DEFINABILITY OF SATISFACTION IN OUTER MODELS

Let M be a transitive model of ZFC. We say that a transitive model of ZFC, N, is an outer model of M if M ⊆ N and ORD ∩ M = ORD ∩ N. The outer model theory of M is the collection of all formulas with parameters from M which hold in all outer models of M (which exist in a universe in which M is countable; this is independent of the choice of such a universe). Satisfaction defined with respect to outer models can be seen as a useful strengthening of first-order logic. Starting from an inaccessible cardinal κ, we show that it is consistent to have a transitive model M of ZFC of size κ in which the outer model theory is lightface definable, and moreover M satisfies V = HOD. The proof combines the infinitary logic L∞,ω, Barwise’s results on admissible sets, and a new forcing iteration of length strictly less than κ+ which manipulates the continuum function on certain regular cardinals below κ. In the appendix, we review some unpublished results of Mack Stanley which are directly related to our topic.

LARGE CARDINALS AND LIGHTFACE DEFINABLE WELL-ORDERS, WITHOUT THE GCH

This paper deals with the question whether the assumption that for every inaccessible cardinal κ there is a well-order of H(κ+) definable over the structure $\langle {\rm{H}}({\kappa ^ + }), \in \rangle$ by a formula without parameters is consistent with the existence of (large) large cardinals and failures of the GCH. We work under the assumption that the SCH holds at every singular fixed point of the ℶ-function and construct a class forcing that adds such a well-order at every inaccessible cardinal and preserves ZFC, all cofinalities, the continuum function, and all supercompact cardinals. Even in the absence of a proper class of inaccessible cardinals, this forcing produces a model of “V = HOD” and can therefore be used to force this axiom while preserving large cardinals and failures of the GCH. As another application, we show that we can start with a model containing an ω-superstrong cardinal κ and use this forcing to build a model in which κ is still ω-superstrong, the GCH fails at κ and there is a well-order of H(κ+) that is definable over H(κ+) without parameters. Finally, we can apply the forcing to answer a question about the definable failure of the GCH at a measurable cardinal.

A Study of L2 Approximations in Atomic Scattering

The approximation of Coulomb continuum functions by an L 2 basis is studied using a Laguerre� function basis which can be extended to completeness. Also studied is the convergence rate of L2 approximations to Born matrix elements for electron impact ionisation as a function of basis�set size. This important class of matrix elements occurs in pseudo�state close-coupling calculations, accounting for scattering to the three�body continuum. Convergence rates in both cases are derived analytically and confirmed numerically. We find that the rate of pointwise convergence of L2 expansions to the continuum function is slow, and of conditional type; however, it is proven that the corresponding ionisation matrix elements converge geometrically, Our result agrees with the behaviour observed in pseudo�state calculations.

Photoionization of magnesium

Photoionization from the 3s subshell of magnesium is investigated in the energy region near threshold where several resonances have been observed. Theoretical cross sections are reported. The latter have been computed using the multi-configuration Hartree–Fock approximation extended to include a continuum function. Results are compared with other theories and experiment.