The purpose of this work is to formulate a general theory of forcing with classes and to solve some of the consistency and independence problems for the impredicative theory of classes, that is, the set theory that uses the full schema of class construction, including formulas with quantification over proper classes. This theory is in principle due to A. Morse [9]. The version I am using is based on axioms by A. Tarski and is essentially the same as that presented in [6, pp. 250–281] and [10, pp. 2–11]. For a detailed exposition the reader is referred there. This theory will be referred to as .

The reflection principle (see [8]), valid for other forms of set theory, is not provable in . Some form of the reflection principle is essential for the proofs in the original version of forcing introduced by Cohen [2] and the version introduced by Mostowski [10]. The same seems to be true for the Boolean valued models methods due to Scott and Solovay [12]. The only suitable form of forcing for found in the literature is the version that appears in Shoenfield [14]. I believe Vopěnka's methods [15] would also be applicable. The definition of forcing given in the present paper is basically derived from Shoenfield's definition. Shoenfield, however, worked in Zermelo-Fraenkel set theory.

I do not know of any proof of the consistency of the continuum hypothesis with assuming only that is consistent. However, if one assumes the existence of an inaccessible cardinal, it is easy to extend Gödel's consistency proof [4] of the axiom of constructibility to .

A hierarchy of systems of quantifier-free elementary recursive arithmetics, based on the Grzegorczyk hierarchy of functions, was set up in [2] and some meta-mathematical properties of these systems were developed. The Grzegorczyk hierarchy has been extended recently, mainly by Löb and Wainer [5], and our metamathematical developments may be similarly extended; the αth member of this hierarchy of formal systems will be denoted -arithmetic throughout. The main result in [2] was: For α > 1, -arithmetic can be proved consistent in -arithmetic. In this paper we shall continue this work in particular, beginning with this consistency result, we shall find (for α > 1) the order type of the weakest simple transfinite induction scheme which is independent of -arithmetic, thus giving a ‘measure of the complexity of derivations’ of these systems. For example we shall show that transfinite induction on a sequence of type ωω is a nonderivable rule of primitive recursive arithmetic (-arithmetic). This particular result was proved by Guard in [4] by a specialisation of a version of Gentzen's proof that ∈0-transfinite induction is independent of some standard formal systems of arithmetic with quantifiers. These methods can be adapted to our hierarchy but require what we might term an ‘ωω-jump’—that is if β is the largest ordinal for which transfinite induction up to β is derivable in the system in question then a scheme of transfinite induction up to β·ωω is independent. The proof presented in this paper requires only an ‘ω-jump’ and allows more precise results to be obtained for the systems in the extended Grzegorczyk hierarchy; it is also more direct and less proof-theoretic in character. We show that the consistency of -arithmetic (for α > 1) can be proved in a system obtained by adding to -arithmetic a transfinite induction scheme up to ω2, and that this induction scheme can be adapted to obtain the required result by increasing the ordinal and simultaneously decreasing the complexity of the functions involved in the induction scheme (detailed definitions of these concepts will be given in the next section).

It will be proven that a set of sentences of infinitary logic is satisfiable iff it is proof theoretically consistent. Since this theorem is known to be false, it must be quickly added that an extended notion of model is being used; truth values may be taken from an arbitrary complete Boolean algebra. We shall give a Henkin style proof of this result which generalizes easily to Boolean valued sets of sentences.

For each infinite candinal number κ the language Lκ is built up from a set of relation symbols together with a constant symbol cα and a variable υα for each α in κ. It contains atomic formulas and is closed under the following rules:

(1) If Γ is a set of formulas of power < κ ∧ Γ is a set of formulas.

(2) If φ is a formula, ¬ φ is also.

(3) If φ is a formula and A Is a subset of κ of power < κ then Aφ is a formula.

∧Γ is meant to be the conjunction of all the formulas in Γ, while Aφ is the universal quantification of all the variables υα for α in A. We let C denote the set of constant symbols in Lκ, the parameter κ must be discovered from the context.

A model is identified with its truth function. Thus a model is a function mapping the sentences of Lκ into a complete Boolean algebra which satisfies the following conditions:

Bachmann, in [2] shows how certain ordinals <Ω(Ω = Ω1 where Ωξ is the (1 + ξ)th infinite initial ordinal) may be described from below using suitable descriptions of ordinals <Ω2. The aim of this paper is to consider another approach to describing ordinal <Ω and compare it with the Bachmann method. Our approach will use functionals of transfinite type based on Ω.

