We show that two abelian-by-finite groups are elementarily equivalent if and only if they satisfy the same sentences with two alternations of quantifiers. We also prove that abelian-by-finite groups satisfy a quantifier elimination property. On the other hand, for each integer n, we give some examples of nilpotent groups which satisfy the same sentences with n alternations of quantifiers and do not satisfy the same sentences with n + 1 alternations of quantifiers.

This paper continues the project initiated in [5]: a model-theoretic study of the concept of cardinality within certain higher-order logics. As recommended by an editor of this Journal, I will digress to say something about the project's motivation. Then I will review some of the basic definitions from [5]; for unexplained notation the reader should consult [5].

The syntax of ordinary usage (with respect to the construction of arguments as well as the construction of individual sentences) makes it natural to classify numerals and expressions of the form ‘the number of F's’ as singular terms, expressions like ‘is prime’ or ‘is divisible by’ as predicates of what Frege called “level one”, and expressions like ‘for some natural number’ as first-order quantifier-phrases. From this syntatic classification, it is a short step—so short as to be frequently unnoticed—to a semantic thesis: that such expressions play the same sort of semantic role as is played by the paradigmatic (and nonmathematical) members of these lexical classes. Thus expressions of the first sort are supposed to designate objects (in post-Fregean terms, entities of type 0), those of the second sort to be true or false of tuples of objects, and those of the third sort to quantify over objects. All this may be summed up in Frege's dictum: “Numbers are objects.”

We establish a surprising connection between Menger's classical covering property and Blass-Laflamme's modern combinatorial notion of groupwise density. This connection implies a short proof of the groupwise density bound on the additivity number for Menger's property.

An important program in the study of the structure the lattice of r.e. sets modulo finite sets, is the classification of the orbits under Aut(), the automorphism group of , If Φ ∈ Aut() and Φ(We) =* Wh(e) for all e ∈ ω, then h is called a presentation of Φ. Define Autχ() to be the class of those elements of Aut() that have a presentation in the class of functions X, where for instance X might be the class of Δn functions for n ∈ ω. If Φ ∈ Autx() then we will say that Φ is an X-automorphism. Note that we only need to consider Δn- and Πn-orbits and automorphisms since if f is a Σn-presentation, then f is a total function and therefore Δn.

Definition. If X is a class of functions and ⊆ {Wi: i < ω}, then is an X-orbit iff is an orbit under Aut() and for all A, B ∈ there is a Φ ∈ Autx() satisfying Φ(A) =* B. (Here Φ: → .)

Harrington proved that the creative sets form a Δ0-orbit and Soare proved that the maximal sets form Δ3-orbit. We will show that there is no Boolean algebra such that {A: A is r.e. and ℒ*(A) ≈ ] forms a Δ2-orbit, where ℒ*(A) is the principal filter of A in ; ℒ*(A) = {B: B ⊆* A & B ∈ }. The idea behind the proof of this theorem is very similar to the proof by Soare that maximal sets do not form a Δ2-orbit (see Soare [1974] or Soare [1987]).

It is a well known fact that Zermelo's Axiom of Choice and Zorn's Lemma are equivalent logical assumptions. However, an investigation from the viewpoint of the hierarchy of types reveals a complication in the nature of this equivalence. In a type-theoretical formalism both postulates enter as spectra (ZAα) and (ZLα) of formulas, where ZAα is Zermelo's Axiom and ZLα is Zorn's Lemma stated for variables of a fixed type α. The complication has its origin in the fact that, although the formula ZAα implies ZLα, in order to deduce ZAα it seems to be necessary to assert Zorn's Lemma ZLβ for a type β which is higher than α. In other words, we don't know a proof assuring the equivalence of the two formulas ZAα and ZLα. Therefore the equivalence of Zermelo's Axiom and Zorn's Lemma has to be understood in the sense that the assertion of the formulas ZAα for all types α implies every one of the formulas ZLβ and conversely the assertion of the spectrum of formulas (ZLα) implies every one of the formulas ZAβ.

In § 3 we shall indicate a type β which makes the deduction of ZAα from ZLβ possible. For this purpose a proof by G. Birkhoff [3] is translated into type-theoretical language.

