Some Aspects and Examples of Infinity Notions

Our main contribution is a formal definition of what could be called a T-notion of infinity, for set theories T extending ZF. Around this definition we organize some old and new notions of infinity; we also indicate some easy independence proofs. Mathematics Subject Classification: 03E25, 03E20.     

On a problem inspired by determinacy

In this paper, we construct a modelN in which ℵ₁, the only regular uncountable cardinal, is measurable via the club filter. Thus,N is a model for the theory “ZF+κ is regular iffκ is measurable”. This research in this paper was partially supported by NSF Grant DMS-8413736.     

Long Borel hierarchies

We show that there is a model of ZF in which the Borel hierarchy on the reals has length ω₂. This implies that ω₁ has countable cofinality, so the axiom of choice fails very badly in our model. A similar argument produces models of ZF in which the Borel hierarchy has exactly λ + 1 levels for any given limit ordinal λ less than ω₂. We also show that assuming a large cardinal hypothesis there are models of ZF in which the Borel hierarchy is arbitrarily long. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Hahn-Banach Property and the Axiom of Choice

We work in set theory ZF without axiom of choice. Though the Hahn-Banach theorem cannot be proved in ZF, we prove that every Gateaux-differentiable uniformly convex Banach space E satisfies the following continuous Hahn-Banach property: if p is a continuous sublinear functional on E, if F is a subspace of E, and if f: F → ? is a linear functional such that f ≤ p|F then there exists a linear functional g : E → ? such that g extends f and g ≤ p. We also prove that the continuous Hahn-Banach property on a topological vector space E is equivalent to the classical geometrical forms of the Hahn-Banach theorem on E. We then prove that the axiom of Dependent choices DC is equivalent to Ekeland's variational principle, and that it implies the continuous Hahn-Banach property on Gateaux-differentiable Banach spaces. Finally, we prove that, though separable normed spaces satisfy the continuous Hahn-Banach property, they do not satisfy the whole Hahn-Banach property in ZF+DC.     

Odd‐sized partitions of Russell‐sets

In the setting of ZF, i.e., Zermelo–Fraenkel set theory without the Axiom of Choice (AC), we study partitions of Russell‐sets into sets each with exactly n elements (called n ‐ary partitions), for some integer n. We show that if n is odd, then a Russell‐set X has an n ‐ary partition if and only if |X | is divisible by n. Furthermore, we establish that it is relative consistent with ZF that there exists a Russell‐set X such that |X | is not divisible by any finite cardinal n > 1 (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Amphi-ZF : axioms for Conway games

A theory of two-sided containers, denoted ZF₂, is introduced. This theory is then shown to be synonymous to ZF in the sense of Visser (2006), via an interpretation involving Quine pairs. Several subtheories of ZF₂, and their relationships with ZF, are also examined. We include a short discussion of permutation models (in the sense of Rieger–Bernays) over ZF₂. We close with highlighting some areas for future research, mostly motivated by the need to understand non-wellfounded games.     

A note on Grzegorczyk's logic

Grzegorczyk's modal logic (Grz) corresponds to the class of upwards well‐founded partially ordered Kripke frames, however all known proofs of this fact utilize some form of the Axiom of Choice; G. Boolos asked in [1], whether it is provable in plain ZF. We answer his question negatively: Grz corresponds (in ZF) to a class of frames, which does not provably coincide with upwards well‐founded posets in ZF alone. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An Interpretation of the Zermelo-Fraenkel Set Theory and the Kelley-Morse Set Theory in a Positive Theory

An interesting positive theory is the GPK theory. The models of this theory include all hyperuniverses (see [5] for a definition of these ones). Here we add a form of the axiom of infinity and a new scheme to obtain GPK_∞⁺. We show that in these conditions, we can interprete the Kelley-Morse theory (KM) in GPK_∞⁺ (Theorem 3.7). This needs a preliminary property which give an interpretation of the Zermelo-Fraenkel set theory (ZF) in GPK_∞⁺. We also see what happens in the original GPK theory. Before doing this, we first need to study the basic properties of the theory. This is done in the first two sections.     