The Bachmann method consists in denning a hierarchy of normal functions ϕδ: Ω → Ω (i.e. continuous and strictly increasing) for δ ≤ η0 < Ω2, starting with ϕ0(λ) = ω1 + λ. The definition of depends on a suitable description of the ordinals ≤ η0. This is obtained by defining a hierarchy 〈Fδ ∣ δ ≤ Ω2〉 of normal functions Fδ: Ω2 → Ω2 analogously to the definition of the initial segment 〈ϕδ ∣ δ ≤ Ω〉 of . The ordinal η0 is .

Note. Our description of Bachmann's hierarchies will differ slightly from those in Bachmann's paper. Let and denote the hierarchies in [2]. Then as Bachmann's normal functions are not defined at 0 we let for λ, δ < Ω2. Bachmann defines for 0 < λ < Ω2 but it seems more natural to omit this so that we let . The situation is analogous for and leads to the following definitions:

where n < ω and ξ is a limit number of cofinality Ω, and

Although there is no smallest ω model [6], Putnam and Gandy independently proved about 1963 that the class of ramified analytical sets, as defined by Cohen [3], form the smallest β model of analysis [2], [8]. This paper will consider other restricted classes of models, namely the βn models for integers n > 1 [10], and prove under appropriate assumptions the existence of minimum such models. In fact we shall construct the minimum βn model in a fashion similar to the procedure yielding the class of ramified analytical sets, but adding at each stage a segment of the sets in those already obtained (for n a fixed integer > 1).

Roughly speaking a βn model is an ω model absolute for n-set-quantifier assertions about its subsets of natural numbers. A β model is simply a β1 model. See Enderton and Friedman [4] for a further investigation of βn models.

The complexity of a computable function can be measured by considering the time or space required to compute its values. Particular notions of time and space arising from variants of Turing machines have been investigated by R. W. Ritchie [14], Hartmanis and Stearns [8], and Arbib and Blum [1], among others. General properties of such complexity measures have been characterized axiomatically by Rabin [12], Blum [2], Young [16], [17], and McCreight and Meyer [10].

In this paper the speed-up and super-speed-up theorems of Blum [2] are generalized to speed-up by arbitrary total effective operators. The significance of such theorems is that one cannot equate the complexity of a computable function with the running time of its fastest program, for the simple reason that there are computable functions which in a very strong sense have no fastest programs.

Let φi be the ith partial recursive function of one variable in a standard Gödel numbering of partial recursive functions. A family Φ0, Φ1, … of functions of one variable is called a Blum measure on computation providing

(1) domain (φi) = domain (Φi), and

(2) the predicate [Φi(x) = m] is recursive in i, x and m.

Typical interpretations of Φi(x) are the number of steps required by the ith Turing machine (in a standard enumeration of Turing machines) to converge on input x, the space or number of tape squares required by the ith Turing machine to converge on input x (with the convention that Φi(x) is undefined even if the machine fails to halt in a finite loop), and the length of the shortest derivation of the value of φi(x) from the ith set of recursive equations.

Let T be a theory in a first-order language L. Let L have a predicate ν0 ≺ ν1 such that in every model of T, the interpretation of ≺ is a linear ordering with infinite field. The order-type of this ordering will be called the order-type of the model .

Several recent theorems have the following form: if T has a model of order-type ξ then T has a model of order-type ζ (see [1]). We shall add one to the list. The new feature of our result is that the order-type ζ may be in a sense “opposite” to ξ. Silver's Theorem 2.24 of [3] is a corollary of Theorem 1 below.

Theorem 1. Let κ be a strong limit number (i.e. μ < κ implies 2μ < κ). Suppose λ < κ, and suppose that for every cardinal μ < κ, T has a model with where the order-type of contains no descending well-ordered sequences of length λ. Then for every cardinal μ ≥ the cardinality ∣L∣ of the language L, T has models and such that

(a) the field of is the union of ≤ ∣L∣ well-ordered (inversely well-ordered) parts;

(b) .

The proof is by Ehrenfeucht-Mostowski models; we presuppose [2].

In this paper we shall construct nonregular ultrafilters showing many of the model-theoretic properties of their regular counterparts. The crucial idea in these constructions is to replace the use of regularity by independent functions. We shall use the notation and terminology of [1], our fundamental concepts being defined as follows:

Definition 1.1. (1) A uniform ultrafilter D over a cardinal κ is regular if there is a family {Xα ∣ α < κ} so that every infinite intersection of these Xα's is empty.

