This paper presents a class of plausible definitions for validity of formulas in the infinitely-many-valued extension of the Łukasiewicz predicate calculus, and shows that all of them are equivalent. This extended system is discussed in some form in [3] and [4]; the questions discussed here are raised rather briefly in the latter.

We first describe the formal framework for the validity definition. The symbols to be used are the following: the connectives + and ‐, which are the strong disjunction B of [2] and negation respectively; the predicate variables Pi for i ∈ I, where I may be taken as the integers; the existential quantifiers E(J), where J⊆I, and I may be thought of as the index set on the individual variables, which however do not appear explicitly in this formulation.

For a ring with unit R, which need not be associative, denote the group of upper unitriangular 3 × 3 matrices over R by UT3(R). Let e1 = (1,0,0), e2 = (0,1,0), where (α, β, γ) denotes the matrix

Denote the expanded group (UT3(R), e1, e2) by (R). A. 1. Mal′cev [M] gave an algebraic characterization of the expanded groups of the form (R) as follows. Let h1, h2 be elements of a group H; then (H, h1, h2) is isomorphic to (R), for some R, if and only if

(i) H is 2-step nilpotent;

(ii) CH(hi) are abelian, i = 1,2;

(iii) CH(h1) ∩ CH(h2) = Z(H);

(iv) [CH(h1),h2] = [h1, CH(h2)] = Z(H);

(v) Z(H) is a direct summand in both CH(hi).

(In [M] condition (v) is a bit stronger; the version above is presented in [B2].)

A pair (h1, h2) of elements of a group H is said to be a base if (H, h1, h2) satisfies the conditions (i)–(iv). A. I. Mal′cev [M] found a uniform way of first order interpreting a ring Ring(H, h1, h2) in any group with a base (H, h1, h2); in particular, Ring((R)) ≃ R.

We prove that in many situations it is consistent with ZFC that part of the invariants involved in Cichoń's diagram are equal to κ while the others are equal to λ, where κ < λ are both arbitrary regular uncountable cardinals. We extend some of these results to the case when λ is singular. We also show that cf is consistent with ZFC.

We prove there are theories (stable or countable) for which over every A there is a prime model but it is not necessarily unique. We also give a simplified proof of the uniqueness theorem for countable stable theories.

In this paper, we prove a result that J. Y. Girard has conjectured for the last few years. That is,

This result is part of a study of ordinal notations. For the proof, we use some concepts introduced in Parts I, II and III of this article (see [GV1], [GV2] and [V]); in particular, in Part III (see [V]) we defined the category GAR of gardens (of type Ω), and we have shown that there is an isomorphism between GAR and the category DIL Ω of dilators which send Ω into Ω. To describe this isomorphism, we defined a functor SYN from GAR to DIL Ω and a functor DEC from DIL Ω to GAR which are inverse to each other. The functor SYN is constructed by induction on the height x of the garden Jx, iterating, at each step of cofinality Ω, the operation UN (unification of variables; see [G1, Chapter 3]). Here, we define the functor S⊿ from GAR to DIL Ω: this construction is very close to that of SYN, but we iterate, at each step of cofinality Ω, the operation ⊿ (identification of variables; see [G, Chapter 3]). Technically, the definition of S⊿ is easier and more natural than that of SYN (as ⊿ is easier than UN) but is not inversible (as ⊿); for example, if we consider the (regular) garden GεΩ+1 introduced in [GV2],S⊿(GεΩ+1) is exactly the dilator N = ⋃0<n<ωNn, with N1 = 2 + Id, and , but we are incapable of expressing SYN(GεΩ+1).

In Jaśkowski [1], some properties of the consequences of formulas of the sentential calculus with no more than two occurrences of any variable have been settled. Analogous problems may be raised concerning the consequences of equalities in which the same restriction relates to individual variables. In the present paper, problems of this kind are solved and more general theorems are proved about the logical consequences of some formulas, called here multiconditionals.

We consider two axioms of second-order arithmetic. These axioms assert, in two different ways, that infinite but narrow binary trees always have infinite paths. We show that both axioms are strictly weaker than Weak König's Lemma, and incomparable in strength to the dual statement (WWKL) that wide binary trees have paths.

We say that a set A of reals is recursive in a real y together with a set B of reals if one can imagine a computing machine with an ability to perform a countably infinite sequence of program steps in finite time and with oracles for B and y so that decides membership in A for any real x input to by way of an oracle for x. We write A ≤ yB. A precise definition of this notion of recursion was first considered in Kleene [9]. In the notation of that paper, A ≤yB if there is an integer e so that χA(x) = {e}(x y, χB, 2E). Here χA is the characteristic function of A. Thus Kleene would say that A is recursive in (y, B, 2E), where 2E is the existential integer quantifier.

