We separate many of the basic fragments of classical logic which are used in reverse constructive mathematics. A group of related Kripke and topological models is used to show that various fragments of the Weak Law of the Excluded Middle, the Limited Principle of Omniscience, and Markov’s Principle, including Weak Markov’s Principle, do not imply each other.

A new method for blowing up the power of a singular cardinal is presented. It allows to blow up the power of a singular in the core model cardinal of uncountable cofinality. The method makes use of overlapping extenders.

We prove that the property “P doesn't make the old reals Lebesgue null” is preserved under countable support iterations of proper forcings, under the additional assumption that the forcings are nep (a generalization of Suslin proper) in an absolute way. We also give some results for general Suslin ccc ideals.

The notations and the definitions used here are identical to those in [B.L.]. We also assume that the reader is familiar with the properties of the Rudin-Keisler order on types [L.].

Let T denote an ℵ0-stable countable theory. The main result in [B.L.] is: All countable models of T are almost homogeneous if and only if T satisfies

(*) For all models M of T and all ā ∈ M, if p ∈ S1(ā) is strongly regular multidimensional, then Dim(p; M) ≥ ℵ0.

In this paper we investigate the case of a theory T which does not satisfy condition (*) and, under certain additional assumptions, we construct nonisomorphic and non-almost-homogeneous countable models. The same type of construction has been used before to show that if T is multidimensional, then for all α ≥ 1, T has at least nonisomorphic models of cardinality ℵα [La.], [Sh.]. As a corollary of our main theorem (Theorem 6) and of the previous result in [B.L.], we prove Vaught's Conjecture (and, in fact, Martin's Strong Conjecture) for theories T with αT finite.

Although we are here interested in countable models, we can also note that our construction proves that theories satisfying the assumptions in Theorem 6 have at least nonisomorphic models of cardinality ℵα.

In well-known papers ([A-K1], [A-K2], and [E]) J. Ax, S. Kochen, and J. Ershov prove a transfer theorem for henselian valued fields. Here we prove an analogue for henselian valued and ordered fields. The orders for which this result apply are the usual orders and also the higher level orders introduced by E. Becker in [Bl] and [B2]. With certain restrictions, two henselian valued and ordered fields are elementarily equivalent if and only if their value groups (with a little bit more structure) and their residually ordered residue fields (a henselian valued and ordered field induces in a natural way an order in its residue field) are elementarily equivalent. Similar results are proved for elementary embeddings and ∀-extensions (extensions where the structure is existentially closed).

We use model-theoretic methods described in [3] to obtain ordinal analyses of a number of theories of first- and second-order arithmetic, whose proof-theoretic ordinals are less than or equal to Γ0.

We note here, in answer to a question of Poizat, that the Morley and Lascar ranks need not coincide in differentially closed fields. We will approach this through the (perhaps) more fundamental issue of the variation of Morley rank in families. We will be interested here only in sets of finite Morley rank. Section 1 consists of some general lemmas relating the above issues. Section 2 points out a family of sets of finite Morley rank, whose Morley rank exhibits discontinuous upward jumps. To make the base of the family itself have finite Morley rank, we use a theorem of Buium.

This paper contains an example of a decidable theory which has

1) only a countable number of countable models (up to isomorphism);

2) a decidable saturated model; and

3) a countable homogeneous model that is not decidable.

By the results in [1] and [2], this can happen if and only if the set of types realized by the homogeneous model (the type spectrum of the model) is not .

If Γ and Σ are types of a theory T, define Γ ◁ Σ to mean that any model of T realizing Γ must realize Σ. In [3] a decidable theory is constructed that has only countably many countable models, only recursive types, but whose countable saturated model is not decidable. This is easy to do if the restriction on the number of countable models is lifted; the difficulty arises because the set of types must be recursively complex, and yet sufficiently related to control the number of countable models. In [3] the desired theory T is such that

is a linear order with order type ω*. Also, the set of complete types of T is not . The last feature ensures that the countable saturated model is not decidable; the first feature allows the number of countable models to be controlled.

In part I of the present paper axiom schemes and rules of inference were defined for m-valued functional calculi of first order with s(m > s > 1) designated truth-values. A proof of plausibility was given, and it was shown that it is not difficult to extend to m-valued functional calculi of first order certain concepts that are closely analogous to the ordinary 2-valued notions of “consistency with respect to an operator” and “absolute consistency.” The purpose of the present paper is to show that the concept of “deductive completeness” may be extended to m-valued functional calculi of first order. For this purpose we define “analytic formula” for the m-valued case and show that if a formula is analytic, then it is provable in our formalization of m-valued functional calculi of first order.

In proving deductive completeness for the m-valued case, it is possible to use a method which is analogous to that used by Gödel in establishing the completeness of 2-valued functional calculi of first order. However, in the present paper we will indicate only very briefly how the Gödel procedure may be extended to the m-valued case. Our chief concern will be the problem of extending to our formalization of m-valued functional calculi the more elegant proof of deductive completeness for the 2-valued case which has recently been developed by Leon Henkin.

We prove that Standardization fails in every nontrivial universe definable in the nonstandard set theory BST, and that a natural characterization of the standard universe is both consistent with and independent of BST. As a consequence we obtain a formulation of nonstandard class theory in the ∈-language.

We prove the elimination of Magidor-Malitz quantifiers for R-modules relative to certain -core sentences and positive primitive formulas. For complete extensions of the elementary theory of R-modules it follows that all Ramsey quantifiers (ℵ0-interpretation) are eliminable. By a result of Baldwin and Kueker [1] this implies that there is no R-module having the finite cover property.

