Determinacy and weakly Ramsey sets in Banach spaces

We give a sufficient condition for a set of block subspaces in an infinite-dimensional Banach space to be weakly Ramsey. Using this condition we prove that in the Levy-collapse of a Mahlo cardinal, every projective set is weakly Ramsey. This, together with a construction of W. H. Woodin, is used to show that the Axiom of Projective Determinacy implies that every projective set is weakly Ramsey. In the case of we prove similar results for a stronger Ramsey property. And for hereditarily indecomposable spaces we show that the Axiom of Determinacy plus the Axiom of Dependent Choices imply that every set is weakly Ramsey. These results are the generalizations to the class of projective sets of some theorems from W. T. Gowers, and our paper ``Weakly Ramsey sets in Banach spaces.'

     

Beyond Borel-amenability: Scales and superamenable reducibilities

We analyze the degree-structure induced by large reducibilities under the Axiom of Determinacy. This generalizes the analysis of Borel reducibilities given in Alessandro Andretta and Donald A. Martin (2003) [1], Luca Motto Ros (2009) [6] and Luca Motto Ros. (in press) [5] e.g. to the projective levels.     

On the Deductive Strength of Various Distributivity Axioms for Boolean Algebras in Set Theory

In this article, we shall show the generalized notions of distributivity of Boolean algebras have essential relations with several axioms and properties of set theory, say the Axiom of Choice, the Axiom of Dependence Choice, the Prime Ideal Theorems, Martin's axioms, Lebesgue measurability and so on.     

Closure operations that induce big Cohen–Macaulay algebras

We study closure operations over a local domain R that satisfy a set of axioms introduced by Geoffrey Dietz. The existence of a closure operation satisfying the axioms (called a Dietz closure) is equivalent to the existence of a big Cohen–Macaulay module for R. When R is complete and has characteristic $p>0$, tight closure and plus closure satisfy the axioms.We give an additional axiom (the Algebra Axiom), such that the existence of a Dietz closure satisfying this axiom is equivalent to the existence of a big Cohen–Macaulay algebra. We prove that many closure operations satisfy the Algebra Axiom, whether or not they are Dietz closures. We discuss the smallest big Cohen–Macaulay algebra closure on a given ring, and show that every Dietz closure satisfying the Algebra Axiom is contained in a big Cohen–Macaulay algebra closure. This leads to proofs that in rings of characteristic $p>0$, every Dietz closure satisfying the Algebra Axiom is contained in tight closure, and there exist Dietz closures that do not satisfy the Algebra Axiom.     

Projective-type axioms for the hyperbolic plane

It is shown that the Laws of Pappus and Desargues may replace the Axiom of Projectivities in Menger's development of hyperbolic geometry from axioms of alignment.     

Highway toll pricing

For a tolled highway where consecutive segments allow vehicles to enter and exit unrestrictedly, we propose a simple toll pricing method. It is shown that the method is the unique method satisfying the classical axioms of Additivity and Dummy in the cost sharing literature, and the axioms of Toll Upper Bound for Local Traffic and Routing-proofness. We also show that the toll pricing method is the only method satisfying Routing-proofness Axiom and Cost Recovery Axiom. The main axiom in the characterizations is Routing-proofness which says that no vehicle can reduce its toll charges by exiting and re-entering intermediately. In the special case when there is only one unit of traffic (vehicle) for each (feasible) pair of entrance and exit, we show that our toll pricing method is the Shapley value of an associated game to the problem. In the case when there is one unit of traffic entering at each entrance but they all exit at the last exit, our toll pricing method coincides with the well-known airport landing fee solution-the Sequential Equal Contribution rule of Littlechild and Owen (1973).     

