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 .     

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 .     

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.     

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.

     

Subtlety and partition relations

We study the subtlety of a cardinal κ and of . We show that, under a certain large cardinal assumption, it is consistent that is subtle for some but κ is not subtle, and the consistency of such a situation is much stronger than the existence of a subtle cardinal. We show that the subtlety of can be characterized by a certain partition relation on . We also study the property of faintness of κ, and the subtlety of with the strong inclusion.     

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.     

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.     

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 .     

The strong reflecting property and Harrington's Principle

In this paper we characterize the strong reflecting property for ‐cardinals for all , characterize Harrington's Principle and its generalization and discuss the relationship between the strong reflecting property for ‐cardinals and Harrington's Principle .     

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.     

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.     

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 .     

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 .     

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).     

Derived models and supercompact measures on

The main result of this paper is Theorem 1.1 , which shows that it is possible for derived models to satisfy “ω₁ is ‐supercompact”. Other constructions of models of this theory are also discussed; in particular, Theorem 4.1 constructs a normal fine measure on and hence a model of “Θ is regular”+“ω₁ is ‐supercompact” from a model of “Θ is measurable”.     

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 .     

Continued fractions of primitive recursive real numbers

A theorem, proven in the present author's Master's thesis 2 states that a real number is ‐computable, whenever its continued fraction is in (the third Grzegorczyk class). The aim of this paper is to settle the matter with the converse of this theorem. It turns out that there exists a real number, which is ‐computable, but its continued fraction is not primitive recursive, let alone in . A question arises, whether some other natural condition on the real number can be combined with ‐computability, so that its continued fraction has low complexity. We give two such conditions. The first is ‐irrationality, based on a notion of Péter, and the second is polynomial growth of the terms of the continued fraction. Any of these two conditions, combined with ‐computability gives an (elementary) continued fraction. We conclude that all irrational algebraic real numbers and the number π have continued fractions in . All these results are generalized to higher levels of Grzegorczyk's hierarchy as well.