Gandy [5] observes that the halting problem for infinitary machines such as , as in the case of Turing machines, gives rise to a jump operator for higher type recursion. Thus given a set B of reals, the superjump B′ of B is defined to be the set of all triples 〈e, x, y〉 such that the eth machine with oracles for y and B eventually halts when given input x. A set A is said to be semirecursive in y together with B if for some integer e, A is the cross section {x: 〈e, x, y 〉 ∈ B′}. In Kleene [9] it is demonstrated that a set A is semirecursive in y alone if and only if it is

It is possible to define a combinatorial closure on alternating bilinear maps with few relations similar to that in [2]. For the ℵ0-categorical case we show that this closure is part of the algebraic closure.

Let S be the group of all permutations of the set of natural numbers. The cofinality spectrum CF(S) of S is the set of all regular cardinals λ such that S can be expressed as the union of a chain of λ proper subgroups. This paper investigates which sets C of regular uncountable cardinals can be the cofinality spectrum of S. The following theorem.is the main result of this paper.

Theorem. Suppose that V ⊨ GCH. Let C be a set of regular uncountable cardinals which satisfies the following conditions.

(a) C contains a maximum element.

(b) If μ is an inaccessible cardinal such thatμ = sup(C ∩ μ), thenμ ∈ C.

(c) if μ is a singular cardinal such thatμ = sup(C ∩ μ), thenμ+ ∈ C.

Then there exists a c.c.c. notion of forcing ℙ such that Vℙ ⊨ CF(S) = C.

We shall also investigate the connections between the cofinality spectrum and pcf theory; and show that CF(S) cannot be an arbitrarily prescribed set of regular uncountable cardinals.

We discuss the connection between decidability of a theory of a large algebraic extensions of ${\Bbb Q}$ and the recursiveness of the field as a subset of a fixed algebraic closure. In particular, we prove that if an algebraic extension K of ${\Bbb Q}$ has a decidable existential theory, then within any fixed algebraic closure $\widetilde{\Bbb Q}$ of ${\Bbb Q}$, the field K must be conjugate over ${\Bbb Q}$ to a field which is recursive as a subset of the algebraic closure. We also show that for each positive integer e there are infinitely many e-tuples $\sigma \in {\text{Gal}}\left( {\Bbb Q} \right)^e $ such that the field $\widetilde{\Bbb Q}\left( \sigma \right)$ is primitive recursive in $\widetilde{\Bbb Q}$ and its elementary theory is primitive recursively decidable. Moreover, $\widetilde{\Bbb Q}\left( \sigma \right)$ is PAC and ${\text{Gal}}\left( {\widetilde{\Bbb Q}\left( \sigma \right)} \right)$ is isomorphic to the free profinite group on e generators.

A Partial Combinatory Algebra is completable if it can be extended to a total one. In [1] it is asked (question 11, posed by D. Scott, H. Barendregt, and G. Mitschke) if every PCA can be completed. A negative answer to this question was given by Klop in [12, 11]; moreover he provided a sufficient condition for completability of a PCA (M, •, K,S) in the form of ten axioms (inequalities) on terms of M. We prove that just one of these axiom (the so called Barendregt's axiom) is sufficient to guarantee (a slightly weaker notion of) completability.

We construct a Galois connection between closure and interior operators on a given set. All arguments are intuitionistically valid. Our construction is an intuitionistic version of the classical correspondence between closure and interior operators via complement.

We construct a fibered dimension function in some topological differential fields.

A relational structure is called reversible iff each bijective endomorphism (condensation) of that structure is an automorphism. We show that reversibility is an invariant of some forms of L∞ω −bi-interpretability, implying that the condensation monoids of structures are topologically isomorphic. Applying these results, we prove that, in particular, all orbits of ultrahomogeneous tournaments and reversible ultrahomogeneous m-uniform hypergraphs are reversible relations and that the same holds for the orbits of reversible ultrahomogeneous digraphs definable by formulas which are not R-negative.

In this article, we study some new characterizations of primitive recursive functions based on restricted forms of primitive recursion, improving the pioneering work of R. M. Robinson and M. D. Gladstone. We reduce certain recursion schemes (mixed/pure iteration without parameters) and we characterize one-argument primitive recursive functions as the closure under substitution and iteration of certain optimal sets.

We classify irreducible actions of connected groups of finite Morley rank on abelian groups of Morley rank 3.