We give some sufficient conditions for a predicate P in a complete theory T to be “stably embedded”. Let be P with its “induced ∅-definable structure”. The conditions are that (or rather its theory) is “rosy”. P has NIP in T and that P is stably 1-embedded in T. This generalizes a recent result of Hasson and Onshuus [6] which deals with the case where P is o-minimal in T. Our proofs make use of the theory of strict nonforking and weight in NIP theories ([3], [10]).

In this paper, we show the normalization of proofs of NF (Quine's New Foundations; see [15]) minus extensionality. This system, called SF (Stratified Foundations) differs in many respects from the associated system of simple type theory. It is written in a first order language and not in a multi-sorted one, and the formulas need not be stratifiable, except in the instances of the comprehension scheme. There is a universal set, but, for a similar reason as in type theory, the paradoxical sets cannot be formed.

It is not immediately apparent, however, that SF is essentially richer than type theory. But it follows from Specker's celebrated result (see [16] and [4]) that the stratifiable formula (extensionality → the universe is not well-orderable) is a theorem of SF.

It is known (see [11]) that this set theory is consistent, though the consistency of NF is still an open problem.

The connections between consistency and cut-elimination are rather loose. Cut-elimination generally implies consistency. But the converse is not true. In the case of set theory, for example, ZF-like systems, though consistent, cannot be freed of cuts because the separation axioms allow the formation of sets from unstratifiable formulas. There are nevertheless interesting partial results obtained when restrictions are imposed on the removable cuts (see [1] and [9]). The systems with stratifiable comprehension are the only known set-theoretic systems that enjoy full cut-elimination.

In this paper we present two calculi for intuitionistic logic. The first one. IG, is characterized by the fact that every proof-search terminates and termination is reached without jeopardizing the subformula property. As to the second one, SIC, proof-search terminates, the subformula property is preserved and moreover proof-search is performed without any recourse to metarules, in particular there is no need to back-track. As a consequence, proof-search in the calculus SIC is accomplished by a single tree as in classical logic.

The purpose of this paper is to give an axiom system for quantum logic. Here quantum logic is considered to have the structure of an orthomodular lattice. Some authors assume that it has the structure of an orthomodular poset.

In finding this axiom system the implication algebra given in Finch [1] has been very useful. Finch shows there that this algebra can be produced from an orthomodular lattice and vice versa.

Definition. An orthocomplementation N on a poset (partially ordered set) whose partial ordering is denoted by ≤ and which has least and greatest elements 0 and 1 is a unary operation satisfying the following:

(1) the greatest lower bound of a and Na exists and is 0,

(2) a ≤ b implies Nb ≤ Na,

(3) NNa = a.

Definition. An orthomodular lattice is a lattice with meet ∧, join ∨, least and greatest elements 0 and 1 and an orthocomplementation N satisfying

where a ≤ b means a ∧ b = a, as usual.

Definition. A Finch implication algebra is a poset with a partial ordering ≤, least and greatest elements 0 and 1 which is orthocomplemented by N. In addition, it has a binary operation → satisfying the following:

An orthomodular lattice gives a Finch implication algebra by defining → by

A Finch implication algebra can be changed into an orthomodular lattice by defining the meet ∧ and join ∨ by

The orthocomplementation is unchanged in both cases.

Generic automorphisms of certain homogeneous structures are considered, for instance, the rationals as an ordered set, the countable universal homogeneous partial order, and the random graph. Two of these cases were discussed in [7], where it was shown that there is a generic automorphism of the second in the sense introduced in [10], In this paper, I study various possible definitions of ‘generic’ and ‘mutually generic’, and discuss the existence of mutually generic automorphisms in some cases. In addition, generics in the automorphism group of the rational circular order are considered.

In this article propositions will be identified with a certain kind of changes of belief. The intended interpretation is that a proposition is characterised by the change it would induce if added to a state of belief. Propositions will thus be defined as functions from states of belief to states of belief. A set of postulates concerning the properties and existence of propositions will be formulated. A proposition will be said to be a tautology iff it is the identity function on states of belief. The main result is that the logic determined by the set of postulates is intuitionistic propositional logic.

