# Abstracts of 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.

• " Some Remarks on the Maximality of Inner Models "
We consider maximality properties of Inner Models, Elementary Embeddings between them, and survey some connections through the concept of J\'{o}nsson cardinal. In particular we give proofs of:
Theorem Assume there is no inner model with a strong cardinal and $K$ is the core model. If $\alpha$ is a regular J\'{o}nsson cardinal, then

(i) $\alpha^+ = \alpha^{+K}$;

(ii) $\{\beta < \alpha | \beta \mbox{ regular, }\beta^+ = \beta^{+K}\}$ is stationary in $\alpha$;

(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 $\alpha$;

• On Gupta-Belnap Revision Theories of truth, Kripkean Fixed points, and the Next Stable Set.
• We consider various concepts associated with the revision theory of truth of Gupta and Belnap. We categorize the notions definable using their {\em theory of circular definitions} as those notions universally definable over {\em the next stable set}. We give a simplified (in terms of definitional complexity) account of {\em varied revision sequences} - as a {\em generalised algorithmic theory of truth}. This enables something of a unification with the Kripkean theory of truth using supervaluation schemes.

• On Unfoldable Cardinals, omega-closed cardinals, and the begiunning of the Inner Model hierarchy
• Let $\kappa$ be a cardinal, and let $H_\kappa$ be the class of sets of hereditary cardinality less than $\kappa$; let $\tau(\kappa)> \kappa$ be the height of the smallest transitive admissible set containing every element of $H_\kappa$. We show that a $ZFC$-definable notion of long unfoldability, a generalisation of weak compactness, implies in the core model $K$, that the mouse order restricted to $H_\kappa$ is as long as $\tau$. (It is known that some weak large cardinal property is necessary for the latter to hold.) In other terms we delimit its strength as follows:

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 $\lambda <\kappa$.

• On Possible Non-homeomorphic Substructures of the Real Line.
• We consider the problem, raised by Kunen and Tall, of whether the real continuum can have non-homeomorphic versions in different submodels of the universe of all sets. We remark that this indeed requires large cardinals, and we obtain: {\bf Theorem} The following are equiconsistent: (i) $ZFC + there exists \kappa a Jonsson cardinal; (ii)$ZFC + there exists M \mbox{ a sufficiently elementary submodel of the universe of sets with reals_M not homeomorphic to the reals.

• On Revision Operators
• We look at various notions of a class of definability operations that generalise inductive operations, and are characterised as revision operations''. More particularly we: (i) characterise the {\em revision theoretically definable} subsets of a countable acceptable structure; (ii) show that the categorical truth set of Belnap and Gupta's theory of truth over arithmetic using {\em fully varied revision} sequences yields a complete $\P^1_3$ set of integers; (iii) the set of {\em stably categorical} sentences using their revision operator $\psi$ is similarly $\P^1_3$ and which is complete in G\"odel's universe of constructible sets $L$; (iv) give an alternative account of a theory of truth - {\em realistic variance} that simplifies full variance, whilst at the same time arriving at Kripkean fixed points.

• Possible World Semantics for Predicates
• with Volker Halbach & Hannes Leitgeb
We develop possible worlds semantics for $\Box$ as a predicate rather than as an operator of sentences. The unary predicate symbol $\Box$ is added to the language of arithmetic (or an extension thereof); this yields the language $\mlsq$. Every world in our possible worlds semantics is the standard model of arithmetic plus an interpretation of $\Box$. We investigate possible--worlds models where $\Box{A}$ is true at a world $w$ if and only if $A$ is true in all worlds seen by $w$. The paradoxes exclude certain frames from being frames for $\Box$ as a predicate. We provide some sufficient and also some necessary conditions on frames that are allowed to act as frames for the predicate approach. Completeness results for certain infinitary systems corresponding to well known modal operator systems are established. We draw some conclusions concerning the current state of the predicate approach to modalities.

• Bounded Martin's Maximum, weak Erdos cardinals and psi_AC with David Aspero
We prove that a form of Erd\H{o}s property (of consistency strength strictly weaker than the Weak Chang's Conjecture at $\omega_1$), together with Bounded Martin's Maximum implies that Woodin's principle $\psi_{AC}$ holds, and therefore $2^{\aleph_0} = \aleph_2$. We also prove that $\psi_{AC}$ implies that every function $f:\omega_1 \into \omega_1$ is bounded by some canonical function on a club and use this to produce a model of Bounded Semiproper Forcing Axiom in which Bounded Martin's Maximum fails.