(2) A filter F over a cardinal κ is ω1-saturated if there is no family of sets {aα ∣ α < ω1} so that for every α < β < ω1

For more on regular ultrafilters, see [2]. The germinal theorem on the subject of nonregular ultrafilters is the following well-known result of Jack Silver:

Theorem 1.2 [1, Theorem 1.39]. If F is an ω1-saturated κ-complete filter over κ, then any ultrafilter extending F is nonregular.

This result will be the cornerstone of our constructions. Of course, the existence of a filter of the above type depends on large cardinal axioms. For more on un ω1-saturated κ-complete filters, see [1].

First order properties of direct products and direct sums (weak direct products) of relational systems have been studied extensively. For example, in Feferman and Vaught [3] an effective procedure is given for reducing such properties of the product to properties of the factors, and thus in particular elementary equivalence is preserved. We consider here two-sorted relational systems, with the direct product and sum operations keeping one of the sorts stationary. (See Feferman [4] for a similar definition of extensions.)

These considerations are motivated by the example of direct products and sums of modules [8], [9]. In [9] examples are given to show that the direct product of two modules (even having only a finite number of module elements) does not preserve two-sorted (even universal) equivalence for any finite or infinitary language Lκ, λ. So we restrict attention here to direct powers and multiples (many copies of one structure). Also in [9] it is shown (for modules, but the proofs generalize immediately to two-sorted structures with a finite number of relations) that the direct multiple operation preserves first order ∀E-equivalence and the direct power operation preserves first order ∀-equivalence. We show here that these results for general two-sorted structures in a finite first order language are, in a sense, best-possible. Examples are given to show that does not imply , and that does not imply .

Omitting-types theorems have been useful in model theory to construct models with special characteristics. For instance, one method of proving the ω-completeness theorem of Henkin [10] and Orey [20] involves constructing a model that omits the type {x ≠ 0, x ≠ 1, x ≠ 2,···} (i.e., {x ≠ 0, x ≠ 1, x ≠ 2,···} is not satisfiable in the model). Our purpose in this paper is to illustrate uses of omitting-types theorems in recursion theory. The Gandy-Kreisel-Tait Theorem [7] is the most well-known example. This theorem characterizes the class of hyperarithmetical sets as the intersection of all ω-models of analysis (the so-called hard core of analysis). The usual way to prove that a nonhyperarithmetical set does not belong to the hard core is to construct an ω-model of analysis that omits the type representing the set (Application 1). We will find basis results for and s — sets that are stronger than results previously known (Applications 2 and 3). The question of how far the natural hierarchy of hyperjumps extends was first settled by a forcing argument (Sacks) and subsequently by a compactness argument (Kripke, Richter). Another problem solved by a forcing argument (Sacks) and then by a compactness argument (Friedman-Jensen) was the characterization of the countable admissible ordinals as the relativized ω1's. Using omitting-types technique, we will supply a third kind of proof of these results (Applications 4 and 5). S. Simpson made a significant contribution in simplifying the proof of the latter result, with the interesting side effect that Friedman's result on ordinals in models of set theory is immediate (Application 6). One approach to abstract recursiveness and hyperarithmeticity on a countable set is to tenuously identify the set with the natural numbers. This approach is equivalent to other approaches to abstract recursion (Application 7). This last result may also be proved by a forcing method.

In this paper we show how nonrecursive combinatorial functions can be used to obtain another proof of the compactness Theorem 3.1 of [6]. Our method is closely related to the argument used to show that a countable ultraproduct is ℵ1-saturated. What we do is to diagonalize in the most obvious way. The main difficulty with this approach is that the resulting diagonal function need not be recursive. Just to give an idea of how bad nonrecursive combinatorial functions are, we mention that by using frames (cf. [5]) they do not extend to Λ, and that by using normal combinatorial operators (cf. [4]) they do extend to Λ, map Λ into Λ, but in general, composition of such functions does not commute with their extension. We take care of this problem by constructing a large class of isols with respect to which the diagonal function is well behaved. The advantage of our method is that it provides the investigator with a natural and intuitive way of constructing counterexamples in his own research area.

Let A2 be the axiomatic system of second order arithmetic as described in [2]. One of the models of A2 is the “principal model” Mpr consisting of all integers and all sets of integers. Obviously there exist many denumerable ω-models elementarily equivalent to Mpr and we shall deal in this paper with some questions pertaining to this family which we denote by .

