Gap structure after forcing with a coherent Souslin tree

We investigate the effect after forcing with a coherent Souslin tree on the gap structure of the class of coherent Aronszajn trees ordered by embeddability. We shall show, assuming the relativized version PFA(S) of the proper forcing axiom, that the Souslin tree S forces that the class of Aronszajn trees ordered by the embeddability relation is universal for linear orders of cardinality at most ${\aleph_1}$ .     

Characterizations of pretameness and the Ord-cc

It is well known that pretameness implies the forcing theorem, and that pretameness is characterized by the preservation of the axioms of ${\mathsf{ZF}}^{?}$, that is $\mathsf{ZF}$ without the power set axiom, or equivalently, by the preservation of the axiom scheme of replacement, for class forcing over models of $\mathsf{ZF}$. We show that pretameness in fact has various other characterizations, for instance in terms of the forcing theorem, the preservation of the axiom scheme of separation, the forcing equivalence of partial orders and their dense suborders, and the existence of nice names for sets of ordinals. These results show that pretameness is a strong dividing line between well and badly behaved notions of class forcing, and that it is exactly the right notion to consider in applications of class forcing. Furthermore, for most properties under consideration, we also present a corresponding characterization of the Ord-chain condition.     

On the consistency strength of the proper forcing axiom

In recent work, the second author extended combinatorial principles due to Jech and Magidor that characterize certain large cardinal properties so that they can also hold true for small cardinals. For inaccessible cardinals, these modifications have no effect, and the resulting principles still give the same characterization of large cardinals. We prove that the proper forcing axiom PFA implies these principles hold for ω₂. Using this, we argue to show that any of the known methods for forcing models of PFA from a large cardinal assumption requires a strongly compact cardinal. If one forces PFA using a proper forcing, then we get the optimal result that a supercompact cardinal is necessary.     

Global and local choice functions

We prove, by an elementary reflection method, without the use of forcing, that ZFGC (ZF with a global choice function) is a conservative extension of ZFC and that every model of ZFC whose ordinals are cofinal (from the outside) withω can be expanded to a model of ZFGC (without adding new members). The results are then generalized to various weaker forms of the axiom of choice which have global versions.     

Reverse Mathematics and parameter-free Transfer

Recently, conservative extensions of Peano and Heyting arithmetic in the spirit of Nelson's axiomatic approach to Nonstandard Analysis, have been proposed. In this paper, we study the Transfer axiom of Nonstandard Analysis restricted to formulas without parameters. Based on this axiom, we formulate a base theory for the Reverse Mathematics of Nonstandard Analysis and prove some natural reversals, and show that most of these equivalences do not hold in the absence of parameter-free Transfer.     

Injective power objects and the axiom of choice

“The axiom of choice states that any set X of non-empty sets has a choice function—i.e. a function satisfying f(x)∈x for all x∈X. When we want to generalise this to a topos, we have to choose what we mean by non-empty, since in , the three concepts non-empty, inhabited, and injective are equivalent, so the axiom of choice can be thought of as any of the three statements made by replacing “non-empty” by one of these notions.It seems unnatural to use non-empty in an intuitionistic context, so the first interpretation to be used in topos theory was the notion based on inhabited objects. However, Diaconescu (1975) [1] showed that this interpretation implied the law of the excluded middle, and that without the law of the excluded middle, even the finite version of the axiom of choice does not hold! Nevertheless some people still view this as the most appropriate formulation of the axiom of choice in a topos.In this paper, we study the formulation based upon injective objects. We argue that it can be considered a more natural formulation of the axiom of choice in a topos, and that it does not have the undesirable consequences of the inhabited formulation. We show that if it holds for , then it holds in a wide variety of topoi, including all localic topoi. It also has some of the classical consequences of the axiom of choice, although a lot of classical results rely on both the axiom of choice and the law of the excluded middle. An additional advantage of this formulation is that it can be defined for a slightly more general class of categories than just topoi.We also examine the corresponding injective formulations of Zorn’s lemma and the well-order principle. The injective form of Zorn’s lemma is equivalent to the axiom of injective choice, and the injective well-order principle implies the axiom of injective choice.     