模糊粗糙近似算子公理集的独立性

用双论域上的模糊关系定义了广义模糊粗糙近似算子,并讨论了近似算子的性质。用公理刻画了模糊集合值算子,各种公理化的近似算子可以保证找到相应的二元模糊关系,使得由模糊关系通过构造性方法定义的模糊粗糙近似算子恰好就是用公理定义的近似算子。讨论了刻画各种特殊近似算子的公理集的独立性,从而给出各种特殊模糊关系所对应的模糊粗糙近似算子的最小公理集。     

Elementary inductive dichotomy: Separation of open and clopen determinacies with infinite alternatives

We introduce a new axiom called inductive dichotomy, a weak variant of the axiom of inductive definition, and analyze the relationships with other variants of inductive definition and with related axioms, in the general second order framework, including second order arithmetic, second order set theory and higher order arithmetic. By applying these results to the investigations on the determinacy axioms, we show the following. (i) Clopen determinacy is consistency-wise strictly weaker than open determinacy in these frameworks, except second order arithmetic; this is an enhancement of Schweber–Hachtman separation of open and clopen determinacy into the consistency-wise separation. (ii) Hausdorff–Kuratowski hierarchy of differences of opens is faithfully reflected by the hierarchy of consistency strengths of corresponding parameter-free determinacies in the aforementioned frameworks; this result is valid also in second order arithmetic only except clopen determinacy.     

The stationary set splitting game

The stationary set splitting game is a game of perfect information of length ω₁ between two players, unsplit and split, in which unsplit chooses stationarily many countable ordinals and split tries to continuously divide them into two stationary pieces. We show that it is possible in ZFC to force a winning strategy for either player, or for neither. This gives a new counterexample to Σ²₂ maximality with a predicate for the nonstationary ideal on ω₁, and an example of a consistently undetermined game of length ω₁ with payoff de.nable in the second‐order monadic logic of order. We also show that the determinacy of the game is consistent with Martin's Axiom but not Martin's Maximum. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The strength of Mac Lane set theory

Saunders Mac Lane has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, , of Transitive Containment, we shall refer as . His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasises, one that is adequate for much of mathematics. In this paper we show that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke–Platek set theory and even the Axiom of Constructibility to Mac Lane's axioms; our method requires a close study of Axiom H, which was proposed by Mitchell; we digress to apply these methods to subsystems of Zermelo set theory , and obtain an apparently new proof that is not finitely axiomatisable; we study Friedman's strengthening of , and the Forster–Kaye subsystem of , and use forcing over ill-founded models and forcing to establish independence results concerning and ; we show, again using ill-founded models, that proves the consistency of ; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret and Boffa that proves a weak form of Stratified Collection, and that is a conservative extension of for stratified sentences, from which we deduce that proves a strong stratified version of ; we analyse the known equiconsistency of with the simple theory of types and give Lake's proof that an instance of Mathematical Induction is unprovable in Mac Lane's system; we study a simple set theoretic assertion—namely that there exists an infinite set of infinite sets, no two of which have the same cardinal—and use it to establish the failure of the full schema of Stratified Collection in ; and we determine the point of failure of various other schemata in . The paper closes with some philosophical remarks.     

Acute triangulation of a triangle in a general setting revisited

Axiom A16 from Pambuccian (Can. Math. Bull. 53, 534?C541, 2010) is shown to be superfluous as it depends on axioms A1?CA15. This provides a surprisingly simple axiom system in which the acute triangulation with seven triangles can be proved for any triangle, consisting only of A1?CA15 in Pambuccian (Can. Math. Bull. 53, 534?C541, 2010).     

Smoke and mirrors: Combinatorial properties of small cardinals equiconsistent with huge cardinals

Since the work of Godel and Cohen many questions in infinite combinatorics have been shown to be independent of the usual axioms for mathematics, Zermelo Frankel Set Theory with the Axiom of Choice (ZFC). Attempts to strengthen the axioms to settle these problems have converged on a system of principles collectively known as Large Cardinal Axioms.These principles are linearly ordered in terms of consistency strength. As far as is currently known, all natural independent combinatorial statements are equiconsistent with some large cardinal axiom. The standard techniques for showing this use forcing in one direction and inner model theory in the other direction.The conspicuous open problems that remain are suspected to involve combinatorial principles much stronger than the large cardinals for which there is a current fine-structural inner model theory for.The main results in this paper show that many standard constructions give objects with combinatorial properties that are, in turn, strong enough to show the existence of models with large cardinals are larger than any cardinal for which there is a standard inner model theory.     

