Locally finite ω‐languages and effective analytic sets have the same topological complexity

Local sentences and the formal languages they define were introduced by Ressayre in 28 . We prove that locally finite ω‐languages and effective analytic sets have the same topological complexity: the Borel and Wadge hierarchies of the class of locally finite ω‐languages are equal to the Borel and Wadge hierarchies of the class of effective analytic sets. In particular, for each non‐null recursive ordinal there exist some ‐complete and some ‐complete locally finite ω‐languages, and the supremum of the set of Borel ranks of locally finite ω‐languages is the ordinal , which is strictly greater than the first non‐recursive ordinal . This gives an answer to the question of the topological complexity of locally finite ω‐languages, which was asked by Simonnet 30 and also by Duparc, Finkel, and Ressayre in 5 . Moreover we show that the topological complexity of a locally finite ω‐language defined by a local sentence φ may depend on the models of the Zermelo‐Fraenkel axiomatic system . Using similar constructions as in the proof of the above results we also show that the equivalence, the inclusion, and the universality problems for locally finite ω‐languages are ‐complete, hence highly undecidable.     

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.     

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.     

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.     

Characterization of ‐like and c0‐like equivalence relations

In this paper, notions of ‐like and c₀‐like equivalence relations are introduced. We characterize the positions of ‐like and c₀‐like equivalence relations in the Borel reducibility hierarchy by comparing them with equivalence relations and .     

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.     

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 .     

A decidable paraconsistent relevant logic: Gentzen system and Routley‐Meyer semantics

In this paper, the positive fragment of the logic of contraction‐less relevant implication is extended with the addition of a paraconsistent negation connective similar to the strong negation connective in Nelson's paraconsistent four‐valued logic . This extended relevant logic is called , and it has the property of constructible falsity which is known to be a characteristic property of . A Gentzen‐type sequent calculus for is introduced, and the cut‐elimination and decidability theorems for are proved. Two extended Routley‐Meyer semantics are introduced for , and the completeness theorems with respect to these semantics are proved.     

On ‐complete equivalence relations on the generalized Baire space

Working with uncountable structures of fixed cardinality, we investigate the complexity of certain equivalence relations and show that if , then many of them are ‐complete, in particular the isomorphism relation of dense linear orders. Then we show that it is undecidable in whether or not the isomorphism relation of a certain well behaved theory (stable, NDOP, NOTOP) is ‐complete (it is, if , but can be forced not to be).     

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 .     

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 .     

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 .     

The logic Ł•

The algebraic category is the image of MV, the category whose objects are the MV‐algebras, by the equivalence (cf. 7 , 8 ). In this paper we define the logic ?^? whose Lindenbaum algebra is an MV‐algebra (object of ), and establish a link between ?^? and the infinite valued ?ukasiewicz logic ?. We define U‐operators, that have properties of universal quantifiers, and establish a bijection that maps an MV‐algebra endowed with a U‐operator (cf. 20 - 22 ) into an MV^?‐algebra endowed with a U‐operator. This map extends to a functor that is a categorical equivalence.     

Lipschitz extensions of definable p‐adic functions

In this paper, we prove a definable version of Kirszbraun's theorem in a non‐Archimedean setting for definable families of functions in one variable. More precisely, we prove that every definable function , where and , that is λ‐Lipschitz in the first variable, extends to a definable function that is λ‐Lipschitz in the first variable.     

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.     

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.     

Totally non‐immune sets

Let be a countable first‐order language and be an ‐structure. “Definable set” means a subset of M which is ‐definable in with parameters. A set is said to be immune if it is infinite and does not contain any infinite definable subset. X is said to be partially immune if for some definable A, is immune. X is said to be totally non‐immune if for every definable A, and are not immune. Clearly every definable set is totally non‐immune. Here we ask whether the converse is true and prove that it is false for every countable structure whose class of definable sets satisfies a mild condition. We investigate further the possibility of an alternative construction of totally non‐immune non‐definable sets with the help of a subclass of immune sets, the class of cohesive sets, as well as with the help of a generalization of definable sets, the semi‐definable ones (the latter being naturally defined in models of arithmetic). Finally connections are found between totally non‐immune sets and generic classes in nonstandard models of arithmetic.     

An alternative Gentzenisation of

In this paper, we give a sequent calculus for the positive contraction‐less relevant logic and we give a proof that it is cut‐free without the use of the truth constant t. Based on , we re‐prove the decidability of the logic .     

choice classes

We study the degree spectrum properties of so called choice classes. A choice class is a class in which no two members have the same Turing degree. This definition leads us to some interesting cardinality properties, basis results and technically innovative constructions which might give us an insight to construct new classes. The presented work can be considered as choice class analogue of the work by Kent and Lewis 15 .     

Computable axiomatizability of elementary classes

The goal of this paper is to generalise Alex Rennet's proof of the non‐axiomatizability of the class of pseudo‐o‐minimal structures. Rennet showed that if is an expansion of the language of ordered fields and is the class of pseudo‐o‐minimal ‐structures (‐structures elementarily equivalent to an ultraproduct of o‐minimal structures) then is not computably axiomatizable. We give a general version of this theorem, and apply it to several classes of structures.