The basic epistemic concept is that of a belief model, which is defined as a pair 〈, 〉, where is a nonempty set and is a class of functions from to . The elements in will be called states of belief and they will be denoted K, K′,…. A discussion of the epistemological interpretation of the states of belief can be found in Gärdenfors [2]. Here, no assumptions about the structure of the elements in will be made.

The elements in will be called propositions, and A, B, C, … will be used as variables over . Functions from states of belief to states of belief can be characterised as epistemic inputs. The intended interpretation of the functions in is that they correspond to changes of belief where the new evidence is accepted as “certain” or “known” in the resulting state of belief. This means that not all functions defined on can properly be called propositions.

The theorem proved in this paper answers some transitivity questions (in the geometric sense) for the type free λ-calculus: Which objects can be mapped on all other objects? How much can an object do by applying it to other objects (see footnote 2)?

The main result is that, for closed terms of the λI-calculus, the following conditions are equivalent:

(a) M has a normal form.

(b) FM = I for some λI-term F.

(c) MN1 … Nn = I for some λI-terms N1 …, Nn.

By the same method it follows that if M is a closed term of the λK-calculus having a normal form, then for some λI-terms (sic) N1, …, Nn, MN1… Nn = I is provable in the λK-calculus.

The theorem of Böhm [2] states that if M1, M2 are terms of the λK-calculus having different βη-normal forms, then ∀A1, A2 ∃N1, …, NnMiN1 … Nn = Ai is provable in the λK-βη-calculus for i = 1, 2. As a consequence of this it was shown (implicitly) in [1, 3.2.20 1/2 (1)] that if M has a normal form, then for some λK-terms N1, …, Nn, MN1 … Nn =I is provable in the λK-calculus.

It was not clear that this also could be proved for the λK-calculus since the proof of the theorem of Böhm essentially made use of λK-terms.

We conjecture that, using the results of this paper, the full theorem of Böhm can be proved for the λI-calculus.

The relation ≪ (is a homomorphic image of) between (linear) order types has properties similar to those of the better known relation ≤ (is embeddable in). For example, the order type η of the rationals not only embeds every countable order type but also maps homomorphically onto . If is scattered, then can be embedded in (ω* + ω)α for some α < ω1. In that case, is also a homomorphic image of (ω* + ω)α [Lan 2]. If is uncountable, then for some uncountable ordinal α, α ≪, α* ≪, or η ≪. Proofs of these facts are much the same for ≤ and ≪.

The main theorem of [Lav 1] implies that the embedding relation better-quasiorders the set of countable order types. Our main theorem (§3) states the analogous result for the homomorphism relation. As a consequence, if 0, 1, … is an infinite sequence of countable order types, then there are i, j, i < j, such that i, is a homomorphic image of j. We observed in [Lan 1] that if this is true, then for each countable order type there is a sentence of Lω1ω such that if is a countable order type, then satisfies if and only if is a homomorphic image of . In fact, the motivation for the work leading to this paper came from this observation.

On the negative side, it is pointed out (§3) that our theorem cannot be extended as far as that of [Lav 1].

Does V = L? Is the Axiom of Constructibility true? Most people with an opinion would answer no. But on what grounds? Despite the near unanimity with which V = L is declared false, the literature reveals no clear consensus on what counts as evidence against the hypothesis and no detailed analysis of why the facts of the sort cited constitute evidence one way or another. Unable to produce a well-developed argument one way or the other, some observers despair, retreating to unattractive fall-back positions, e.g., that the decision on whether or not V = L is a matter of personal aesthetics. I would prefer to avoid such conclusions, if possible. If we are to believe that L is not V, as so many would urge, then there ought to be good reasons for this belief, reasons that can be stated clearly and subjected to rational evaluation. Though no complete argument has been presented, the literature does contain a number of varied argument fragments, and it is worth asking whether some of these might be developed into a persuasive case.

One particularly simple approach would be to note that the existence of a measurable cardinal (MC) implies that V ≠ L,1 and to argue that there is a measurable cardinal. The drawback to this approach is that its implying V ≠ L cannot then be counted as evidence in favor of MC, as it often is. Indeed, there seems to have been considerable sentiment against V = L even before the proof of its negation from MC,2 and this sentiment must either be accounted for as reasonable or explained away as an aberration of some kind.

