Given a regular cardinal $\kappa $ such that $\kappa ^{<\kappa }=\kappa $ (or any regular $\kappa $ if the Generalized Continuum Hypothesis holds), we study a class of toposes with enough points, the $\kappa $ -separable toposes. These are equivalent to sheaf toposes over a site with $\kappa $ -small limits that has at most $\kappa $ many objects and morphisms, the (basis for the) topology being generated by at most $\kappa $ many covering families, and that satisfy a further exactness property T. We prove that these toposes have enough $\kappa $ -points, that is, points whose inverse image preserve all $\kappa $ -small limits. This generalizes the separable toposes of Makkai and Reyes, that are a particular case when $\kappa =\omega $ , when property T is trivially satisfied. This result is essentially a completeness theorem for a certain infinitary logic that we call $\kappa $ -geometric, where conjunctions of less than $\kappa $ formulas and existential quantification on less than $\kappa $ many variables is allowed. We prove that $\kappa $ -geometric theories have a $\kappa $ -classifying topos having property T, the universal property being that models of the theory in a Grothendieck topos with property T correspond to $\kappa $ -geometric morphisms (geometric morphisms the inverse image of which preserves all $\kappa $ -small limits) into that topos. Moreover, we prove that $\kappa $ -separable toposes occur as the $\kappa $ -classifying toposes of $\kappa $ -geometric theories of at most $\kappa $ many axioms in canonical form, and that every such $\kappa $ -classifying topos is $\kappa $ -separable. Finally, we consider the case when $\kappa $ is weakly compact and study the $\kappa $ -classifying topos of a $\kappa $ -coherent theory (with at most $\kappa $ many axioms), that is, a theory where only disjunction of less than $\kappa $ formulas are allowed, obtaining a version of Deligne’s theorem for $\kappa $ -coherent toposes from which we can derive, among other things, Karp’s completeness theorem for infinitary classical logic.

We characterize the determinacy of $F_\sigma $ games of length $\omega ^2$ in terms of determinacy assertions for short games. Specifically, we show that $F_\sigma $ games of length $\omega ^2$ are determined if, and only if, there is a transitive model of ${\mathsf {KP}}+{\mathsf {AD}}$ containing $\mathbb {R}$ and reflecting $\Pi _1$ facts about the next admissible set.

As a consequence, one obtains that, over the base theory ${\mathsf {KP}} + {\mathsf {DC}} + ``\mathbb {R}$ exists,” determinacy for $F_\sigma $ games of length $\omega ^2$ is stronger than ${\mathsf {AD}}$ , but weaker than ${\mathsf {AD}} + \Sigma _1$ -separation.

In [1] the ω-consistency of arithmetic was proved by a method which yields fine ordinal bounds for κ-consistency, κ ≥ 1. In this paper these bounds are shown to be best possible. The ω-consistency of a number-theoretic system S can be expressed thus: for all sentences ∃xM,

where ProvS is the proof predicate for S, if n is a nonnegative integer then n is the formal numeral (of S) for n, and if G is a formula then ˹G˺ is the Gödel number of G. The κ-consistency of S is the restriction of (1) to Σκ0 sentences ∃xM. The proof in [1] establishes the no-counterexample interpretation of (1), that is, the existence of a constructive functional Φ such that, for all sentences ∃xM, all numbers p, and all functions f,

(see [1, §2]). A functional Φ is an ω-consistency functional for S if it satisfies (2) for all sentences ∃xM, and a κ-consistency functional for S if it satisfies (2) for all Σκ0 sentences ∃xM.

The systems considered in [1] are those obtained from classical first-order arithmetic Z, including the schema for definition of primitive recursive (p.r.) functions, by adjoining, for some p.r. well-founded ordering ≺ of the nonnegative integers, the axiom schema

that is, the least number principle on ≺; it is equivalent to the schema of transfinite induction on ≺.

In connection with the work of Anscombe, Macpherson, Steinhorn and the present author in [1] we investigate the notion of a multidimensional exact class (R-mec), a special kind of multidimensional asymptotic class (R-mac) with measuring functions that yield the exact sizes of definable sets, not just approximations. We use results about smooth approximation [24] and Lie coordinatization [13] to prove the following result (Theorem 4.6.4), as conjectured by Macpherson: For any countable language $\mathcal {L}$ and any positive integer d the class $\mathcal {C}(\mathcal {L},d)$ of all finite $\mathcal {L}$-structures with at most d 4-types is a polynomial exact class in $\mathcal {L}$, where a polynomial exact class is a multidimensional exact class with polynomial measuring functions.

In [3] we have associated to a structure an ordinal which gives us information about elementary substructures of the structure. For example a structure whose ascending chain number (as we call the ordinal) is ω could be called Noetherian since all ascending elementary chains inside it are finite (and there are arbitrarily large finite chains). Theorem 2 shows that such structures exist. In fact we prove that for any α < ω1 there is a structure whose ascending chain number is α. The construction is based on the existence of a certain group of permutations of ω (see Theorem 1). The second part of this paper deals with the relevance of the chain number to the study of Jonsson algebras.

