Interpretations of the alternative set theory

Summary We show an axiom A such that there is no nontrivial interpretation of the alternative set theory (AST) inAST+A keeping , sets and the class of all standard natural numbers. Furthermore, there is no interpretation ofAST inAST without the prolongation axiom, but there is an interpretation ofAST in the theory having the prolongation axiom and the basic set-theoretical axioms only.     

Some remarks on category of the real line

We find a characterization of the covering number , of the real line in terms of trees. We also show that the cofinality of is greater than or equal to for every where ( is the additivity number of the ideal of all Lebesgue measure zero sets) is the least cardinal number k for which the statement: fails. Received: 19 October 1994 / Revised version: 12 December 1996     

Global singularization and the failure of SCH

We say that κ is μ-hypermeasurable (or μ-strong) for a cardinal μ≥κ⁺ if there is an embedding j:V→M with critical point κ such that H(μ)^V is included in M and j(κ)>μ. Such a j is called a witnessing embedding.Building on the results in [7], we will show that if V satisfies GCH and F is an Easton function from the regular cardinals into cardinals satisfying some mild restrictions, then there exists a cardinal-preserving forcing extension V^∗ where F is realised on all V-regular cardinals and moreover, all F(κ)-hypermeasurable cardinals κ, where F(κ)>κ⁺, with a witnessing embedding j such that either j(F)(κ)=κ⁺ or j(F)(κ)≥F(κ), are turned into singular strong limit cardinals with cofinality ω. This provides some partial information about the possible structure of a continuum function with respect to singular cardinals with countable cofinality.As a corollary, this shows that the continuum function on a singular strong limit cardinal κ of cofinality ω is virtually independent of the behaviour of the continuum function below κ, at least for continuum functions which are simple in that 2^α∈{α⁺,α⁺⁺} for every cardinal α below κ (in this case every κ⁺⁺-hypermeasurable cardinal in the ground model is witnessed by a j with either j(F)(κ)≥F(κ) or j(F)(κ)=κ⁺).     

On a problem of Foreman and Magidor

A question of Foreman and Magidor asks if it is consistent for every sequence of stationary subsets of the _ns for 1n< to be mutually stationary. We get a positive answer to this question in the context of the negation of the Axiom of Choice. We also indicate how a positive answer to a generalized version of this question in a choiceless context may be obtained.The author wishes to thank James Cummings for helpful correspondence on the subject matter of this paper. The author also wishes to thank the referee and Andreas Blass, the corresponding editor, for helpful comments and suggestions that have been incorporated into this version of the paper. 03E35, 03E55 Supercompact cardinal – Indestructibility – Almost huge cardinal – Mutual stationarity – Symmetric inner modelRevised version: 6 June 2004     

A non-implication between fragments of Martin’s Axiom related to a property which comes from Aronszajn trees

We introduce a property of forcing notions, called the anti-, which comes from Aronszajn trees. This property canonically defines a new chain condition stronger than the countable chain condition, which is called the property .In this paper, we investigate the property . For example, we show that a forcing notion with the property does not add random reals. We prove that it is consistent that every forcing notion with the property has precaliber ℵ₁ and for forcing notions with the property fails. This negatively answers a part of one of the classical problems about implications between fragments of .     

Parameterized partition relations on the real numbers

We consider several kinds of partition relations on the set of real numbers and its powers, as well as their parameterizations with the set of all infinite sets of natural numbers, and show that they hold in some models of set theory. The proofs use generic absoluteness, that is, absoluteness under the required forcing extensions. We show that Solovay models are absolute under those forcing extensions, which yields, for instance, that in these models for every well ordered partition of there is a sequence of perfect sets whose product lies in one piece of the partition. Moreover, for every finite partition of there is and a sequence of perfect sets such that the product lies in one piece of the partition, where is the set of all infinite subsets of X. The proofs yield the same results for Borel partitions in ZFC, and for more complex partitions in any model satisfying a certain degree of generic absoluteness. This work was supported by the research projects MTM 2005-01025 of the Spanish Ministry of Science and Education and 2005SGR-00738 of the Generalitat de Catalunya. A substantial part of the work was carried out while the second-named author was ICREA Visiting Professor at the Centre de Recerca Matemàtica in Bellaterra (Barcelona), and also during the first-named author’s stays at the Instituto Venezolano de Investigaciones Científicas and the California Institute of Technology. The authors gratefully acknowledge the support provided by these institutions.     

Models of Cohen measurability

We show that in contrast with the Cohen version of Solovay's model, it is consistent for the continuum to be Cohen-measurable and for every function to be continuous on a non-meagre set.     

The γ-borel conjecture

In this paper we prove that it is consistent that every -set is countable while not every strong measure zero set is countable. We also show that it is consistent that every strong -set is countable while not every -set is countable. On the other hand we show that every strong measure zero set is countable iff every set with the Rothberger property is countable.Thanks to Boise State University for support during the time this paper was written and to Alan Dow for some helpful discussions and to Boaz Tsaban for some suggestions to improve an earlier version.     