This paper introduces an extension of Kleene's axiomatization of Brouwer's intuitionistic analysis, in which the classical arithmetical and analytical hierarchies are faithfully represented as hierarchies of the domains of continuity. A domain of continuity is a relation R(α) on Baire space with the property that every constructive partial functional defined on {α: R(α)} is continuous there. The domains of continuity for coincide with the stable relations (those equivalent in to their double negations), while every relation R(α) is equivalent in to ∃βA(α, β) for some stable A(α, β) (which belongs to the classical analytical hierarchy).

The logic of is intuitionistic. The axioms of include countable comprehension, bar induction, Troelstra's generalized continuous choice, primitive recursive Markov's Principle and a classical axiom of dependent choices proposed by Krauss. Constructive dependent choices, and constructive and classical countable choice, are theorems, is maximal with respect to classical Kleene function realizability, which establishes its consistency. The usual disjunction and (recursive) existence properties ensure that preserves the constructive sense of “or” and “there exists.”

Consider a definably complete uniformly locally o-minimal expansion of the second kind of a densely linearly ordered abelian group. Let $f:X \rightarrow R^n$ be a definable map, where X is a definable set and R is the universe of the structure. We demonstrate the inequality $\dim (f(X)) \leq \dim (X)$ in this paper. As a corollary, we get that the set of the points at which f is discontinuous is of dimension smaller than $\dim (X)$. We also show that the structure is definably Baire in the course of the proof of the inequality.

In this paper we investigate the computational complexity of deciding if the variety generated by a given finite idempotent algebra satisfies a special type of Maltsev condition that can be specified using a certain kind of finite labelled path. This class of Maltsev conditions includes several well known conditions, such as congruence permutability and having a sequence of n Jónsson terms, for some given n. We show that for such “path defined” Maltsev conditions, the decision problem is polynomial-time solvable.

Let be a countable saturated structure, and assume that D(v) is a strongly minimal formula (without parameter) such that is the algebraic closure of D(). We will prove the two following theorems:

Theorem 1. If G is a subgroup of Aut() of countable index, there exists a finite set A in such that every A-strong automorphism is in G.

Theorem 2. Assume that G is a normal subgroup of Aut() containing an element g such that for all n there exists X ⊆ D() such that Dim(g(X)/X) > n. Then every strong automorphism is in G.

The price is failure on a class of inductive inference problems that are easily solved, in contrast, by nonBayesian mechanical learners. By “mechanical” is meant “simulable by Turing machine”.

One of the central tenets of Bayesianism, which is common to the heterogeneous collection of views which fall under this rubric, is that hypothesis change proceeds via conditionalization on accumulated evidence, the posterior probability of a given hypothesis on the evidence being computed using Bayes's theorem. We show that this strategy for hypothesis change precludes the solution of certain problems of inductive inference by mechanical means—problems which are solvable by mechanical means when the restriction to this Bayesian strategy is lifted. Our discussion proceeds as follows. After some technical preliminaries, the concept of (formal) learner is introduced along with a criterion of inferential success. Next we specify a class of inductive inference problems, and then define the notion of “Bayesian behavior” on those problems. Finally, we exhibit an inductive inference problem from the specified class such that (a) some nonmechanical Bayesian learner solves the problem, (b) some nonBayesian mechanical learner solves the problem, (c) some mechanical learner manifests Bayesian behavior on the problem, but (d) no mechanical Bayesian learner solves the problem.

Insofar as possible terminology and notation are drawn from Osherson, Stob, and Weinstein [1986].

Which groups are isomorphic to automorphism groups of models of Peano Arithmetic? It will be shown here that any group that has half a chance of being isomorphic to the automorphism group of some model of Peano Arithmetic actually is.

For any structure , let Aut() be its automorphism group. There are groups which are not isomorphic to any model = (N, +, ·, 0, 1, ≤) of PA. For example, it is clear that Aut(N), being a subgroup of Aut((, <)), must be torsion-free. However, as will be proved in this paper, if (A, <) is a linearly ordered set and G is a subgroup of Aut((A, <)), then there are models of PA such that Aut() ≅ G.

