We prove an isomorphism theorem between the canonical denotation systems for large natural numbers and large countable ordinal numbers, linking two fundamental concepts in Proof Theory. The first one is fast-growing hierarchies. These are sequences of functions on $\mathbb {N}$ obtained through processes such as the ones that yield multiplication from addition, exponentiation from multiplication, etc. and represent the canonical way of speaking about large finite numbers. The second one is ordinal collapsing functions, which represent the best-known method of describing large computable ordinals.

We observe that fast-growing hierarchies can be naturally extended to functors on the categories of natural numbers and of linear orders. The isomorphism theorem asserts that the categorical extensions of binary fast-growing hierarchies to ordinals are isomorphic to denotation systems given by cardinal collapsing functions. As an application of this fact, we obtain a restatement of the subsystem $\Pi ^1_1$-${\mathsf {CA_0}}$ of analysis as a higher-type well-ordering principle asserting that binary fast-growing hierarchies preserve well-foundedness.

It is shown that the determinacy of $G_{\delta \sigma }$ games of length $\omega ^2$ is equivalent to the existence of a transitive model of ${\mathsf {KP}} + {\mathsf {AD}} + \Pi _1\textrm {-MI}_{\mathbb {R}}$ containing $\mathbb {R}$ . Here, $\Pi _1\textrm {-MI}_{\mathbb {R}}$ is the axiom asserting that every monotone $\Pi _1$ operator on the real numbers has an inductive fixpoint.

We isolate two abstract determinacy theorems for games of length $\omega_1$ from work of Neeman and use them to conclude, from large-cardinal assumptions and an iterability hypothesis in the region of measurable Woodin cardinals that

1. (1) if the Continuum Hypothesis holds, then all games of length $\omega_1$ which are provably $\Delta_1$ -definable from a universally Baire parameter (in first-order or $\Omega $ -logic) are determined;

2. (2) all games of length $\omega_1$ with payoff constructible relative to the play are determined; and

3. (3) if the Continuum Hypothesis holds, then there is a model of ${\mathsf{ZFC}}$ containing all reals in which all games of length $\omega_1$ definable from real and ordinal parameters are determined.

We characterize the determinacy of $F_\sigma $ games of length $\omega ^2$ in terms of determinacy assertions for short games. Specifically, we show that $F_\sigma $ games of length $\omega ^2$ are determined if, and only if, there is a transitive model of ${\mathsf {KP}}+{\mathsf {AD}}$ containing $\mathbb {R}$ and reflecting $\Pi _1$ facts about the next admissible set.

As a consequence, one obtains that, over the base theory ${\mathsf {KP}} + {\mathsf {DC}} + ``\mathbb {R}$ exists,” determinacy for $F_\sigma $ games of length $\omega ^2$ is stronger than ${\mathsf {AD}}$ , but weaker than ${\mathsf {AD}} + \Sigma _1$ -separation.

Since the beginning of the Holocene, hunter-gatherers have occupied the central-south Brazilian coast, as it was a very productive estuarine environment. Living as fishers and mollusk gatherers, they built prehistoric shellmounds, known as sambaqui, up to 30 m high, which can still be found today from the Espírito Santo (21°S) to Rio Grande do Sul (32°S) states, constituting an important testimony of paleodiversity and Brazilian prehistory. The chronology of the Sambaqui da Tarioba, situated in Rio das Ostras, Rio de Janeiro, is discussed herein. Selected well-preserved shells of Iphigenia brasiliana and charcoal from fireplaces in sequential layers were used for radiocarbon dating analysis. Based on a statistical model developed using OxCal software, the results indicate that the settlement occupation begun most probably around 3800 cal BP and lasted for up to 5 centuries.

We present the first results from our next-generation microlensing survey, the SuperMACHO project. We are using the CTIO 4m Blanco telescope and the MOSAIC imager to carry out a search for microlensing toward the Large Magellanic Cloud (LMC). We plan to ascertain the nature of the population responsible for the excess microlensing rate seen by the MACHO project. Our observing strategy is optimized to measure the differential microlensing rate across the face of the LMC. We find this derivative to be relatively insensitive to the details of the LMC's internal structure but a strong discriminant between Galactic halo and LMC self lensing. In December 2003 we completed our third year of survey operations. 2003 also marked the first year of real-time microlensing alerts and photometric and spectroscopic followup. We have extracted several dozen microlensing candidates, and we present some preliminary light curves and related information. Similar to the MACHO project, we find SNe behind the LMC to be a significant contaminant - this background has not been completely removed from our current single-color candidate sample. Our follow-up strategy is optimized to discriminate between SNe and true microlensing.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html