Undecidability results for restricted universally quantified formulae of set theory

The problem of establishing whether there are sets satisfying a formula in the first order set theoretic language ??_? based on =,?, which involves only restricted quantifiers and has an equivalent ??-prenex form ((??)₀-formula), is neither decidable nor semidecidable. In fact, given any ω-model of ZF – FA, where FA denotes the Foundation Axiom, the set of existential closures of (??)₀-formulae true in the model is a productive set. Undecidability arises even when dealing with restricted universal quantifiers only, provided a predicate is_a_pair(x), meaning that x is a pair of distinct sets, is added to ??_?. If satisfiability refers to ω-models of ZF – FA in which a form of Boffa's antifoundation axiom holds, then semidecidability fails as well; in fact, given any such model, the set of existential closures of formulae involving only restricted quantifiers and the predicate is_a_pair which are true in it, is a productive set. These results are all proved by making use of appropriate codings of Turing machine computations in the set theoretic language.     

Approximation properties of zonal function networks using scattered data on the sphere

A zonal function (ZF) network is a function of the form x↦∑ _k=1 ⁿ c _k (x · y _k), where x and the y _k's are on the unit sphere in q+1 dimensional Euclidean space, and where the y _k's are scattered points. In this paper, we study the degree of approximation by ZF networks. In particular, we compare this degree of approximation with that obtained with the classical spherical harmonics. In many cases of interest, this is the best possible for a given amount of information regarding the target function. We also discuss the construction of ZF networks using scattered data. Our networks require no training in the traditional sense, and provide theoretically predictable rates of approximation. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the monadic theory ofω 1 without A.C.

Buchi inLecture Notes in Mathematics, Decidable Theories II (1973) by using A.C. characterized the theoriesMT[β, <] forβ<ω ₁ and showed thatMT[ω ₁, <] is decidable. We extend Buchi’s results to a larger class of models of ZF (without A.C.) by proving the following under ZF only: (1) There is a choice function which chooses a “good” run of an automaton on countable input (Lemma 5.1). It follows that Buchi’s results cocerning countable ordinals are provable within ZF. (2) Let U.D. be the assertion that there exists a uniform denumeration ofω ₁ (i.e. a functionf: ω ₁ → ω ₁ ^ω such that for everyα<ω ₁,f(α) is a function fromω ontoα). We show that U.D. can be stated as a monadic sentence, and thereforeω ₁ is characterizable by a sentence. (3) LetF be the filter of the cofinal closed subsets ofω ₁. We show that if U.D. holds thenMT[ω ₁, <] is recursive in the first order theory of the boolean algebraP (ω ₁)/F. (We can effectively translate each monadic sentence Σ to a boolean sentenceσ such that [ω ₁, <] ⊨ Σ iffP(ω ₁)/F⊨σ). (4) As every complete boolean algebra theory is recursive we have that in every model of ZF+U.D.,MT[ω ₁, <] is recursive. All our proofs are within ZF. Buchi’s work is often referred to. Following Buchi, the main tool is finite automata. We don’t deal withMT[ω ₁, <] forω ₁ which doesn’t satisfy U.D. The results in this paper appeared in the author’s M.Sc. thesis, which was prepared at the Hebrew University under the supervision of Professor M. Rabin.     

On the topology of a generic fibre of a polynomial function

In this work we study the topologies of the fibres of some families of complex polynomial functions with isolated critical points. We consider polynomials with some transversality conditions at infinity and compute explicitly its global Milnor number μ(f). the invariant λ(f) and therefore the Euler characteristic of its generic fibre. We show that under some mild ransversality condition (transversal at infinity) the behavior of f at infinity is good and the topology of the generic fibre is determined by the two homogeneous parts of higher degree of f Finally we study families of polynomials, called two-term polynomials. This polynomials may have atypical values at infinity. Given such a two-term polynomial f we characterize its atypical values by some invariants of f. These polynomials are a source of interesting examples.     

