18,906 research outputs found
A note on parameter free Î 1-induction and restricted exponentiation
We characterize the sets of all Π2 and all equation image (= Boolean combinations of Σ1) theorems of IΠ−1 in terms of restricted exponentiation, and use these characterizations to prove that both sets are not deductively equivalent. We also discuss how these results generalize to n > 0. As an application, we prove that a conservation theorem of Beklemishev stating that IΠ−n + 1 is conservative over IΣ−n with respect to equation image sentences cannot be extended to Πn + 2 sentences.Ministerio de Educación y Ciencia MTM2005-08658Ministerio de Educación y Ciencia MTM2008-0643
A note on ordinal exponentiation and derivatives of normal functions
Michael Rathjen and the present author have shown that -bar
induction is equivalent to (a suitable formalization of) the statement that
every normal function has a derivative, provably in . In this
note we show that the base theory can be weakened to . Our
argument makes crucial use of a normal function with and . We will also exhibit a
normal function with and
Intermediate arithmetic operations on ordinal numbers
There are two well-known ways of doing arithmetic with ordinal numbers: the
"ordinary" addition, multiplication, and exponentiation, which are defined by
transfinite iteration; and the "natural" (or Hessenberg) addition and
multiplication (denoted and ), each satisfying its own set of
algebraic laws. In 1909, Jacobsthal considered a third, intermediate way of
multiplying ordinals (denoted ), defined by transfinite iteration of
natural addition, as well as the notion of exponentiation defined by
transfinite iteration of his multiplication, which we denote
. (Jacobsthal's multiplication was later rediscovered by
Conway.) Jacobsthal showed these operations too obeyed algebraic laws. In this
paper, we pick up where Jacobsthal left off by considering the notion of
exponentiation obtained by transfinitely iterating natural multiplication
instead; we will denote this . We show that
and that
;
note the use of Jacobsthal's multiplication in the latter. We also demonstrate
the impossibility of defining a "natural exponentiation" satisfying reasonable
algebraic laws.Comment: 18 pages, 3 table
Exponentiation of the Drell-Yan cross section near partonic threshold in the DIS and MSbar schemes
It has been observed that in the DIS scheme the refactorization of the
Drell-Yan cross section leading to exponentiation of threshold logarithms can
also be used to organize a class of constant terms, most of which arise from
the ratio of the timelike Sudakov form factor to its spacelike counterpart. We
extend this exponentiation to include all constant terms, and demonstrate how a
similar organization may be achieved in the MSbar scheme. We study the
relevance of these exponentiations in a two-loop analysis.Comment: 20 pages, JHEP style, no figure
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