Closure Spaces of Finite Type

As well known in a closure space ${(M, \mathfrak{D})}$ satisfying the exchange axiom and the finiteness condition we can complete each independent subset of a generating set of M to a basis of M (Theorem A) and any two bases have the same cardinality (Theorem B) (cf. [1,3,4,7]). In this paper we consider closure spaces of finite type which need not satisfy the finiteness condition but a weaker condition (cf. Theorem 3.5). We prove Theorems A and B for a closure space of finite type satisfying a stronger exchange axiom. An example is given satisfying this strong exchange axiom, but not Theorems A and B.     

Martin’s maximum revisited

We present several results relating the general theory of the stationary tower forcing developed by Woodin with forcing axioms. In particular we show that, in combination with class many Woodin cardinals, the forcing axiom MM⁺⁺ makes the \({\Pi_2}\)-fragment of the theory of \({H_{\aleph_2}}\) invariant with respect to stationary set preserving forcings that preserve BMM. We argue that this is a promising generalization to \({H_{\aleph_2}}\) of Woodin’s absoluteness results for \({L(\mathbb{R})}\). In due course of proving this, we shall give a new proof of some of these results of Woodin. Finally we relate our generic absoluteness results with the resurrection axioms introduced by Hamkins and Johnstone and with their unbounded versions introduced by Tsaprounis.     

Proper forcing axiom and selective separability

We continue the study of Selectively Separable (SS) and, a game-theoretic strengthening, strategically selectively separable spaces (SS⁺) (see Barman, Dow (2011) [1]). The motivation for studying SS⁺ is that it is a property possessed by all separable subsets of Cp(X) for each σ-compact space X. We prove that the winning strategy for countable SS⁺ spaces can be chosen to be Markov. We introduce the notion of being compactlike for a collection of open sets in a topological space and with the help of this notion we prove that there are two countable SS⁺ spaces such that the union fails to be SS⁺, which contrasts the known result about SS spaces. We also prove that the product of two countable SS⁺ spaces is again countable SS⁺. One of the main results in this paper is that the proper forcing axiom, PFA, implies that the product of two countable Fréchet spaces is SS, a statement that was shown in Barman, Dow (2011) [1] to consistently fail. An auxiliary result is that it is consistent with the negation of CH that all separable Fréchet spaces have π-weight at most ω₁.     

Final compactness and separability in regular symmetrizable spaces

In 1966 A. V. Arkhangel'skii posed the following question: Is it true that every regular finally compact symmetrizable space is separable? S. I. Nedev soon showed that a regular finally compact symmetrizable space is hereditarily finally compact. Consequently any counterexample to Arkhangel'skii's conjecture must be an L-space. Applying the technique of iterated forcing we prove that in the axiom systemZFC for set theory it is consistent to assume the existence of a regular (hereditarily) finally compact symmetrizable space X that is nonseparable. Thus it is impossible to prove using the axiom systemZFC that every regular finally compact symmetrizable space is separable. The space X has additional properties as well: it has a basis consisting of open/closed sets (i.e., it is zero-dimensional in the sense ofind, it can be mapped continuously and one-to-one onto a separable metric space, it is α-left and has cardinality ω₁. Bibliography: 25 titles.     

Bounded Namba forcing axiom may fail

We show that in a σ‐closed forcing extension, the bounded forcing axiom for Namba forcing fails. This answers a question of J. T. Moore.     

Some weak fragments of Martin’s axiom related to the rectangle refining property

We introduce the anti-rectangle refining property for forcing notions and investigate fragments of Martin’s axiom for ℵ₁ dense sets related to the anti-rectangle refining property, which is close to some fragment of Martin’s axiom for ℵ₁ dense sets related to the rectangle refining property, and prove that they are really weaker fragments. T. Yorioka was partially supported by Grant-in-aids for Scientific Research No.16340022 and No.18840022.     

Distributive proper forcing axiom and a left-right dichotomy of Cichon's diagram

In this paper, we study distributive proper forcing axiom(DPFA) and prove its consistency with a dichotomy of the Cichon's diagram, relative to certain large cardinal assumption. Namely, we evaluate the cardinal invariants in Cichon's diagram with the first two uncountable cardinals in the way that the left-hand side has the least possible cardinality while the right-hand side has the largest possible value, and preserve the evaluation along the way of forcing DPFA.     

