Matrices of completely Ramsey sets with infinitely many rows

The main result of the present article is the following: Let N be an infinite subset of , , and let be a matrix with infinitely many rows of completely Ramsey subsets of such that for every n, . Then there exist , a sequence of nonempty finite subsets of N, and an infinite subset T of such that for every infinite subset I of . We also give an application of this result to partitions of an uncountable analytic subset of a Polish space X into sets belonging to the σ‐algebra generated by the analytic subsets of X.     

From Bolzano‐Weierstraß to Arzelà‐Ascoli

We show how one can obtain solutions to the Arzelà‐Ascoli theorem using suitable applications of the Bolzano‐Weierstraß principle. With this, we can apply the results from 10 and obtain a classification of the strength of instances of the Arzelà‐Ascoli theorem and a variant of it. Let be the statement that each equicontinuous sequence of functions contains a subsequence that converges uniformly with the rate and let be the statement that each such sequence contains a subsequence which converges uniformly but possibly without any rate. We show that is instance‐wise equivalent, over , to the Bolzano‐Weierstraß principle and that is instance‐wise equivalent, over , to , and thus to the strong cohesive principle (). Moreover, we show that over the principles , and are equivalent.     

Completeness of intermediate logics with doubly negated axioms

Let denote a first‐order logic in a language that contains infinitely many constant symbols and also containing intuitionistic logic . By , we mean the associated logic axiomatized by the double negation of the universal closure of the axioms of plus . We shall show that if is strongly complete for a class of Kripke models , then is strongly complete for the class of Kripke models that are ultimately in .     

On Martin's Axiom and Forms of Choice

Martin's Axiom is the statement that for every well‐ordered cardinal , the statement holds, where is “if is a c.c.c. quasi order and is a family of dense sets in P, then there is a ‐generic filter of P”. In , the fragment is provable, but not in general in . In this paper, we investigate the interrelation between and various choice principles. In the choiceless context, it makes sense to drop the requirement that the cardinal κ be well‐ordered, and we can define for any (not necessarily well‐ordered) cardinal the statement to be “if is a c.c.c. quasi order with , and is a family of dense sets in P, then there is a ‐generic filter of P”. We then define to be the statement that for every (not necessarily well‐ordered) cardinal , we have that holds. We then investigate the set‐theoretic strength of the principle .     

Forcing a countable structure to belong to the ground model

Suppose that P is a forcing notion, L is a language (in ), a P‐name such that “ is a countable L‐structure”. In the product , there are names such that for any generic filter over , and . Zapletal asked whether or not implies that there is some such that . We answer this question negatively and discuss related issues.     

On sequentially closed subsets of the real line in

We show:

• (i) iff every countable product of sequential metric spaces (sequentially closed subsets are closed) is a sequential metric space iff every complete metric space is Cantor complete.
• (ii) Every infinite subset X of has a countably infinite subset iff every infinite sequentially closed subset of includes an infinite closed subset.
• (iii) The statement “ is sequential” is equivalent to each one of the following propositions:

• (a) Every sequentially closed subset A of includes a countable cofinal subset C,
• (b) for every sequentially closed subset A of , is a meager subset of ,
• (c) for every sequentially closed subset A of , ,
• (d) every sequentially closed subset of is separable,
• (e) every sequentially closed subset of is Cantor complete,
• (f) every complete subspace of is Cantor complete.

     

On the set‐theoretic strength of the existence of disjoint cofinal sets in posets without maximal elements

In set theory without the Axiom of Choice , we study the deductive strength of the statements (“Every partially ordered set without a maximal element has two disjoint cofinal subsets”), (“Every partially ordered set without a maximal element has a countably infinite disjoint family of cofinal subsets”), (“Every linearly ordered set without a maximum element has two disjoint cofinal subsets”), and (“Every linearly ordered set without a maximum element has a countably infinite disjoint family of cofinal subsets”). Among various results, we prove that none of the above statements is provable without using some form of choice, is equivalent to , + (Dependent Choices) implies , does not imply in (Zermelo‐Fraenkel set theory with the Axiom of Extensionality modified in order to allow the existence of atoms), does not imply in (Zermelo‐Fraenkel set theory minus ) and (hence, ) is strictly weaker than in .     

The space of minimal structures

For a signature L with at least one constant symbol, an L‐structure is called minimal if it has no proper substructures. Let be the set of isomorphism types of minimal L‐structures. The elements of can be identified with ultrafilters of the Boolean algebra of quantifier‐free L‐sentences, and therefore one can define a Stone topology on . This topology on generalizes the topology of the space of n‐marked groups. We introduce a natural ultrametric on , and show that the Stone topology on coincides with the topology of the ultrametric space iff the ultrametric space is compact iff L is locally finite (that is, L contains finitely many n‐ary symbols for any ). As one of the applications of compactness of the Stone topology on , we prove compactness of certain classes of metric spaces in the Gromov‐Hausdorff topology. This slightly refines the known result based on Gromov's ideas that any uniformly totally bounded class of compact metric spaces is precompact.     

Definably extending partial orders in totally ordered structures

We show, for various classes of totally ordered structures , including o‐minimal and weakly o‐minimal structures, that every definable partial order on a subset of extends definably in  to a total order. This extends the result proved in 5 for and o‐minimal.     

Some combinatorial principles for trees and applications to tree families in Banach spaces

Suppose that is a normalized family in a Banach space indexed by the dyadic tree S. Using Stern's combinatorial theorem we extend important results from sequences in Banach spaces to tree‐families. More precisely, assuming that for any infinite chain β of S the sequence is weakly null, we prove that there exists a subtree T of S such that for any infinite chain β of T the sequence is nearly (resp., convexly) unconditional. In the case where is a family of continuous functions, under some additional assumptions, we prove the existence of a subtree T of S such that for any infinite chain β of T, the sequence is unconditional. Finally, in the more general setting where for any chain β, is a Schauder basic sequence, we obtain a dichotomy result concerning the semi‐boundedly completeness of the sequences .     