The topology at infinity of Coxeter groups and buildings

The connectivity at infinity of a finitely generated Coxeter group W is completely determined by topological properties of its nerve L (a finite simplicial complex). For example, W is simply connected at infinity if and only if L and the subcomplexes (where ranges over all simplices in L) are simply connected. This characterization extends to locally finite buildings. Received: May 3, 2001     

Asymptotics of Multivariate Randomness Statistics

Kiefer considered the asymptotics of q-sample Cramer-Von Mises statistics for a fixed q and sample sizes tending to infinity. For univariate observations, McDonald proved the asymptotic normality of these statistics when q goes to infinity while the sample sizes stay fixed. Here we define a class of multivariate randomness statistics that generalizes the class considered by McDonald. We also prove the asymptotic normality of such statistics when the sample sizes stay fixed while q tends to infinity.     

Restricted versions of the Tukey-Teichmüller theorem that are equivalent to the Boolean prime ideal theorem

We formulate a restricted version of the Tukey-Teichmüller Theorem that we denote by (rTT). We then prove that (rTT) and (BPI) are equivalent in ZF and that (rTT) applies rather naturally to several equivalent forms of (BPI): Alexander Subbase Theorem, Stone Representation Theorem, Model Existence and Compactness Theorems for propositional and first-order logic. We also give two variations of (rTT) that we denote by (rTT)⁺ and (rTT)⁺⁺; each is equivalent to (rTT) in ZF. The variation (rTT)⁺⁺ applies rather naturally to various Selection Lemmas due to Cowen, Engeler, and Rado.Dedicated to W.W. Comfort on the occasion of his seventieth birthday.     

Littlewood–Paley Theory and Function Spaces with Alocp Weights

Characterizations via convolutions with smooth compactly supported kernels and other distinguished properties of the weighted Besov–Lipschitz and Triebel–Lizorkin spaces on ℝⁿ with weights that are locally in A_p but may grow or decrease exponentially at infinity are investigated. Square–function characterizations of the weighted L^p and Hardy spaces with the above class of weights are also obtained. A certain local variant of the Calderón reproducing formula is constructed and widely used in the proofs.     

A Semiparametric Model of Estimating Volatility of Diffusion Processes

This article provides a semiparametric model to estimate the diffusion coefficient of a stochastic differential equation from discretely observed data without assuming any functional form of the diffusion coefficient. It is shown that the model has the consistency such that estimated states of the diffusion coefficient converge to the true ones as the number of observations (N) goes to infinity and the sampling time interval (Δt) goes to zero while NΔt going to infinity.     

Strict L<Subscript>∞</Subscript> Isotonic Regression

Given a function f and weights w on the vertices of a directed acyclic graph G, an isotonic regression of (f,w) is an order-preserving real-valued function that minimizes the weighted distance to f among all order-preserving functions. When the distance is given via the supremum norm there may be many isotonic regressions. One of special interest is the strict isotonic regression, which is the limit of p-norm isotonic regression as p approaches infinity. Algorithms for determining it are given. We also examine previous isotonic regression algorithms in terms of their behavior as mappings from weighted functions over G to isotonic functions over G, showing that the fastest algorithms are not monotonic mappings. In contrast, the strict isotonic regression is monotonic.     

Evolution of harmonic maps on manifolds flat at infinity

The aim of this article is to prove a global existence result with small data for the heat flow for harmonic maps from a manifold flat at infinity into a compact manifold. By flat at infinity we mean that the growth rate of the volumes of the balls on the manifold is the same as in the flat space. This is true for any manifold for small enough radius, but is in general not true when the radius of the ball grows. So prescribing such a growth rate also at infinity selects a class of manifolds on which our result holds. In this setting estimates are available for the heat kernel and its gradient on the base manifold. From such estimates it is easy to get L ^p −L ^q bounds for the heat kernel. A contraction principle argument then yields a local existence result in a suitable Sobolev space and a global existence result for small data.     

Estimation of a smooth quantile function under the proportional hazards model

The problem of estimating a smooth quantile function, Q(·), at a fixed point p, 0 < p < 1, is treated under a nonparametric smoothness condition on Q. The asymptotic relative deficiency of the sample quantile based on the maximum likelihood estimate of the survival function under the proportional hazards model with respect to kernel type estimators of the quantile is evaluated. The comparison is based on the mean square errors of the estimators. It is shown that the relative deficiency tends to infinity as the sample size, n, tends to infinity.