9 research outputs found
Variants on the Berz sublinearity theorem
We consider variants on the classical Berz sublinearity theorem, using only DC, the Axiom of Dependent Choices, rather than AC, the Axiom of Choice, which Berz used. We consider thinned versions, in which conditions are imposed on only part of the domain of the function—results of quantifier-weakening type. There are connections with classical results on subadditivity. We close with a discussion of the extensive related literature
A standard model of Peano arithmetic with no conservative elementary extension
AbstractThe principal result of this paper answers a long-standing question in the model theory of arithmetic [R. Kossak, J. Schmerl, The Structure of Models of Peano Arithmetic, Oxford University Press, 2006, Question 7] by showing that there exists an uncountable arithmetically closed family A of subsets of the set ω of natural numbers such that the expansion NA≔(N,A)A∈A of the standard model N≔(ω,+,×) of Peano arithmetic has no conservative elementary extension, i.e., for any elementary extension NA∗=(ω∗,…) of NA, there is a subset of ω∗ that is parametrically definable in NA∗ but whose intersection with ω is not a member of A. We also establish other results that highlight the role of countability in the model theory of arithmetic.Inspired by a recent question of Gitman and Hamkins, we furthermore show that the aforementioned family A can be arranged to further satisfy the curious property that forcing with the quotient Boolean algebra A/FIN (where FIN is the ideal of finite sets) collapses ℵ1 when viewed as a notion of forcing
Hilbert Spaces Without Countable AC
This article examines Hilbert spaces constructed from sets whose existence is
incompatible with the Countable Axiom of Choice (CC). Our point of view is
twofold: (1) We examine what can and cannot be said about Hilbert spaces and
operators on them in ZF set theory without any assumptions of Choice axioms,
even the CC. (2) We view Hilbert spaces as ``quantized'' sets and obtain some
set-theoretic results from associated Hilbert spaces.Comment: 51 page
Integration on the Surreals
Conway's real closed field No of surreal numbers is a sweeping generalization
of the real numbers and the ordinals to which a number of elementary functions
such as log and exponentiation have been shown to extend. The problems of
identifying significant classes of functions that can be so extended and of
defining integration for them have proven to be formidable. In this paper, we
address this and related unresolved issues by showing that extensions to No,
and thereby integrals, exist for most functions arising in practical
applications. In particular, we show they exist for a large subclass of the
resurgent functions, a subclass that contains the functions that at infinity
are semi-algebraic, semi-analytic, analytic, meromorphic, and Borel summable as
well as generic solutions to linear and nonlinear systems of ODEs possibly
having irregular singularities. We further establish a sufficient condition for
the theory to carry over to ordered exponential subfields of No more generally
and illustrate the result with structures familiar from the surreal literature.
We work in NBG less the Axiom of Choice (for both sets and proper classes),
with the result that the extensions of functions and integrals that concern us
here have a "constructive" nature in this sense. In the Appendix it is shown
that the existence of such constructive extensions and integrals of
substantially more general types of functions (e.g. smooth functions) is
obstructed by considerations from the foundations of mathematics.Comment: This paper supersedes the positive portion of O. Costin, P. Ehrlich
and H. Friedman, "Integration on the surreals: a conjecture of Conway,
Kruskal and Norton", arXiv:1505.02478v3, 24 Aug 2015. A separate paper
superseding the negative portion of the earlier arXiv preprint is in
preparation by H. Friedman and O. Costi
Arrow's theorem, ultrafilters, and reverse mathematics
This paper initiates the reverse mathematics of social choice theory, studying Arrow's impossibility theorem and related results including Fishburn's possibility theorem and the Kirman–Sondermann theorem within the framework of reverse mathematics. We formalise fundamental notions of social choice theory in second-order arithmetic, yielding a definition of countable society which is tractable in RCA0. We then show that the Kirman–Sondermann analysis of social welfare functions can be carried out in RCA0. This approach yields a proof of Arrow's theorem in RCA0, and thus in PRA, since Arrow's theorem can be formalised as a Π01 sentence. Finally we show that Fishburn's possibility theorem for countable societies is equivalent to ACA0 over RCA0
Set theory and the analyst
This survey is motivated by specific questions arising in the similarities and contrasts between (Baire) category and (Lebesgue) measure - category-measure duality and non-duality, as it were. The bulk of the text is devoted to a summary, intended for the working analyst, of the extensive background in set theory and logic needed to discuss such matters: to quote from the Preface of Kelley [Kel]: "what every young analyst should know"