Ever since Cantor, we have known that the reals and the rationals are not isomorphic (as equality structures, i.e., sets). Logically speaking, however, they are not all that different; in first-order classical logic they are elementarily equivalent, since the theory of infinite sets is complete. The same holds for ℝ and ℚ as ordered sets; again the theory of dense linear order without end points is complete.

From an intuitionistic point of view these matters are more complicated; e.g., the theory of equality of ℚ is decidable, whereas the one of ℝ patently is not. This, in a roundabout way, shows that ℚ and ℝ are not isomorphic; of course, there is no need for such a detour, as Cantor's original proof [2] is intuitionistically correct, and Brouwer's new proof [1] is another alternative intuitionistic argument.

In view of the fact that ℚ and ℝ behave so strikingly differently with respect to first-order logic, one is easily tempted to look for elementary equivalences among the subsets of ℝ. Until quite recently most model theoretic investigations of intuitionistic theories made use of special (artificial) notions of “model”, e.g., Kripke models, sheaf models,…; but there is no prima facie reason why one should not practice model theory much the same way as traditional model theorists do. That is to say on the basis of a naive set theory, or, in our case, of naive intuitionistic mathematics.

This paper uses the method of (k, p)-isomorphisms of Fraïssé, and it is briefly shown that one half of the Fraïssé theorem holds intuitionistically.

Assume T is a superstable theory with < 2ℵ0 countable models. We prove that any *- algebraic type of -rank > 0 is m-nonorthogonal to a *-algebraic type of -rank 1. We study the geometry induced by m-dependence on a *-algebraic type p* of -rank 1. We prove that after some localization this geometry becomes projective over a division ring . Associated with p* is a meager type p. We prove that p is determined by p* up to nonorthogonality and that underlies also the geometry induced by forking dependence on any stationarization of p. Also we study some *-algebraic *-groups of -rank 1 and prove that any *-algebraic *-group of -rank 1 is abelian-by-finite.

Our framework is ZFC, and we view cardinals as initial ordinals. Baumgartner ([Bal] and [Ba2]) studied properties of large cardinals by considering these properties as properties of normal ideals and not as properties of cardinals alone. In this paper we study these combinatorial properties by defining operations which take as input one or more ideals and give as output an ideal associated with a large cardinal property. We consider four operations T, P, S and C on ideals of a regular cardinal κ, and study the structure of the collection of subsets they give, and the relationships between them.

The operation T is defined using combinatorial properties based on trees 〈X, <T〉 on subsets X ⊆ κ (where α <T β → α < β). Given an ideal I, consider the property *: “every tree on κ with every branching set in I has a branch of size κ” (where a branching set is a maximal set with the same set of <T-predecessors, and a chain is a maximal <T-linearly ordered set; for definitions see §2). Now consider the collection T(I) of all subsets of κ that do not satisfy * (see Definition 2.2 and the introduction to §5). The operation T provides us with the large cardinal property (whether κ ∈ T(I) or not) and it also provides us with the ideal associated with this large cardinal property (namely T(I)); in general, we obtain different notions depending on the ideal I.

We construct a decidable first-order theory T such that the theory of its finite models is undecidable. Moreover, T will be equationally axiomatizable and of finite type.

For a canonical model of set theory whose projective theory of the real numbers is stable under set forcing extensions, a set of reals of minimal complexity is constructed which fails to be universally Baire. The construction uses a general method for generating non-universally Baire sets via the Levy collapse of a cardinal, as well as core model techniques. Along the way it is shown (extending previous results of Steel) how sufficiently iterable fine structure models recognize themselves as global core models.

We study the structure of families of theories in the language of arithmetic extended to allow these families to refer to one another and to themselves. If a theory contains schemata expressing its own truth and expressing a specific Turing index for itself, and contains some other mild axioms, then that theory is untrue. We exhibit some families of true self-referential theories that barely avoid this forbidden pattern.

It will be shown that system Q of [1], with minor corrections, is free from contradiction and can be strengthened in various ways without destroying its consistency.

By the system C*A will be meant the system that results in place of CA if rule 3.4.10 is omitted in [2] from the rules 3.4 defining PA and RA for an arbitrary class A of equations. We let P*A be the theorems of C*A and R*A be the antitheorems of C*A, just as PA and RA are respectively the theorems and antitheorems of CA in [2]. Similarly Γ* is the class that is defined exactly like the class Γ in 4.18 of [2] except that systems of the form C*A replace those of the form CA. The terminology and conventions of [2] will be assumed in what follows.

Using the methods that were used to show in 5.9 of [2] that CΓ is free from contradiction, it is easy to show that C*A is free from contradiction for every class A of equations. (The fact that rule 3.4.10 is omitted causes the proof of consistency to work even for cases where A is not systematically closed in the sense of 4.11 of [2].)

Just as it was shown in 4.19 of [2] that Γ is systematically closed (in the sense of 4.11 of [2]), so also it clearly can be shown that Γ* is systematically closed. This means, in effect, that the rule of substitution of equals for equals holds in the system C*Γ*.

We continue the investigation of Gregory trees and the Cantor Tree Property carried out by Hart and Kunen. We produce models of MA with the Continuum arbitrarily large in which there are Gregory trees, and in which there are no Gregory trees.