Arithmetical summability

Arithmetical Knowledge and Arithmetical Definability: Four Studies

by Sean Walsh The subject of this dissertation is arithmetical knowledge and arithmetical definability. The first two chapters contain respectively a critique of a logicist account of a preferred means by which we may legitimately infer to arithmetical truths and a tentative defense of an empiricist account. According to the logicist account, one may infer from quasi-logical truths to patently ...

Arithmetical Measure

We develop arithmetical measure theory along the lines of Lutz [10]. This yields the same notion of “measure 0 set” as considered before by Martin-Löf, Schnorr, and others. We prove that the class of sets constructible by r.e.-constructors, a direct analogue of the classes Lutz devised his resource bounded measures for in [10], is not equal to RE, the class of r.e. sets, and we locate this clas...

Arithmetical Paraphrases

(2) gitl,b, ••', £r\), Ä(|lJi, 1}2, •••, TJ.) exist and are determinate for all integral values of the x, y in £, 77 respectively; the value of g is unchanged when any one of the £ is replaced by its negative; h changes sign and vanishes with each r\. It is emphasized, once for all, that beyond these restrictions f, g, h are wholly arbitrary. As examples of the bar notation, fix,y\) =f(-x,y\) =...

Arithmetical meadows

An inversive meadow is a commutative ring with identity equipped with a total multiplicative inverse operation satisfying 0 −1 = 0. Previously, inversive meadows were shortly called meadows. A divisive meadow is an inversive meadows with the multiplicative inverse operation replaced by a division operation. In the spirit of Peacock's arithmetical algebra, we introduce variants of inversive and ...

Summability in topological spaces