particular, it uses Solovay and Cohen's original forcing proof that GB is a conservative extension of ZFC (see [2, p. 105] and [5, p. 77]). The method of proof used for pre-tidy models of set theory can be used to obtain an alternate proof of the result for This new proof yields more information. First of all, a condition is obtained which resembles the hypothesis of the “Omitting Types” theorem, and which is sufficient for a theory T to have a model omitting a type Σ and containing an infinite set of indiscernibles. The proof that this condition is sufficient is essentially contained in Morley's proof [9] that the Hanf number for omitting types is so the condition will be called Morley's condition. If T is a pre-tidy theory, Morley's condition guarantees that T will have models omitting Σ in all infinite powers.

The main result of this paper is the fact that for T a complete extension of either P or ZF + V = L, with no new symbols, there is a type which is omitted in a model of T of power ℵ1 but which is realized in all models of higher power. Therefore, the “Hanf number” for omitting types over T is greater than ℵ2. The present section contains some basic definitions and a discussion of related results. §1 discusses generic relations on models of P and ZFC, and describes a special forcing technique due to Addison. In §2, this technique is used to obtain the main result. The use of forcing is not essential. Those who do not wish to read a forcing argument may skip §1. At the point in §2 where forcing is used, a proof without forcing is sketched. This proof was obtained only after a careful study of the forcing argument. The forcing argument is given because it is simple, and because the technique of forcing makes intuitively clear why this and other similar results should be true. All theories in the paper are assumed to be countable and to have infinite models. Let T be a complete theory in a language L. The Hanf number for omitting types over T, denoted by H(T), is the first infinite cardinal κ such that for all L-types, Σ, if T has models omitting Σ in all infinite powers less than κ, then T has models omitting Σ in all infinite powers.

In [4] it is shown that if the structure omits a type Σ, and Σ is complete with respect to Th( ), then there is a proper elementary extension of which omits Σ. This result is extended in the present paper. It is shown that Th( ) has models omitting Σ in all infinite powers. A type is a countable set of formulas with just the variable ν occurring free. A structure is said to omit the type Σ if no element of satisfies all of the formulas of Σ. A type Σ, in the same language as a theory T, is said to be complete with respect to T if (1) T ∪ Σ is consistent, and (2) for every formula φ(ν) of the language of T (with just ν free), either φ or ¬φ is in Σ. The proof of the result of this paper resembles Morley's proof [5] that the Hanf number for omitting types is . It is shown that there is a model of Th( ) which omits Σ and contains an infinite set of indiscernibles. Where Morley used the Erdös-Rado generalization of Ramsey's theorem, a definable version of the ordinary Ramsey's theorem is used here. The “omitting types” version of the ω-completeness theorem ([1], [3], [6]) is used, as it was in Morley's proof and in [4]. In [4], satisfaction of the hypotheses of the ω-completeness theorem followed from the fact that, in , any infinite, definable set can be split into two infinite, definable sets.

In this paper it is shown that, for any complete type Σ omitted in the structure , or in any expansion of having only countably many relations and operations, there is a proper elementary extension of (or of ) which omits Σ. This result (which was announced in [2]) is used to answer a question of Malitz on complete -sentences. The result holds also for countable families of types. A type is a countable set of formulas with just the variable υ free. A structure is said to omit a type Σ if no element of satisfies all of the formulas of Σ. For example, omits the type Σω = {υ ≠ n: n ∈ ω}, since n fails to satisfy υ ≠ n. (Here n is the constant symbol standing for n.) A type Σ is said to be complete with respect to a theory T if the set of sentences T ∪ Σ(e) generates a complete theory, where Σ(e) is the result of replacing υ by the new constant e in all of the formulas of Σ. The type Σω is clearly not complete with respect to Th( ). (For any structure Th( ), Th( ) is the set of all sentences true in .)

In this paper, Cohen's forcing technique is applied to some problems in model theory. Forcing has been used as a model-theoretic technique by several people, in particular, by A. Robinson in a series of papers [1], [10], [11]. Here forcing will be used to expand a family of structures in such a way that weak second-order embeddings are preserved. The forcing situation resembles that in Solovay's proof that for any theorem φ of GB (Godel-Bernays set theory with a strong form of the axiom of choice), if φ does not mention classes, then it is already a theorem of ZFC. (See [3, p. 105] and [2, p. 77].) The first application of forcing here is to the problem (posed by Keisler) of when is it possible to add a Skolem function to a pair of structures, one of which is an elementary substructure of the other, in such a way that the elementary embedding is preserved. It is not always possible to find such a Skolem function. Payne [9] found an example involving countable structures with uncountably many relations. The author [4], [6] found an example involving uncountable structures with only two relations. The problem remains open in case the structures are required both to be countable and to have countable type. Forcing is used to obtain a positive result under some special conditions.