In a previous paper [9] a demonstrably consistent type-free system of combinatory logic CΔ was presented. This system contained a moderately strong exten-sionality principle, the usual equations of combinatory logic, Boolean propositional connectives, unrestricted quantifiers, an unrestricted abstraction principle (“comprehension axiom”), and a limited principle of excluded middle. It also contained elementary arithmetic in its entirety, avoiding arithmetical Gödel incompleteness [14] by being ω-complete. CΔ probably can be shown to contain certain important parts of mathematical analysis, such as the theory of continuous functions, which are contained by the somewhat similar (but nonextensional) system K′ [6], [7]. A stronger system CΓ, having these same properties and more, will be formulated in the present paper. Elsewhere [13] it will be shown that the system Q of my book, Elements of combinatory logic (Yale University Press, New Haven and London, 1974), referred to below as ECL, is essentially a subsystem of CΓ, so that the consistency of CΓ guarantees that of Q.

CΓ will be stronger than both CΔ and Q in having an operator ‘ɿ’ (inverted Greek iota) which denotes the inverse of equality in the sense that ‘ɿ(= a) = a’ is a theorem of the system. Thus ‘ɿ’ denotes a function or operator which operates on a unit class to give the only member of that class. (Here ‘=a’ denotes the unit class whose only member is denoted by ‘a’.) If uninverted Greek iota, ‘ι’, is used as an abbreviation for ‘=’, the above equation could be written as ‘ɿ(ιa) = a’. Here ‘ιa’ is analogous to Bertrand Russell's well-known notation ‘ιa’, denoting a unit class. This same operator ‘ɿ’ makes it possible to transform one-many or many-one relations into the corresponding functions or operators, a kind of transformation not usually available in systems of combinatory logic [1], [2]. It also provides a method for restricting functions by restricting the corresponding relations.

Among various unnatural and technically burdensome effects of the theory of types, one is the unstable character of meaningfulness: a mere permutation of variables is capable of reducing a significant context to meaninglessness. Another effect, and perhaps the most conspicuous, is the systematic reduplication to which the logical constants are subjected; the calculi of classes and relations and even arithmetic lose their unity and generality, and are reproduced anew within each type. The elaborate compensatory manoeuvres which are thus made necessary are familiar to all readers of Principle, mathematica.

In a recent publication, to be cited henceforth as New foundations, I proposed an alternative course which avoids these consequences, but which would seem to offer less assurance of consistency. My efforts to derive a contradiction have delivered none, but they have lent a strange aspect to Cantor's proof that every class has more subclasses than members. This result is the topic of the present paper.

Preparatory to sketching the new system, which I shall call S′, I shall sketch a very similar but contradictory system S. By way of primitives S involves just membership, universal quantification, and alternative denial (Sheffer's stroke function), together with general variables “x”, “y”, …; the adequacy of this equipment as a basis for mathematical logic is made evident by Wiener's and Kuratowski's discovery of methods of constructing relation theory in terms of classes. Thus the formulae or statements and statement forms of S are describable recursively as follows: if a variable is put in each blank of “(ϵ)”, the result is a formula; if a variable enclosed in parentheses is prefixed to a formula, the result is a formula; and if a formula is put in each blank of “(∣)”, the result is a formula. The theorems of S are determined by a postulate and five rules, called P1 and R1-5 in New foundations. P1 and R1-3 specify various formulae as initial theorems, and R4-5 specify inferential connections for deriving further theorems. R1–2 and R4–5 are so fashioned as to provide the “theory of deduction”: they provide as theorems all those formulae which are valid by virtue merely of their structure in terms of alternative denial and quantification.

We show that the 0-1 law does not hold for the class (∀∃∀ without = ) by finding a sentence in this class which almost surely expresses parity. We also show that every recursive real in the unit interval is the asymptotic probability of a sentence in this class. This expands a result by Lidia Tendera, who in 1994 proved that every rational number in the unit interval is the asymptotic probability of a sentence in the class ∀∃∀ with equality.

We show that there is a complete, consistent Borel theory which has no “Borel model” in the following strong sense: There is no structure satisfying the theory for which the elements of the structure are equivalence classes under some Borel equivalence relation and the interpretations of the relations and function symbols are uniformly Borel.

We also investigate Borel isomorphisms between Borel structures.

This paper continues the investigation of inconsistent arithmetical structures. In §2 the basic notion of a model with identity is defined, and results needed from elsewhere are cited. In §3 several nonisomorphic inconsistent models with identity which extend the (=, <) theory of the usual classical denumerable nonstandard model of arithmetic are exhibited. In §4 inconsistent nonstandard models of the classical theory of finite rings and fields modulo m, i.e. Zm, are briefly considered. In §5 two models modulo an infinite nonstandard number are considered. In the first, it is shown how to model inconsistently the arithmetic of the rationals with all names included, a strengthening of earlier results. In the second, all inconsistency is confined to the nonstandard integers, and the effects on Fermat's Last Theorem are considered. It is concluded that the prospects for a good inconsistent theory of fields may be limited.

We show that Sacks' and Shoenfield's analogs of jump inversion fail for both tt- and wtt-reducibilities in a strong way. In particular we show that there is a δ20 set B >tt ∅′ such that there is no c.e. set A with A′ ≡wttB. We also show that there is a Σ20 set C >tt ∅′ such that there is no δ20 set D with D′ ≡wttC.

We study cut-elimination in first-order classical logic. We construct a sequence of polynomial-length proofs having a non-elementary number of different cut-free normal forms. These normal forms are different in a strong sense: they not only represent different Herbrand-disjunctions but also differ in their prepositional structure.

This result illustrates that the constructive content of a proof in classical logic is not uniquely determined but rather depends on the chosen method for extracting it.