If is a structure, then its automorphism group can be considered as a topological group by letting the stabilizers of finite subsets of A be the basic open subgroups. If ′ is an expansion of , then Aut(′) is a closed subgroup of Aut(). Conversely, for any closed subgroup G ≤ Aut() there is an expansion ′ of such that Aut(′) = G. Thus, if is a model of PA, then Aut() is not only a subgroup of Aut((N, <)), but it is even a closed subgroup of Aut((N, ′)).

There is a characterization, due to Cohn [2] and to Conrad [3], of those groups G which are isomorphic to closed subgroups of automorphism groups of linearly ordered sets.

We consider the problem whether the theorems of the fragments form a strictly increasing hierarchy. We shall show a link to some results about the lengths of proofs in predicate logic that supports the conjecture that the hierarchy is strictly increasing.

In this article we introduce and study hyperclass-forcing (where the conditions of the forcing notion are themselves classes) in the context of an extension of Morse-Kelley class theory, called MK**. We define this forcing by using a symmetry between MK** models and models of ZFC− plus there exists a strongly inaccessible cardinal (called SetMK**). We develop a coding between β-models ${\cal M}$ of MK** and transitive models M+ of SetMK** which will allow us to go from ${\cal M}$ to M+ and vice versa. So instead of forcing with a hyperclass in MK** we can force over the corresponding SetMK** model with a class of conditions. For class-forcing to work in the context of ZFC− we show that the SetMK** model M+ can be forced to look like LK*[X], where κ* is the height of M+, κ strongly inaccessible in M+ and $X \subseteq \kappa$. Over such a model we can apply definable class forcing and we arrive at an extension of M+ from which we can go back to the corresponding β-model of MK**, which will in turn be an extension of the original ${\cal M}$. Our main result combines hyperclass forcing with coding methods of [3] and [4] to show that every β-model of MK** can be extended to a minimal such model of MK** with the same ordinals. A simpler version of the proof also provides a new and analogous minimality result for models of second-order arithmetic.

We make a more systematic study of the van Douwen diagram for cardinal coefficients related to combinatorial properties of partitions of natural numbers.

We consider total well-founded orderings on monadic terms satisfying the replacement and full invariance properties. We show that any such ordering on monadic terms in one variable and two unary function symbols must have order type ω, ω2 or ωω. We show that a familiar construction gives rise to continuum many such orderings of order type ω. We construct a new family of such orderings of order type ω2, and show that there are continuum many of these. We show that there are only four such orderings of order type ωω, the two familiar recursive path orderings and two closely related orderings. We consider also total well-founded orderings on Nn which are preserved under vector addition. We show that any such ordering must have order type ωk for some 1 ≤ k ≤ n. We show that if k < n there are continuum many such orderings, and if k = n there are only n!, the n! lexicographic orderings.

In [1], S. Buss introduced the systems of Bounded Arithmetic for (i = 0,1,2,…) which has a close relationship to classes in polynomial hierarchy.

In [4], we defined a very special kind of proof-predicate Prfi for which gives detailed information on bounds of free variables used in the proof. There we also introduced infinitely many Gödel sentences for Prfi (k = 0, 1, 2, …) and showed that the properties of Prfi and are closely related to the P ≠ NP problem. Then we presented many conjectures on Prfi and which imply P ≠ NP.

Now in [2], Feferman emphasized that the arithmetization of metamathematics must be carried out intensionally. Bounded Arithmetic is a very interesting case in this sense.

In this paper, we also introduce the usual proof-predicate PRFi for and infinitely many Gödel sentences for PRFi(k= 0, 1, 2, …). Then we show that (Prfi, )and (PRFi, ) form a good contrast, this contrast is also closely related to the P ≠ NP problem, and present more conjectures which imply P ≠ NP.

As in [4] we define to be the following extension of Buss' original .

(1) We add finitely many function symbols which express polynomial time computable functions to Buss' original language of .

(2) All basic axioms on function symbols and ≤ can be expressed by initial sequents without logical symbols.