In §1 we define a rather natural relation ε between two denumerable families of sets of integers. From the upward Skolem-Löwenheim theorem it follows easily that there exists a family ordered by ε in the type ω1, but it is not immediately obvious whether there exist a subfamily of not well-ordered by ε. In the present paper we construct such a family of type η. ω1 where η is the order type of rationals and indicate some applications to hyperdegrees.

We adopt the terminology and notation of [2], with the only change that we adjoin to the language of A2 the constants ν0, ν1, … for the consecutive numerals 0, 1, 2, … and axioms which characterise them:

Also we modify the axioms of A2 given in [2] by prefixing them by general quantifiers bounded either to S or to N.

If two nonstandard models of complete arithmetic are elementarily embedded in a third, then their intersection may be considerably smaller than either of them; indeed, the intersection may be only the standard model. For example, if D and E are nonprincipal ultrafilters on ω, then the nonstandard models D-prod and E-prod (where is the standard model) have canonical elementary embeddings into D-prod (E-prod , and the intersection of their images is easily seen to be the (canonical image of the) standard model. In this paper, we shall prove that, under certain conditions, this phenomenon will not occur. Our main result (Theorem 3) is that the intersection of countably many pairwise cofinal models is itself cofinal with these models, provided that at least one of them is generated by a single element. (Precise definitions will be given below.)

The theorems in this paper were first formulated in terms of ultrafilters, then rephrased (using the methods of Chapter III of [1]) as statements about ultra-powers of , and finally generalized to their present form. Since the theorems and their proofs are now entirely model-theoretic, they are presented here separately from the study of ultrafilters in which they originated. That study, including applications of the present results, will appear in [2].

Let L be the first-order language whose n-place relation symbols are all the relations R ⊆; ωn and whose n-place function symbols are all the functions f: ωn → ω. Let be the standard model for L; its universe is ω and every nonlogical symbol of L denotes itself. Let be an elementary extension of . The relation (or function) denoted by R (or f) in will be called *R (or *f).

If T is a complete first-order totally transcendental theory then over every T-structure A there is a prime model unique up to isomorphism over A. Moreover M is a prime model over A iff: (1) every finite sequence from M realizes an isolated type over A, and (2) there is no uncountable indiscernible set over A in M.

The existence of prime models was proved by Morley [3] and their uniqueness for countable A by Vaught [9]. Sacks asked (see Chang and Keisler [1, question 25]) whether the prime model is unique. After proving this I heard Ressayre had proved that every two strictly prime models over any T-structure A are isomorphic, by a strikingly simple proof. From this follows

If T is totally transcendental, M a strictly prime model over A then every elementary permutation of A can be extended to an automorphism of M. (The existence of M follows by [3].)

By our results this holds for any prime model. On the other hand Ressayre's result applies to more theories. For more information see [6, §0A]. A conclusion of our theorem is the uniqueness of the prime differentially closed field over a differential field. See Blum [8] for the total transcendency of the theory of differentially closed fields.

We can note that the prime model M over A is minimal over A iff in M there is no indiscernible set over A (which is infinite).

In this paper we show (cf. Theorem 22) that in a language L* with equality, whose relation symbols denote arbitrary relations over ω* (=rational integers) and whose function symbols denote (= ∃∀ definable in the arithmetic hierarchy) functions over ω*, (i) a positive sentence is true in Λ* (= isolic integers) iff some Horn reduct is true in ω* with Skolem functions. We also show (cf Theorem 20) that (ii) a universally quantified sentence is true in Λ* iff some Horn reduct is true in Λ*.The latter result is nontrivial because our relations are arbitrary and our functions are In order to obtain (i) it was necessary to generalize the frame extensions of [7]. This is done in §2. Our extension procedure agrees with that of [7] for recursive relations (cf. Theorem 12), and is certainly more general for − relations. What happens in the case is still open. In §3 we develop the basic properties of our extension so that in §4 we can prove a metatheorem (cf. Theorems 8 and 10) about Λ (=isols), in a language L with equality whose relation symbols denote arbitrary relations over ω (=nonnegative integers) and whose function symbols denote almost R↑ combinatorial functions. In Theorem 11 this is generalized to infinitary universal sentences. In §5 generic isols are introduced. These are used (cf. Theorems 16–19) to generalize and simplify the “fundamental lemma” of [8]. The basic induction is patterned after Lemma 4.1 of [8], but is stronger in that any sufficiently generic assignment attainable from a frame yields Skolem functions. Finally in §6 these results are applied to Λ*, yielding the titled result (i) of our paper. Immediately following Theorem 15 there is a discussion which attempts to justify the way we extend relations to Λ.