In § 2 it is shown that ZAα implies ZLα. In this proof no use is made of the notion of well-ordering. Specifically, we do not first deduce the Well-Ordering Theorem as it is usually done in proving Zorn's Lemma from Zermelo's Axiom. However, the method is suggested by Zermelo's second proof of the Well-Ordering Theorem [2].

We prove the statement in the title.

Throughout this paper denotes a nonprincipal ultrafilter and ℐ denotes the dual ideal. ℙ(ℐ) is the poset of all partial functions p: ω → 2 such that dom(p) ∈ ℐ. In [2], Grigorieff proved that ω1 is preserved in the corresponding generic extension if and only if is a P-point. Later, when Shelah introduced the notion of a proper poset, many people observed that if is a P-point, then ℙ(ℐ) is proper. One way of proving this is to show that player II has a winning strategy in the game for ℙ(ℐ) (see [3, p. 91].)

The notion of an Axiom A poset was introduced by Baumgartner [1]. If a poset satisfies Axiom A, then player II has a winning strategy in the game for ℙ, and thus, ℙ is proper. Indeed most of the naturally occurring proper posets satisfy Axiom A (e.g., Mathias's poset and Laver's poset). Thus, it is natural to ask whether or not ℙ(ℐ) satisfies Axiom A. The main result of this paper is a negative answer to this question. We will prove this by introducing another game and showing that is a P-point if and only if the corresponding game is undetermined. We will then show that if ℙ(ℐ) satisfied Axiom A, then player II would have a winning strategy in the corresponding game .

We let [X]<ω = {s ⊆ X∣ ∣s∣ < ω}. We let Seq(X) denote the set of finite sequences of elements of X. If s = 〈x0, …, xn〉 ∈ Seq(X) and y ∈ X, then s * 〈y〉 = 〈x0,…,xn,y〉 ∈ Seq(X).

We consider a weak version of Schindler’s remarkable cardinals that may fail to be ${{\rm{\Sigma }}_2}$-reflecting. We show that the ${{\rm{\Sigma }}_2}$-reflecting weakly remarkable cardinals are exactly the remarkable cardinals, and that the existence of a non-${{\rm{\Sigma }}_2}$-reflecting weakly remarkable cardinal has higher consistency strength: it is equiconsistent with the existence of an ω-Erdős cardinal. We give an application involving gVP, the generic Vopěnka principle defined by Bagaria, Gitman, and Schindler. Namely, we show that gVP + “Ord is not ${{\rm{\Delta }}_2}$-Mahlo” and ${\text{gVP}}(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\Pi } _1 )$ + “there is no proper class of remarkable cardinals” are both equiconsistent with the existence of a proper class of ω-Erdős cardinals, extending results of Bagaria, Gitman, Hamkins, and Schindler.

The language of linear temporal logic can be interpreted on the class of dynamic topological systems, giving rise to the intuitionistic temporal logic ${\sf ITL}^{\sf c}_{\Diamond \forall }$ , recently shown to be decidable by Fernández-Duque. In this article we axiomatize this logic, some fragments, and prove completeness for several familiar spaces.

We present three examples of countable homogeneous structures (also called Fraïssé limits) whose automorphism groups are not universal, namely, fail to contain isomorphic copies of all automorphism groups of their substructures.

Our first example is a particular case of a rather general construction on Fraïssé classes, which we call diversification, leading to automorphism groups containing copies of all finite groups. Our second example is a special case of another general construction on Fraïssé classes, the mixed sums, leading to a Fraïssé class with all finite symmetric groups appearing as automorphism groups and at the same time with a torsion-free automorphism group of its Fraïssé limit. Our last example is a Fraïssé class of finite models with arbitrarily large finite abelian automorphism groups, such that the automorphism group of its Fraïssé limit is again torsion-free.

