A dual open coloring axiom

We discuss a dual of the Open Coloring Axiom introduced by Abraham et al. [U. Abraham, M. Rubin, S. Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of ₁-dense real order types, Ann. Pure Appl. Logic 29 (2) (1985) 123–206] and show that it follows from a statement about continuous colorings on Polish spaces that is known to be consistent. We mention some consequences of the new axiom and show that implies that all cardinal invariants in Cichoń’s diagram are at least ₂.     

A poset hierarchy

This article extends a paper of Abraham and Bonnet which generalised the famous Hausdorff characterisation of the class of scattered linear orders. They gave an inductively defined hierarchy that characterised the class of scattered posets which do not have infinite incomparability antichains (i.e. have the FAC). We define a larger inductive hierarchy ^κℌ* which characterises the closure of the class of all κ-well-founded linear orders under inversions, lexicographic sums and FAC weakenings. This includes a broader class of “scattered” posets that we call κ-scattered. These posets cannot embed any order such that for every two subsets of size < κ, one being strictly less than the other, there is an element in between. If a linear order has this property and has size κ it is unique and called ℚ(κ). Partial orders such that for every a < b the set {x: a < x < b} has size ≥ κ are called weakly κ-dense, and posets that do not have a weakly κ-dense subset are called strongly κ-scattered. We prove that ^κℌ* includes all strongly κ-scattered FAC posets and is included in the class of all FAC κ-scattered posets. For κ = ℵ₀ the notions of scattered and strongly scattered coincide and our hierarchy is exactly aug(ℌ) from the Abraham-Bonnet theorem. The authors warmly thank Uri Abraham for his many useful suggestions and comments. Mirna Džamonja thanks EPSRC for their support on an EPSRC Advanced Fellowship.     

The set of axiom A diffeomorphisms with no cycles

LetM be aC ^∞ closed manifold and Diff¹ (M) be the space of diffeomorphisms ofM endowed with theC ¹ topology. This paper contains an affirmative answer to the following conjecture raised by Mañé, which is an extension of the stability and Ω-stability conjectures of Palis and Smale, as follows: theC ¹ interior of the subset of diffeomorphism such that all the periodic points are hyperbolic is characterized as the set of diffeomorphisms satisfying Axiom A and the no-cycles condition. Moreover, it is showed that theC ¹ interior of the set of all Kupka-Smale diffeomorphisms coincides with the set of all diffeomorphisms satisfying Axiom A and the strong transversality condition.     

Splitting the Pasch axiom

On the basis of the theory ^– of Pasch-free 2-dimensional geometry, Pasch's axiom is shown to be equivalent to the conjunction of the following two axioms: In any right triangle the hypotenuse is greater than the leg and If AOB is right, B lies between O and C, and D is the footpoint of the perpendicular from B to AC, then the segment OA is greater than the segment BD. This represents an attempt to split the Pasch axiom with respect to ^–. Only the question whether the second of the above two axioms is really weaker than Pasch's axiom, remains open.     

The grounded Martin's axiom

We introduce a variant of Martin's axiom, called the grounded Martin's axiom, or , which asserts that the universe is a c.c.c. forcing extension in which Martin's axiom holds for posets in the ground model. This principle already implies several of the combinatorial consequences of . The new axiom is shown to be consistent with the failure of and a singular continuum. We prove that is preserved in a strong way when adding a Cohen real and that adding a random real to a model of preserves (even though it destroys itself). We also consider the analogous variant of the proper forcing axiom.     

The complete tunnel axiom

The complete tunnel axiom (abbreviated CTA) is that there is a continuous function from the Stone-Čech remainder ω¹ of the integers onto a linearly ordered space, such that every point- inverse has empty interior. The concept of a complete tunnel through a space is defined and its presence in ω¹ is shown equivalent to the CTA. Various modifications of this concept are defined and some are shown equivalent to statements about ω¹ and about compactifications of the Mrówka-Isbell space Ψ. The strongest of these statements is shown to follow from CH, but the weakest is negated by the Proper Forcing Axiom (PFA).     

On the axiom of union

In this paper, we study the union axiom of ZFC. After a brief introduction, we sketch a proof of the folklore result that union is independent of the other axioms of ZFC. In the third section, we prove some results in the theory T:= ZFC minus union. Finally, we show that the consistency of T plus the existence of an inaccessible cardinal proves the consistency of ZFC.     

Parametric axiom of associated consistency

It is proved that the equal allocation of non-separable costs value is characterized if associated consistency is invoked for parametric values $\lambda =\frac{1}{k},k=2,3,\dots ,n?1$, even without the axiom of continuity imposed in Hwang (2006).     

A generalized Feiner hierarchy

We study some properties of a generalization of the Feiner hierarchy of Δ _ω ⁰ -computable functions and consider various conditions for the inclusion of one class of this hierarchy into another.