Abstracts of some Preprints and Papers

The following abstracts of papers or preprints are available.

Following Magidor's work "Representing sets of ordinals as countable unions of sets in the Core Model" we ask when a set of ordinals X closed under the canonical Sigma_1 Skolem functions for K_\alpha can be decomposed as a countable union of sets in K. This is always possible if no countably closed filter is put on the K-sequence, but not otherwise. This proviso holds if there is no inner model of a weak Erdos type property. Available as a postscript file.

Equivalences have been established between the determinacy of games played at various intervals of the Hausdorff Difference Hierarchy of Co-analytic sets, with embeddings of inner models, by the work of Martin. Taken together with a theorem of Harrington, these yield a strictly level-by-level description at most levels. We complete this analysis by establishing such equivalences for the remaining cases. Namely we show that $\omega^2\alpha - \Pi^1_1$-Determinacy, for any (recursive) $\alpha$ is equivalent to the existence of a generalised "sharp", generating embeddings of particular inner models. For $\alpha = \delta + 1$, such determinacy follows from (but is strictly weaker than) the existence of an inner model with $\delta$ measurable cardinals, with an $\omega_1$-Erdos cardinal above their supremum. Available as a postscript file.

We consider the following question of Kunen: Does Con(ZFC + "there exists M a transitive inner model and a non-trivial elementary embedding j:M --> V" ) imply Con(ZFC + "there exists a measurable cardinal")? We use core model theory to investigate consequences of the existence of such a j: M --> V. We prove, amongst other things, the existence of such an embedding implies that the core model K is a model of ``there exists a proper class of almost Ramsey cardinals''. Conversely, if On is Ramsey, then such a j,M are definable. We construe this as a negative answer to the question above. We consider further the consequences of strengthening the closure assumption on j. Available as a postscript file.

We show that the halting times of infinite time Turing Machines (considered as ordinals) are themselves all halting outputs of such machines. This gives a clarification of the nature of ``supertasks" or infinite time computations. The proof further yields that the class of sets coded by outputs of halting computations coincides with a level of Goedel's constructible hierarchy: namely that of $L(\lambda)$ where $\lambda$ is the supremum of halting times. A number of other open questions are thereby answered. Available as a postscript file.

We characterise explicitly the decidable predicates on integers of Infinite Time Turing machines, in terms of admissibility theory and the constructible hierarchy. We do this by pinning down $\zeta$, the least ordinal not the length of any eventual output of an Infinite Time Turing machine (halting or otherwise); using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that the natural ordinals associated with the jump operator satisfy a Spector criterion, and correspond to the $L_\zeta$-stables. It also implies that the machines devised are ``$\Sigma_2$ Complete" amongst all such other possible machines. It is shown that least upper bounds of an ``eventual jump" hierarchy exist on an initial segment.

We show that the length of the naturally occurring jump hierarchy of the infinite time Turing degrees is precisely $\omega$, and construct continuum many incomparable such degrees which are minimal over {\bm $0$}. We show that we can apply an argument going back to that of H. Friedman to prove that the set $\infty$-degrees of certain $\Sigma^1_2$-correct $KP$-models of the form $L_\sigma (\sigma < \omega^L_1)$ have minimal upper bounds.