Properties of almost all graphs and complexes

There are many results in the literature asserting that almost all or almost no graphs have some property. Our object is to develop a general logical theorem that will imply almost all of these results as corollaries. To this end, we propose the first-order theory of almost all graphs by presenting Axiom n which states that for each sequence of 2n distinct vertices in a graph (u₁, …, u_n, v₁, …, v_n), there exists another vertex w adjacent to each u₁ and not adjacent to any v_i. A simple counting argument proves that for each n, almost all graphs satisfy Axiom n. It is then shown that any sentence that can be stated in terms of these axioms is true in almost all graphs or in almost none. This has several immediate consequences, most of which have already been proved separately including: (1) For any graph H, almost all graphs have an induced subgraph isomorphic to H. (2) Almost no graphs are planar, or chordal, or line graphs. (3) Almost all grpahs are connected wiht diameter 2. It is also pointed out that these considerations extend to digraphs and to simplicial complexes.     

Characterizing properties of approximate solutions for optimization problems

Approximate solutions for optimization problems become of interest if the ‘true’ optimum cannot be found: this may happen for the simple reason that an optimum does not exist or because of the ‘bounded rationality’ (or bounded accuracy) of the optimizer. This paper characterizes several approximate solutions by means of consistency and additional requirements. In particular we consider invariance properties. We prove that, where the domain contains optimization problems without maximum, there is no non-trivial consistent solution satisfying non-emptiness, translation and multiplication invariance. Moreover, we show that the class of ‘satisficing’ solutions is obtained, if the invariance axioms are replaced with Chernoff’s Choice Axiom.     

On Stone's theorem and the Axiom of Choice

It is a well established fact that in Zermelo-Fraenkel set theory, Tychonoff's Theorem, the statement that the product of compact topological spaces is compact, is equivalent to the Axiom of Choice. On the other hand, Urysohn's Metrization Theorem, that every regular second countable space is metrizable, is provable from just the ZF axioms alone. A. H. Stone's Theorem, that every metric space is paracompact, is considered here from this perspective. Stone's Theorem is shown not to be a theorem in ZF by a forcing argument. The construction also shows that Stone's Theorem cannot be proved by additionally assuming the Principle of Dependent Choice.

     

Games of length

This note combines an unpublished theorem of Woodin's about and Uniformisation with combinatorial arguments of Blass' to get a startling consequence for games on of length : The determinacy of these games is equivalent to the Axiom of Real Determinacy.

     

The Bounded Axiom A Forcing Axiom

We introduce the Bounded Axiom A Forcing Axiom (BAAFA). It turns out that it is equiconsistent with the existence of a regular ∑₂‐correct cardinal and hence also equiconsistent with BPFA. Furthermore we show that, if consistent, it does not imply the Bounded Proper Forcing Axiom (BPFA) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coherent sequences and threads

The combinatorial principle □(λ) says that there is a coherent sequence of length λ that cannot be threaded. If λ=κ⁺, then the related principle κ_□ implies □(λ). Let κ?ℵ₂ and X⊆κ. Assume both □(κ) and κ_□ fail. Then there is an inner model N with a proper class of strong cardinals such that X∈N. If, in addition, κ?ℵ₀² and n<ω, then there is an inner model Mn(X) with n Woodin cardinals such that X∈Mn(X). In particular, by Martin and Steel, Projective Determinacy holds. As a corollary to this and results of Todorcevic and Velickovic, the Proper Forcing Axiom for posets of cardinality ⁺(ℵ₀²) implies Projective Determinacy.