In [10, §0, E), 5)] Shelah states using the proofs of 7.9 and 6.9 in [9] it is possible to prove that if a countable first-order theory T is ℵ0-stable (totally transcendental) and not ℵ1-categorical, then it has at least ∣1 + α∣ models of power ℵα.

In this note we will give a new proof of this theorem using the work of Baldwin and Lachlan [1]. Our original proof used the generalized continuum hypothesis (GCH). We are indebted to G. E. Sacks for suggesting that the notions of ℵ0-stability and ℵ1-categoricity are absolute, and that consequently our use of GCH was eliminable [8]. Routine results from model theory may be found, e.g. in [2].

Proof (with GCH). In the proof of Theorem 3 of [1] Baldwin and Lachlin show of power ℵα such that there is a countable definable subset in . Let B0 be such a subset. Say . We will give by transfinite induction an elementary chain of models of T of power ℵα such that B[i1 … in] has power ℵβ and such that every infinite definable subset of has power ≥ℵβ. This clearly suffices.

We investigate extensions of Heyting's predicate calculus (HPC). We relate geometric properties of the trees of Kripke models (see [2]) with schemas of HPC and thus obtain completeness theorems for several intermediate logics defined by schemas. Our main results are:

(a) ∼(∀x ∼ ∼ϕ(x) Λ ∼∀xϕ(x)) is characterized by all Kripke models with trees T with the property that every point is below an endpoint. (From this we shall deduce Glivenko type theorems for this extension.)

(b) The fragment of HPC without ∨ and ∃ is complete for all Kripke models with constant domains.

We assume familiarity with Kripke [2]. Our notation is different from his since we want to stress properties of the trees. A Kripke model will be denoted by (Aα, ≤ 0), α ∈ T where (T, ≤, 0) is the tree with the least member 0 ∈ T and Aα, α ∈ T, is the model standing at the node α. The truth value at α of a formula ϕ(a1 … an) under the indicated assignment at α is denoted by [ϕ(a1 … an)]α.

A Kripke model is said to be of constant domains if all the models Aα, α ∈ T, have the same domain.

Let {Wi} be a standard enumeration of all recursively enumerable (r.e.) sets, and for any class A of r.e. sets, let θA denote the index set of A = {n ∣ Wn ∈ A}. (Clearly, .) In [1], the index sets of nonempty finite classes of finite sets were classified under one-one reducibility into an increasing sequence {Ym}, 0 ≤ m < ∞. In this paper we examine further properties of this sequence within the partial ordering of one-one degrees of index sets. The main results are as follows: (1) For each m, Ym < Ym + 1 and < Ym + 1; (2) Ym is incomparable to ; (3) Ym + 1 and ; are immediate successors (among index sets) of Ym and m; (4) the pair (Ym + 1, ) is a “least upper bound” for the pair (Ym, ) in the sense that any successor of both Ym and is ≥ Ym + 1or; (5) the pair (Ym, ) is a “greatest lower bound” for the pair (Ym + 1, ) in the sense that any predecessor of both Ym + 1 and is ≤ Ym or . Since and all Ym are in the bounded truth-table degree of K, this yields some local information about the one-one degrees of index sets which are “at the bottom” in the one-one ordering of index sets.

Although we believe the results reported below to have direct philosophical import, we shall for the most part confine our remarks to the realm of mathematics. The reader is referred to [4] for a philosophically oriented discussion, comprehensible to mathematicians, of tense logic.

The “minimal” tense logic T0 is the system having connectives ∼, →, F (“at some future time”), and P (“at some past time”); the following axioms:

(where G and H abbreviate ∼F∼ and ∼P∼ respectively); and the following rules:

(8) from α and α → β, infer β,

(9) from α, infer any substitution instance of α,

(10) from α, infer Gα,

(11) from α, infer Hα.

A tense logic is a system T whose language is that of T0 and whose axioms and rules include (1)–(11). The axioms and rules of T other than (1)–(11) are called proper axioms and rules.

We shall investigate three systems of semantics for tense logics, i.e. three notions of structure and three relations ⊧ which stand between structures and formulas. One reads ⊧ α as “α is valid in .” A structure is a model of a tense logic T if every formula provable in T is valid in . A semantics is adequate for T if the set of models of T in the semantics is characteristic for T, i.e. if whenever T ∀ α then there is a model of T in the semantics such that ∀ α. Two structures and , possibly from different semantics, are called equivalent ( ∼ ) if exactly the same formulas are valid in as in .