Laver and set theory

In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip

Measurable cardinals and the cardinality of Lindel\"of spaces

If it is consistent that there is a measurable cardinal, then it is consistent that all points g-delta Rothberger spaces have "small" cardinality.Comment: 9 pag

Quantitative testing semantics for non-interleaving

This paper presents a non-interleaving denotational semantics for the ?-calculus. The basic idea is to define a notion of test where the outcome is not only whether a given process passes a given test, but also in how many different ways it can pass it. More abstractly, the set of possible outcomes for tests forms a semiring, and the set of process interpretations appears as a module over this semiring, in which basic syntactic constructs are affine operators. This notion of test leads to a trace semantics in which traces are partial orders, in the style of Mazurkiewicz traces, extended with readiness information. Our construction has standard may- and must-testing as special cases

Games with creatures

Many forcing notions obtained using the creature technology are naturally connected with certain integer games

Maximal almost disjoint families, determinacy, and forcing

We study the notion of $\mathcal J$-MAD families where $\mathcal J$ is a Borel ideal on $\omega$. We show that if $\mathcal J$ is an arbitrary $F_\sigma$ ideal, or is any finite or countably iterated Fubini product of $F_\sigma$ ideals, then there are no analytic infinite $\mathcal J$-MAD families, and assuming Projective Determinacy there are no infinite projective $\mathcal J$-MAD families; and under the full Axiom of Determinacy + $V=\mathbf{L}(\mathbb{R})$ there are no infinite $\mathcal J$-mad families. These results apply in particular when $\mathcal J$ is the ideal of finite sets $\mathrm{Fin}$, which corresponds to the classical notion of MAD families. The proofs combine ideas from invariant descriptive set theory and forcing.Comment: 40 page

Games with Filters

This paper has two parts. The first is concerned with a variant of a family of games introduced by Holy and Schlicht, that we call \emph{Welch games}. Player II having a winning strategy in the Welch game of length $\omega$ on $\kappa$ is equivalent to weak compactness. Winning the game of length $2^\kappa$ is equivalent to $\kappa$ being measurable. We show that for games of intermediate length $\gamma$, II winning implies the existence of precipitous ideals with $\gamma$-closed, $\gamma$-dense trees. The second part shows the first is not vacuous. For each $\gamma$ between $\omega$ and $\kappa^+$, it gives a model where II wins the games of length $\gamma$, but not $\gamma^+$. The technique also gives models where for all $\omega_1< \gamma\le\kappa$ there are $\kappa$-complete, normal, $\kappa^+$-distributive ideals having dense sets that are $\gamma$-closed, but not $\gamma^+$-closed

Logical Dreams

We discuss the past and future of set theory, axiom systems and independence results. We deal in particular with cardinal arithmetic