A note on tall cardinals and level by level equivalence

Starting from a model “κ is supercompact” + “No cardinal is supercompact up to a measurable cardinal”, we force and construct a model such that “κ is supercompact” + “No cardinal is supercompact up to a measurable cardinal” + “δ is measurable iff δ is tall” in which level by level equivalence between strong compactness and supercompactness holds. This extends and generalizes both [ 4 , Theorem 1] and the results of 5 .     

Precisely controlling level by level behavior

We construct four models containing one supercompact cardinal in which level by level equivalence between strong compactness and supercompactness and level by level inequivalence between strong compactness and supercompactness are precisely controlled at each non‐supercompact measurable cardinal. In these models, no cardinal κ is ‐supercompact, where is the least inaccessible cardinal greater than κ.     

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 .     

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.     

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”.     

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.     

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.     

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 .     

Inconsistency lemmas in algebraic logic

In this paper, the inconsistency lemmas of intuitionistic and classical propositional logic are formulated abstractly. We prove that, when a (finitary) deductive system is algebraized by a variety , then has an inconsistency lemma—in the abstract sense—iff every algebra in has a dually pseudo‐complemented join semilattice of compact congruences. In this case, the following are shown to be equivalent: (1)  has a classical inconsistency lemma; (2)  has a greatest compact theory and is filtral, i.e., semisimple with EDPC; (3) the compact congruences of any algebra in form a Boolean lattice; (4) the compact congruences of any constitute a Boolean sublattice of the full congruence lattice of . These results extend to quasivarieties and relative congruences. Except for (2), they extend even to protoalgebraic logics, with deductive filters in the role of congruences. A protoalgebraic system with a classical inconsistency lemma always has a deduction‐detachment theorem (DDT), while a system with a DDT and a greatest compact theory has an inconsistency lemma. The converses are false.     

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.     

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.     

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 ).     

The Boolean prime ideal theorem and products of cofinite topologies

We show:

1. The Boolean Prime Ideal theorem is equivalent to each one of the statements:
   1. “For every family of compact spaces, for every family of basic closed sets of the product with the fip there is a family of subbasic closed sets () with the fip such that for every ”.
   2. “For every compact Loeb space (the family of all non empty closed subsets of has a choice function) and for every set X the product is compact”.
2. (: the axiom of choice restricted to families of finite sets) implies “every well ordered product of cofinite topologies is compact” and “every well ordered basic open cover of a product of cofinite topologies has a finite subcover”.
3. (: the axiom of choice restricted to countable families of finite sets) iff “every countable product of cofinite topologies is compact”.
4. (: every filter of extends to an ultrafilter) is equivalent to the proposition “for every compact Loeb space having a base of size and for every set X of size the product is compact”.

     

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.     

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.

     

Forcing a set model of + Harrington's Principle

Let denote third order arithmetic. Let Harrington's Principle, , denote the statement that there is a real x such that every x‐admissible ordinal is a cardinal in . In this paper, assuming there exists a remarkable cardinal with a weakly inaccessible cardinal above it, we force a set model of via set forcing without reshaping.     

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 .     

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.     

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.