Continuum function

Abstract: Previous Hylleraas calculations of $\mathcal{E}=l\mathrm{QHQ}g$ for the lowest $^{1}P$ autoionization state of helium are here supplemented by calculations of the shift ($\ensuremath{\Delta}$), width ($\ensuremath{\Gamma}$), and shape parameter ($q$) using a ($ls,2\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{p}$) pseudostate nonresonant continuum function. The function is constructed to eliminate the dominant ($2s,2p$) configuration of the autoionization state, while at the same time containing three variationally determined radial functions. Both $\ensuremath{\Delta}$ and $q$ are also shown to contain contributions from the discrete part of the nonresonant spectrum, although quantitatively that contribution is found to be small. Final results change previous polarized-orbital results minimally, which means that the resonance position, $\mathcal{E}=\mathrm{E}+\ensuremath{\Delta}$, continues to be on the edge of the experimental error, and $q$ remains somewhat outside the experimental result. Further relativistic corrections are briefly discussed, but a simple argument indicates that they are not likely to explain the differences with experiment. It is concluded that more-accurate experiments should be carried out.

Abstract: §1. Introduction. Among the most remarkable discoveries in set theory in the last quarter century is the rich structure of the arithmetic of singular cardinals, and its deep relationship to large cardinals. The problem of finding a complete set of rules describing the behavior of the continuum function 2ℵα for singular ℵα's, known as the Singular Cardinals Problem, has been attacked by many different techniques, involving forcing, large cardinals, inner models, and various combinatorial methods. The work on the singular cardinals problem has led to many often surprising results, culminating in a beautiful theory of Saharon Shelah called the pcf theory (“pcf” stands for “possible cofinalities”). The most striking result to date states that if 2ℵ n < ℵω for every n = 0, 1, 2, …, then 2ℵω < ℵω4. In this paper we present a brief history of the singular cardinals problem, the present knowledge, and an introduction into Shelah's pcf theory. In Sections 2, 3 and 4 we introduce the reader to cardinal arithmetic and to the singular cardinals problems. Sections 5, 6, 7 and 8 describe the main results and methods of the last 25 years and explain the role of large cardinals in the singular cardinals problem. In Section 9 we present an outline of the pcf theory. §2. The arithmetic of cardinal numbers. Cardinal numbers were introduced by Cantor in the late 19th century and problems arising from investigations of rules of arithmetic of cardinal numbers led to the birth of set theory. The operations of addition, multiplication and exponentiation of infinite cardinal numbers are a natural generalization of such operations on integers. Addition and multiplication of infinite cardinals turns out to be simple: when at least one of the numbers κ, λ is infinite then both κ + λ and κ · λ are equal to max {κ, λ}. In contrast with + and ·, exponentiation presents fundamental problems. In the simplest nontrivial case, 2 κ represents the cardinal number of the power set P(κ), the set of all subsets of κ. (Here we adopt the usual convention of set theory that the number κ is identified with a set of cardinality κ, namely the set of all ordinal numbers smaller than κ.