Nous dirons qu'un corps (commutatif) K est un corps de courbe réelle sur le corps k si et seulement si c'est une extension de k, ordonnable, finiment engendrée et de degré de transcendance 1 sur k, telle que k soit relativement algébriquement clos dans K. Un tel corps est le corps d'une courbe (plane) définie sur k. De plus si k est réel-clos, toute courbe définie sur k dont K est le corps, a un point régulier rationnel sur k; en effet soit I ⊂ k[X0,…, Xn] l'idéal d'une telle courbe Γ; on a donc: K = k[x0,…, xn] où x0,…, xn sont les classes d'équivalence de X0,…, Xn; (x0,…, xn) est un point régulier de Γ dans la clôture réelle de K, et par modèle-complétude, Γ a un point régulier rationnel sur k.

Notre but est d'étendre les résultats de Raphael Robinson [6] sur l'indécidabilité des corps de fractions rationnelles sur un corps ordonnable (dans le langage des corps). Comme lui, nous donnerons une méthode qui ne s'applique que lorsque le corps de base k est réel-clos, et une seconde méthode générate.

A realizability notion that employs only primitive recursive functions is defined, and, relative to it, the soundness of the fragment of Heyting Arithmetic (HA) in which induction is restricted to formulae is proved. A dual concept of falsifiability is proposed and an analogous soundness result is established for a further restricted fragment of HA.

Finitely axiomatizable ℵ1 categorical theories are locally modular.

Mathematicians have one over on the physicists in that they already have a unified theory of mathematics, namely, set theory. Unfortunately, the plethora of independence results since the invention of forcing has taken away some of the luster of set theory in the eyes of many mathematicians. Will man's knowledge of mathematical truth be forever limited to those theorems derivable from the standard axioms of set theory, ZFC? This author does not think so, he feels that set theorists' intuition about the universe of sets is stronger than ZFC. Here in this paper, using part of this intuition, we introduce some axiom schemata which we feel are very natural candidates for being considered as part of the axioms of set theory. These schemata assert the existence of many generics over simple inner models. The main purpose of this article is to present arguments for why the assertion of the existence of such generics belongs to the axioms of set theory.

Our central guiding principle in justifying the axioms is what Maddy called the rule of thumb maximize in her survey article on the axioms of set theory, [8] and [9]. More specifically, our intuition conforms with that expressed by Mathias in his article What is Maclane Missing? challenging Mac Lane's view of set theory.

This paper deals with some structural properties of the sequent calculus and describes strong symmetries between cut-free derivations and derivations, which do not make use of identity axioms. Both of them are discussed from a semantic and syntactic point of view.

Identity axioms and cuts are closely related to the treatment of negation in the sequent calculus, so the results of this article explain some nice symmetries of negation.

Let L0 be a language on ℕ consisting of 0, 1, +, ∸, ·, ⌊½a⌋, ∣a∣, #, ∧(a,b), ∨(a,b), ¬(a), ≤ (a,b), and μx ≤ ∣s∣t(x). Here μx ≤ ∣s∣t(x) is the smallest number x ≤ ∣s∣ satisfying t(x) > 0 and 0 if there exist no such x and we stipulate that if s and t(a) are terms in Lo, then μx ≤ ∣s∣t(x) is also a term in Lo. The defining axioms of functions ∧(a,b), ∨(a,b), ¬(a), ≤ (a,b) are as follows:

Let L a language on ℕ with only predicate constant = and L0 ⊆ L. Let f (b, a1,…,am) be a function for ℕm+1 into ℕ. We say “f is weakly expressed by terms t1(b, a1,…, am),…, tr(b, a1,…, am) in L” if for every b, a1,…, am ∈ ℕ, f (b, a1,…,am) is equal to one of ti(b, a1, …, am). The critical number of b in f with respect to L is the minimum number n such that whenever f(b, a1,…, an) is weakly expressed by terms t1(b, a1, …, an),…, the number of occurrences of b in some ti(b, a1,…, an) is at least n.

As is defined in [1], a function f is defined by a limited iteration from g and h with respect to L iff the following holds: Let τ be defined as

with the condition ; and is defined by

where and are terms in L. We say “f is defined by a short limited iteration from g and h” if is defined by

where τ, s, t are the same as above satisfying the condition .

We reduce to a standard circuit-size complexity problem a relativisation of the P=NP question that we believe to be connected with the same question in the model for computation over the reals defined by L. Blum. M. Shub, and S. Smale. On this occasion, we set the foundations of a general theory for computation over an arbitrary structure, extending what these three authors did in the case of rings.