A note on equivalence relations

We consider the equivalence relations on induced by the Banach subspaces . We show that if , then there is no Borel reduction from the equivalence relation , where X is a Banach space, to .     

Two‐cardinal diamond star

Our main results are: (A) It is consistent relative to a large cardinal that holds but fails. (B) If holds and are two infinite cardinals such that and λ carries a good scale, then holds. (C) If are two cardinals such that κ is λ‐Shelah and , then there is no good scale for λ.     

Maximal cofinitary groups revisited

Let κ be an arbitrary regular infinite cardinal and let denote the set of κ‐maximal cofinitary groups. We show that if holds and C is a closed set of cardinals such that

• 1. , ,
• 2. if then ,
• 3. ,

then there is a generic extension in which cofinalities have not been changed and such that . The theorem generalizes a result of Brendle, Spinas and Zhang (cf. 4 ) regarding the possible sizes of maximal cofinitary groups. Our techniques easily modify to provide analogous results for the spectra of maximal κ‐almost disjoint families in , maximal families of κ‐almost disjoint permutations on κ and maximal families of κ‐almost disjoint functions in . In addition we construct a κ‐Cohen indestructible κ‐maximal cofinitary group and so establish the consistency of , which for is due to Yi Zhang (cf. 10 ).     

Iteration on notation and unary functions

In the first half of the 1990s, Clote and Takeuti characterized several function complexity classes by means of the concatenation recursion on notation operators. In this paper, we borrow from computability theory well‐known techniques based on pairing functions to show that , , and functions can be characterized by means of concatenation iteration on notation. Indeed, a function class satisfying simple constraints and defined by using concatenation recursion on notation is inductively characterized by means of concatenation iteration on notation. Furthermore, , , and unary functions are inductively characterized using addition, composition, and concatenation iteration on notation.     

Peano Corto and Peano Basso: A Study of Local Induction in the Context of Weak Theories

In this paper we study local induction w.r.t. Σ₁‐formulas over the weak arithmetic . The local induction scheme, which was introduced in 7 , says roughly this: for any virtual class that is progressive, i.e., is closed under zero and successor, and for any non‐empty virtual class that is definable by a Σ₁‐formula without parameters, the intersection of and is non‐empty. In other words, we have, for all Σ₁‐sentences S, that S implies , whenever is progressive. Since, in the weak context, we have (at least) two definitions of Σ₁, we obtain two minimal theories of local induction w.r.t. Σ₁‐formulas, which we call Peano Corto and Peano Basso. In the paper we give careful definitions of Peano Corto and Peano Basso. We establish their naturalness both by giving a model theoretic characterization and by providing an equivalent formulation in terms of a sentential reflection scheme. The theories Peano Corto and Peano Basso occupy a salient place among the sequential theories on the boundary between weak and strong theories. They bring together a powerful collection of principles that is locally interpretable in . Moreover, they have an important role as examples of various phenomena in the metamathematics of arithmetical (and, more generally, sequential) theories. We illustrate this by studying their behavior w.r.t. interpretability, model interpretability and local interpretability. In many ways the theories are more like Peano arithmetic or Zermelo Fraenkel set theory, than like finitely axiomatized theories as Elementary Arithmetic, and . On the one hand, Peano Corto and Peano Basso are very weak: they are locally cut‐interpretable in . On the other hand, they behave as if they were strong: they are not contained in any consistent finitely axiomatized arithmetical theory, however strong. Moreover, they extend , the theory of parameter‐free Π₁‐induction.     

Hyperhypersimple sets and Q1‐reducibility

We prove that the c.e. Q₁‐degrees are not dense, and there exists a c.e. Q₁‐degree with no minimal c.e. predecessors. It is proved that if M₁ and M₂ are maximal sets such that then or . We also show that there exist infinite collections of Q₁‐degrees and such that the following hold: (i) for every , , , and , (ii) each consists entirely of maximal sets; and (iii) each consists entirely of non‐maximal hyperhypersimple sets.     

Paraconsistent double negation as a modal operator

A paraconsistent modal‐like logic, , is defined as a Gentzen‐type sequent calculus. The modal operator in the modal logic can be simulated by the paraconsistent double negation in . Some theorems for embedding into a Gentzen‐type sequent calculus for and vice versa are proved. The cut‐elimination and completeness theorems for are also proved.     

Automorphisms of η‐like computable linear orderings and Kierstead's conjecture

We develop an approach to the longstanding conjecture of Kierstead concerning the character of strongly nontrivial automorphisms of computable linear orderings. Our main result is that for any η‐like computable linear ordering , such that has no interval of order type η, and such that the order type of is determined by a ‐limitwise monotonic maximal block function, there exists computable such that has no nontrivial automorphism.     

Bounded dagger principles

For an uncountable cardinal κ, let be the assertion that every ω₁‐stationary preserving poset of size is semiproper. We prove that is a strong principle which implies a strong form of Chang's conjecture. We also show that implies that is presaturated.     

Asymptotic density and the Ershov hierarchy

We classify the asymptotic densities of the sets according to their level in the Ershov hierarchy. In particular, it is shown that for , a real is the density of an n‐c.e. set if and only if it is a difference of left‐ reals. Further, we show that the densities of the ω‐c.e. sets coincide with the densities of the sets, and there are ω‐c.e. sets whose density is not the density of an n‐c.e. set for any .