On forcing matching number of boron-nitrogen fullerene graphs

Let G be a graph that admits a perfect matching M. A forcing set S for a perfect matching M is a subset of M such that it is contained in no other perfect matchings of G. The smallest cardinality of forcing sets of M is called the forcing number of M. Computing the minimum forcing number of perfect matchings of a graph is an NP-complete problem. In this paper, we consider boron-nitrogen (BN) fullerene graphs, cubic 3-connected plane bipartite graphs with exactly six square faces and other hexagonal faces. We obtain the forcing spectrum of tubular BN-fullerene graphs with cyclic edge-connectivity 3. Then we show that all perfect matchings of any BN-fullerene graphs have the forcing number at least two. Furthermore, we mainly construct all seven BN-fullerene graphs with the minimum forcing number two.     

Distributive proper forcing axiom and a left-right dichotomy of Cichoń’s diagram

In this paper, we study distributive proper forcing axiom (DPFA) and prove its consistency with a dichotomy of the Cichoń’s diagram, relative to certain large cardinal assumption. Namely, we evaluate the cardinal invariants in Cichoń’s diagram with the first two uncountable cardinals in the way that the left-hand side has the least possible cardinality while the right-hand side has the largest possible value, and preserve the evaluation along the way of forcing DPFA.     

Duality,area-considerations,and the Kalai–Smorodinsky solution

We introduce a new solution concept for $2$-person bargaining problems, which can be considered as the dual of the Equal-Area solution (EA) (see Anbarc? and Bigelow (1994)). Hence, we call it the Dual Equal-Area solution (DEA). We show that the point selected by the Kalai–Smorodinsky solution (see Kalai and Smorodinsky (1975)) lies in between those that are selected by EA and DEA. We formulate an axiom–area-based fairness–and offer three characterizations of the Kalai–Smorodinsky solution in which this axiom plays a central role.     

Bounds on the forcing numbers of bipartite graphs

The forcing number of a perfect matching M of a graph G is the cardinality of the smallest subset of M that is contained in no other perfect matching of G. In this paper, we demonstrate several techniques to produce upper bounds on the forcing number of bipartite graphs. We present a simple method of showing that the maximum forcing number on the 2m×2n rectangle is mn, and show that the maximum forcing number on the 2m×2n torus is also mn. Further, we investigate the lower bounds on the forcing number and determine the conditions under which a previously formulated lower bound is sharp; we provide an example of a family of graphs for which it is arbitrarily weak.     

On the axioms of planes in quaternionic geometry

Summary The axioms of planes in Riemannian geometry and Kaehlerian geometry have been largely studied. In this paper we study axioms for three kinds of planes in Quaternionic geometry: the axiom of quaternionic 4-planes, the axiom of half-quaternionic planes and the axiom of totally real planes. We also give a characterization of quaternionspaceforms in terms of the constancy of the totally real sectional curvatures.     

Bredon homology and ramified covering G-maps

Let G be a finite group. The objective of this paper is twofold. First we prove that the cellular Bredon homology groups with coefficients in an arbitrary coefficient system M are isomorphic to the homotopy groups of certain topological abelian group. And second, we study ramified covering G-maps of simplicial sets and of simplicial complexes. As an application, we construct a transfer for them in Bredon homology, when M is a Mackey functor. We also show that the Bredon-Illman homology with coefficients in M satisfies the equivariant weak homotopy equivalence axiom in the category of G-spaces.     

A preference-based approach to spanning trees and shortest paths problems****

Comparison of solutions in combinatorial problems is often based on an additive cost function inducing a complete order on solutions. We investigate here a generalization of the problem, where preferences take the form of a quasi-transitive binary relation defined on the solutions space. We first propose preference-based search algorithms for two classical combinatorial problems, namely the preferred spanning trees problem (a generalization of the minimum spanning tree problem) and the preferred paths problem (a generalization of the shortest path problem). Then, we introduce a very useful axiom for preference relations called independence. Using this axiom, we establish admissibility results concerning our preference-based search algorithms. Finally, we address the problem of dealing with non-independent preference relations and provide different possible solutions for different particular problems (e.g. lower approximation of the set of preferred solutions for multicriteria spanning trees problems, or relaxation of the independence axiom for interval-valued preferred path problems).