Kernel contraction is a natural nonrelational generalization of safe contraction. All partial meet contractions are kernel contractions, but the converse relationship does not hold. Kernel contraction is axiomatically characterized. It is shown to be better suited than partial meet contraction for formal treatments of iterated belief change.

Let κ be a regular cardinal, and let B be a subalgebra of an interval algebra of size κ. The existence of a chain or an antichain of size κ in ℬ is due to M. Rubin (see [7]). We show that if the density of B is countable, then the same conclusion holds without this assumption on κ. Next we also show that this is the best possible result by showing that it is consistent with 2ℵ0 = ℵω1 that there is a boolean algebra B of size ℵω1 such that length(B) = ℵω1 is not attained and the incomparability of B is less than ℵω1. Notice that B is a subalgebra of an interval algebra. For more on chains and antichains in boolean algebras see. e.g., [1] and [2].

Partial models of the theory New Foundations (NF) introduced by Quine have already appeared in the literature, but in every model the membership set of NF is missing. On the other hand, Jensen showed that “NF + Urelements” is consistent with respect to ZF and, in the model built there, the membership set of the theory exists, Here we build a partial model of NF from the one of Jensen in which the membership set exists.

A generator program for a computable function (by definition) generates an infinite sequence of programs all but finitely many of which compute that function. Machine learning of generator programs for computable functions is studied. To motivate these studies partially, it is shown that, in some cases, interesting global properties for computable functions can be proved from suitable generator programs which cannot be proved from any ordinary programs for them. The power (for variants of various learning criteria from the literature) of learning generator programs is compared with the power of learning ordinary programs. The learning power in these cases is also compared to that of learning limiting programs, i.e., programs allowed finitely many mind changes about their correct outputs.

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.

There is a family of questions in relativized complexity theory—weak analogs of the Friedberg Jump-Inversion Theorem—that are resolved by 1-generic sets but which cannot be resolved by essentially any weaker notion of genericity. This paper defines aw2-generic sets, i.e., sets which meet every dense set of strings that is re. in some incomplete r.e. set. Aw2-generic sets are very close to 1-generic sets in strength, but are too weak to resolve these questions. In particular, it is shown that for any set X there is an aw2-generic set G such that NPG ∩ co-NPG ⊈ PG⊕x. (On the other hand, if G is 1-generic, then NPG ∩ co-NPG ⊆ PG⊕SAT, where SAT is the NP-complete satisfiability problem [6].) This result runs counter to the fact that most finite extension constructions in complexity theory can be made effective. These results imply that any finite extension construction that ensures any of the Friedberg analogs must be noneffective, even relative to an arbitrary incomplete r.e. set. It is then shown that the recursion theoretic properties of aw2-generic sets differ radically from those of 1-generic sets: every degree above 0′ contains an aw2-generic set: no aw2-generic set exists below any incomplete r.e. set: there is an aw2-generic set which is the join of two Turing equivalent aw2-generic sets. Finally, a result of Shore is presented [30] which states that every degree above 0′ is the jump of an aw2-generic degree.

With a certain graphic interpretation in mind, we say that a function whose value at every point in its domain is a nonempty set of real numbers is an Abacus. It is shown that to every collection C of abaci there corresponds a logic, called an abacus logic, i.e.. a certain set of propositions partially ordered by generalized implication. It is also shown that to every collection C of abaci there corresponds a theory Jc in a classical propositional calculus such that the abacus logic determined by C is isomorphic to the poset of Jc. Two examples are given. In both examples abacus logic is a lattice in which there happens to be an operation of orthocomplementation. In the first example abacus logic turns out to be the Lindenbaum algebra of Jc. In the second example abacus logic is a lattice isomorphic to the ortholattice of subspaces of a Hilbert space. Thus quantum logic can be regarded as an abacus logic. Without suggesting “hidden variables” it is finally shown that the Lindenbaum algebra of the theory in the second example is a subalgebra of the abacus logic B of the kind studied in example 1. It turns out that the “classical observables” associated with B and the “quantum observables” associated with quantum logic are not unrelated. The value of a classical observable contains, in coded form, information about the “uncertainty” of a quantum observable. This information is retrieved by decoding the value of the corresponding classical observable.

Using a result of Gurevich and Lewis on the word problem for finite semigroups, we give short proofs that the following theories are hereditarily undecidable: (1) finite graphs of vertex-degree at most 3; (2) finite nonvoid sets with two distinguished permutations; (3) finite-dimensional vector spaces over a finite field with two distinguished endomorphisms.