The differences between Kurepa trees and Jech-Kunen trees

Summary By an ₁ we mean a tree of power ₁ and height ₁. An ₁-tree is called a Kurepa tree if all its levels are countable and it has more than ₁ branches. An ₁-tree is called a Jech-Kunen tree if it has branches for some strictly between ₁ and . In Sect. 1, we construct a model ofCH plus , in which there exists a Kurepa tree with not Jech-Kunen subtrees and there exists a Jech-Kunen tree with no Kurepa subtrees. This improves two results in [Ji1] by not only eliminating the large cardinal assumption for [Ji1, Theorem 2] but also handling two consistency proofs of [Ji1, Theorem 2 and Theorem 3] simultaneously. In Sect. 2, we first prove a lemma saying that anAxiom A focing of size ₁ over Silver's model will not produce a Kurepa tree in the extension, and then we apply this lemma to prove that, in the model constructed for Theorem 2 in [Ji1], there exists a Jech-Kunen tree and there are no Kurepa trees.     

A model of Cummings and Foreman revisited

This paper concerns the model of Cummings and Foreman where from ω   supercompact cardinals they obtain the tree property at each _ℵn${\mathrm{\aleph }}_n$ for 2≤n<ω$2\le n<\omega$. We prove some structural facts about this model. We show that the combinatorics at _ℵω+1${\mathrm{\aleph }}_{\omega +1}$ in this model depend strongly on the properties of _ω1${\omega }_1$ in the ground model. From different ground models for the Cummings–Foreman iteration we can obtain either _ℵω+1∈I[_ℵω+1]${\mathrm{\aleph }}_{\omega +1}\in I[{\mathrm{\aleph }}_{\omega +1}]$ and every stationary subset of _ℵω+1${\mathrm{\aleph }}_{\omega +1}$ reflects or there are a bad scale at _ℵω${\mathrm{\aleph }}_{\omega }$ and a non-reflecting stationary subset of _ℵω+1∩cof(_ω1)${\mathrm{\aleph }}_{\omega +1}\cap \mathrm{cof}({\omega }_1)$. We also prove that regardless of the ground model a strong generalization of the tree property holds at each _ℵn${\mathrm{\aleph }}_n$ for n≥2$n\ge 2$.     

Undecidability of automorphism groups

We give a simple (and easy to apply) technique that gives the undecidability of the theory of many automorphism groups: Let G be a group of automorphisms of a structure. Suppose that is not the identity and has no non-singleton finite orbits. If the centraliser of g is transitive on the support of g and satisfies a further technical condition, then the subgroup generated by g is equal to the double centraliser of g. Thus if G contains such an element g that is conjugate to all its positive powers, then one can interpret addition and multiplication of natural numbers in the theory of G using the parameter g; consequently, G has undecidable theory. Received: 9 October 2000 / in final form: 2 October 2001 / Published online: 29 April 2002     

Universality of Embeddability Relations for Coloured Total Orders

Some examples of Σ₁ ¹-universal preorders are presented, in the form of various relations of embeddability between countable coloured total orders. As an application, strengthening a theorem of (Marcone, A. and Rosendal, C.: The Complexity of Continuous Embeddability between Dendrites, J. Symb. Log. 69 (2004), 663–673), the Σ₁ ¹-universality of continuous embeddability for dendrites whose branch points have order 3 is obtained.     

On guessing generalized clubs at the successors of regulars

König, Larson and Yoshinobu initiated the study of principles for guessing generalized clubs, and introduced a construction of a higher Souslin tree from the strong guessing principle.Complementary to the author’s work on the validity of diamond and non-saturation at the successor of singulars, we deal here with a successor of regulars. It is established that even the non-strong guessing principle entails non-saturation, and that, assuming the necessary cardinal arithmetic configuration, entails a diamond-type principle which suffices for the construction of a higher Souslin tree.We also establish the consistency of GCH with the failure of the weakest form of generalized club guessing. This, in particular, settles a question from the original paper.     

Cardinal characteristics and projective wellorders

Using countable support iterations of S-proper posets, we show that the existence of a definable wellorder of the reals is consistent with each of the following: d<c, b<a=s and b<g.     

Generic embeddings associated to an indestructibly weakly compact cardinal

I use generic embeddings induced by generic normal measures on P_κ(λ) that can be forced to exist if κ is an indestructibly weakly compact cardinal. These embeddings can be applied in order to obtain the forcing axioms in forcing extensions. This has consequences in : The Singular Cardinal Hypothesis holds above κ, and κ has a useful Jónsson-like property. This in turn implies that the countable tower Q_<κ works much like it does when κ is a Woodin limit of Woodin cardinals. One consequence is that every set of reals in the Chang model is Lebesgue measurable and has the Baire Property, the Perfect Set Property and the Ramsey Property. So indestructible weak compactness has effects on cardinal arithmetic high up and also on the structure of sets of real numbers, down low, similar to supercompactness.     

Measurable cardinals and the cardinality of Lindelöf spaces

We obtain from the consistency of the existence of a measurable cardinal the consistency of “small” upper bounds on the cardinality of a large class of Lindelöf spaces whose singletons are Gδ sets.     

Constructibility in higher order arithmetics

Summary We define and investigate constructibility in higher order arithmetics. In particular we get an interpretation ofn-order arithmetic inn-order arithmetic without the scheme of choice such that and the property to be a well-ordering are absolute in it and such that this interpretation is minimal among such interpretations.