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.
(i) $\alpha^+ = \alpha^{+K}$;
(ii) $\{\beta < \alpha | \beta \mbox{ regular, }\beta^+ = \beta^{+K}\}$ is stationary in $\alpha$;
(iii) $\all A \subseteq \alpha$, $A^\#$ exists.
Theorem Assume there is no inner model with a Woodin cardinal, there is a measurable cardinal $\Omega$ and $K$ is the Steel core model. If $\alpha < \Omega$ is a regular J\'{o}nsson cardinal, then
(i) $\alpha^+ = \alpha^{+K}$;
(ii) $\{\beta < \alpha | \beta \mbox{ regular, }\beta^+ = \beta^{+K}\}$ is stationary in $\alpha$;
Theorem
$Con(ZFC+ \omega^2$-$\Pi^1_1$-Determinacy)
$\Imp$\\$\Imp Con(ZFC +
V = K + \ex \mbox{ a long unfoldable cardinal} \Imp$ \\
$\Imp
Con (ZFC + \forall C(X^{\#}$ exists) +
``$\forall D \subseteq \omega_{1}$ $(D$ is
universally Baire $\Leftrightarrow \exists r \in
{\Bbb R} (D \in L(r)))$'', and this is
set-generically absolute).
We isolate a notion of $\omega$-closed cardinal
which is weaker than an $\omega_1$-Erdos cardinal, and
show that this bounds the first long unfoldable:
Theorem Let $\kappa$ be $\omega$-closed. Then there is a long unfoldable $\lambda <\kappa$.