Let CKDT be the assertion that for every countably infinite bipartite graph G, there exist a vertex covering C of G and a matching M in G such that C consists of exactly one vertex from each edge in M. (This is a theorem of Podewski and Stefifens [12].) Let ATR0 be the subsystem of second-order arithmetic with arithmetical transfinite recursion and restricted induction. Let RCA0 be the subsystem of second-order arithmetic with recursive comprehension and restricted induction. We show that CKDT is provable in ATR0. Combining this with a result of Aharoni, Magidor, and Shore [2], we see that CKDT is logically equivalent to the axioms of ATR0, the equivalence being provable in RCA0.

This paper was inspired by Lerman [14] in which he proved various properties of upper bounds for the arithmetical degrees. The degrees which are upper bounds for the arithmetical degrees were first studied by Hodes [5] and Enderton and Putnam [5]. Enderton and Putnam [5] showed that if a is an upper bound for the arithmetical degrees, then a(2) ≥ 0(ω). This area was further studied by Hodes [5], Knight, Lachlan and Soare [12], Jockusch and Simpson [12], and Sacks [17].

Enderton and Putnam [5] and Sacks [17] have combined to show that 0(ω) is the least degree in {a(2): a is an upper bound for the arithmetical degrees}. So there seem to be some analogies between the degrees of upper bounds for the arithmetical degrees and the degrees below 0(2). But Sacks [17] showed an important difference between these two structures; namely, the Turing jumps of upper bounds for the arithmetical degrees have no least element. In Lerman [14], a systematic investigation of properties of the jumps of upper bounds for the arithmetical degrees was suggested, which could lead to a definition of the jump operator over the elementary theory of the partial ordering of the degrees. Although Cooper [5] found a degree-theoretic definition of the jump operator, Lerman's plan is still interesting.

We say a degree a is generic if there is a representative A of a such that A is Cohen generic for the arithmetical sentences. By Jockusch [5], the statement that A is Cohen generic for the arithmetical sentences is equivalent to saying that for any arithmetical set S of binary strings, there is a σ extended by A such that σ is in S or no extension of σ is in S. First, we investigate the relation between generic degrees and upper bounds for the arithmetical degrees. In the case D(≤ 0′), the set of degrees below 0′, it is well known that any nonrecursive r.e. degree bounds a 1-generic degree (Cohen generic for Σ1 sentences). Jockusch and Ponser [12] showed that any degree a with a″ >T (a ∪ 0′)′ bounds a 1-generic degree.

Given a finite relational language L is there an algorithm that, given two finite sets and of structures in the language, determines how many homogeneous L structures there are omitting every structure in and embedding every structure in ?

For directed graphs this question reduces to: Is there an algorithm that, given a finite set of tournaments Γ, determines whether Γ, the class of finite tournaments omitting every tournament in Γ. is well-quasi-order?

First, we give a nonconstructive proof of the existence of an algorithm for the case in which Γ consists of one tournament. Then we determine explicitly the set of tournaments each of which does not have an antichain omitting it. Two antichains are exhibited and a summary is given of two structure theorems which allow the application of Kruskal's Tree Theorem. Detailed proofs of these structure theorems will be given elsewhere.

The case in which Γ consists of two tournaments is also discussed.

Fitting, in [1] and [2], investigated two families of many-valued modal logics. The first, which is somewhat familiar in the literature, is that of the logics characterized using a many-valued version of the Kripke model (binary modal model in his terminology) with a two-valued accessibility relation. On the other hand, those logics which are characterized using another many-valued version of the Kripke model (implicational modal model), with a many-valued accessibility relation, form the second family. Although he gave a sequent calculus for each of these logics, it is far from having the cut-elimination property (CEP) or the subformula property. So we will give a substitute for his system enjoying the subformula property, though it is not of ordinary sequent calculus but of the many-valued version of sequent calculus initiated by Takahashi [7] and Rousseau [3].

The author, unaware of the deduction systems with CEP, had given in [8] and [9], after Rousseau [4], the deduction systems for the intuitionistic many-valued logics which enjoy CEP only for a certain restricted class of proofs. Then in [10], he gave for three-valued modal logics the ones with CEP, but these systems have a rule of inference which is unnecessary if the Cut rule is present. Why are we particular about CEP? The author's answer is that a cut-free proof is easy to examine since it is composed solely of subformulas of the formulas which form its conclusion. In this direction, the author has given, for modal logics with the Brouwerian axiom [11], the ones without CEP which nevertheless enjoy the subformula property. This paper is a sequel to the study in [11].

We present a sequent calculus for the modal logic S4, and building on some relevant features of this system (the absence of contraction rules and the confinement of weakenings into axioms and modal rules) we show how S4 can easily be translated into full prepositional linear logic, extending the Grishin-Ono translation of classical logic into linear logic. The translation introduces linear modalities (exponentials) only in correspondence with S4 modalities. We discuss the complexity of the decision problem for several classes of linear formulas naturally arising from the proposed translations.