We define and study a metric independence notion in a homogeneous metric abstract elementary class with perturbations that is dp-superstable (superstable wrt. the perturbation topology), weakly simple and has complete type spaces and we give a new example of such a class based on B. Zilber’s approximations of Weyl algebras. We introduce a way to measure the dependence of a tuple a from a set B over another set A. We prove basic properties of the notion, e.g., that a is independent of B over A in the usual sense of homogeneous model theory if and only if the measure of dependence is < ε for all ε > 0. In well behaved situations, the measure corresponds to the distance to a free extension. As an example of our measure of dependence we show a connection between the measure and entropy in models from quantum mechanics in which the spectrum of the observable is discrete. As an application, we show that weak simplicity implies a very strong form of simplicity and study the question of when the dependence inside a set of all realisations of some type can be seen to arise from a pregeometry in cases when the type is not regular. In the end of the paper, we demonstrate our notions and results in one more example: a class built from the p-adic integers.

We improve on and generalize a 1960 result of Maltsev. For a field F, we denote by $H(F)$ the Heisenberg group with entries in F. Maltsev showed that there is a copy of F defined in $H(F)$ , using existential formulas with an arbitrary non-commuting pair of elements as parameters. We show that F is interpreted in $H(F)$ using computable $\Sigma _1$ formulas with no parameters. We give two proofs. The first is an existence proof, relying on a result of Harrison-Trainor, Melnikov, R. Miller, and Montalbán. This proof allows the possibility that the elements of F are represented by tuples in $H(F)$ of no fixed arity. The second proof is direct, giving explicit finitary existential formulas that define the interpretation, with elements of F represented by triples in $H(F)$ . Looking at what was used to arrive at this parameter-free interpretation of F in $H(F)$ , we give general conditions sufficient to eliminate parameters from interpretations.

We refer to the axioms in Quine's book, Mathematical logic, New York, 1940.

To prove the independence of *200, give xϵ α the truth value F in all cases and give (x)ϕ the same truth value as ϕ. Then clearly all formulas derivable from the other axioms besides *200 have the value T, whereas from *200 one can derive (∃x)(∃α)(xϵ α) which has the value F. This method of proving independence amounts to taking for a model a universe consisting of the single object Λ.

For *201 we prove a contingent independence. That is, we prove that if Quine's system is consistent, then *201 is independent. The line of argument is the following. Suppose *201 can be derived from the other axioms. Let us replace xϵ α by throughout all the axioms. Then what *201 becomes can be derived from what the other axioms become. However what *201 becomes will lead to a contradiction in Quine's system whereas the rules which the other axioms become are valid in Quine's system.

We now get down to technical details. Let us refer to the replacement of xϵ α by throughout an expression as an r replacement. Denote the result of performing an r replacement on ϕ by ϕr. Let Wα denote

Then

Note that if x and y are variables, then by D10,

In analogy to ω-logic, one defines M-logic for an arbitrary structure M (see [5],[6]). In M-logic only those structures are considered in which a special part, determined by a fixed unary predicate, is isomorphic to M. Let L be the similarity type of M and T its complete theory. We say that M-logic is κ-compact if it satisfies the compactness theorem for sets of < κ sentences. In this paper we introduce the related notion of compactness for expandability: a model M is κ-compactly expandable if for every extension T′ ⊇ T of cardinality < κ, if every finite subset of T′ can be satisfied in an expansion of M, then T′ can also be satisfied in an expansion of M. Moreover, M is compactly expandable if it is ∥M∥+-compactly expandable. It turns out that M-logic is κ-compact iff M is κ-compactly expandable.

Whereas for first-order logic consistency and finite satisfiability are the same, consistency with T and finite satisfiability in M are, in general, no longer the same thing. We call the model Mκ-expandable if every consistent extension T′ ⊇ T of cardinality < κ can be satisfied in an expansion of M. We say that M is expandable if it is ∥M∥+-expandable. Here we study the relationship between saturation, expandability and compactness for expandability. There is a close parallelism between our results about compactly expandable models and some theorems of S. Shelah about expandable models, which are in fact expressed in terms of categoricity of PC-classes (see [7, Th. VI.5.3, VI.5.4 and VI.5.5]). Our results could be obtained directly from these theorems of Shelah if expandability and compactness for expandability were the same notion.

We prove that if V = L then there is a maximal orthogonal (i.e., mutually singular) set of measures on Cantor space. This provides a natural counterpoint to the well-known theorem of Preiss and Rataj [16] that no analytic set of measures can be maximal orthogonal.