• Possible World Semantics for Modalities conceived as Predicates
• with Volker Halbach & Hannes Leitgeb
If $\Box$ is conceived as an operator, i.e., an expression that gives applied to a sentence another sentence, the expressive power of the language is severely restricted if compared to a language where $\Box$ is conceived as a predicate, i.e., an expression that yields a sentence if it is applied to a closed term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possible worlds semantics for predicates of sentences has not been developed very far. By introducing possible worlds semantics for the language of arithmetic plus the unary predicate $\Box$, we tackle both problems. Given a frame $\fra$ consisting of a set $W$ of worlds and a binary relation $R$ of $W$, we investigate whether we can interpret $\Box$ at every world in such a way that $\Box{A}$ holds at a world $w\in W$ if and only if $A$ holds at every world $v\in W$ such that $wRv$. The arithmetical vocabulary is interpreted by the standard model at every world. Several paradoxes' (like Montague's Theorem, G\"odel's Second Incompleteness Theorem, McGee's Theorem on the $\omega$-inconsistency of certain truth theories etc.) show that many frames, e.g., reflexive frames do not allow for such an interpretation. We present sufficient and necessary conditions for the existence of a suitable interpretation of $\Box$ at any world. Sound and complete semi--formal systems, corresponding to the modal systems $\textsf{K}$ and $\textsf{K}_4$, for the class of all possible worlds models for predicates and all transitive possible worlds models are presented. We apply our account also to nonstandard models of arithmetic and other languages than the language of arithmetic.

• Post's and other problems in higher type supertasks
• We consider the theory of infinite time Turing machines, both considered as computing functions on integers, but more particularly as computing functions on reals. A number of open problems are raised. In particular we consider what kind of large cardinals are necessary in order that Post's problem for the semi-decidable sets of reals has a negative answer.

• Ultimate truth vis à vis stable truth.
• We show that the set of ultimately true sentences in Hartry Field's Revenge-immune solution to the semantic paradoxes is recursively isomorphic to the set of stably true sentences obtained in Hans Herzberger's revision sequence starting from the null hypothesis. We further remark that this shows that a substantial subsystem of second order number theory is needed to establish the semantic values of sentences over the ground model of the standard natural numbers: \Delta^1_3-CA_0 (second order number theory with a \Delta^1_3-Comprehension Axiom scheme) is insufficient.

• Weak systems of determinacy and arithmetical quasi-inductive definitions.
• We show that the theories: \Pi^1_3-CA_0, \Delta^1_3-CA_0+ \Sigma^0_3-Determinacy, \Delta^1_3-CA_0 + AQI, and \Delta^1_3-CA_0 are in strictly descending order of strength. (Here AQI is the assertion that every arithmetical quasi-inductive definition converges.) More precisely, we locate winning strategies for various \Sigma^0_3-games in the L-hierarchy in order to prove the following:

Theorem 1 KP \proves \Sigma_2 - Separation implies that there exists a \beta-model of \Delta^1_3-CA_0 + \Sigma^0_3-Determinacy.

Alternatively put: \Pi^1__3-CA_0 proves there is a \beta-model of \Delta^1_3-CA_0 + \Sigma^0_3}-Determinacy. The implication is not reversible.

Theorem 2 KP + \Delta_2-Comprehension + \Sigma_2-Collection + AQI does not prove \Sigma^0_3-Determinacy.

Alternatively: \Delta^1_3-CA_0 + AQI does not prove \Sigma^0_3-Determinacy.

• On the possibility, or otherwise, of hypercomputation'.
• We claim that a recent article of P. Cotogno in this journal is based on an incorrect argument concerning the non-computability of diagonal functions. The point is that whilst diagonal functions are not computable by any function of the class over which they diagonalise, there is no "logical impossibility" in their being computed over a wider class. Hence this "logical impossibility" regrettably cannot be used in his argument that no hypercomputation can compute the Halting problem. This seems to lead him into a further error in his analysis of the supposed conventional status of the Infinite Time Turing machines of Hamkins and Lewis. Theorem 1 refutes this directly.