We show in ZFC that if there is no proper class inner model with a Woodin cardinal, then there is an absolutely definablecore modelthat is close toVin various ways.

In this paper we shall answer some questions in the set theory of L, the universe of all sets constructible from the reals. In order to do so, we shall assume ADL, the hypothesis that all 2-person games of perfect information on ω whose payoff set is in L are determined. This is by now standard practice. ZFC itself decides few questions in the set theory of L, and for reasons we cannot discuss here, ZFC + ADL yields the most (...) interesting “completion” of the ZFC-theory of L.ADL implies that L satisfies “every wellordered set of reals is countable”, so that the axiom of choice fails in L. Nevertheless, there is a natural inner model of L, namely HODL, which satisfies ZFC.. The superscript “L” indicates, here and below, that the notion in question is to be interpreted in L.) HODL is reasonably close to the full L, in ways we shall make precise in § 1. The most important of the questions we shall answer concern HODL: what is its first order theory, and in particular, does it satisfy GCH?These questions first drew attention in the 70's and early 80's. (shrink)

From the early ninth century until about eight centuries later, the Middle East witnessed a series of both simple and systematic astronomical observations for the purpose of testing contemporary astronomical tables and deriving the fundamental solar, lunar, and planetary parameters. Of them, the extensive observations of lunar eclipses available before 1000 AD for testing the ephemeredes computed from the astronomical tables are in a relatively sharp contrast to the twelve lunar observations that are pertained to the four extant accounts of (...) the measurements of the basic parameters of Ptolemaic lunar model. The last of them are Taqī al-Dīn Muḥammad b. Ma‘rūf’s trio of lunar eclipses observed from Istanbul, Cairo, and Thessalonica in 1576–1577 and documented in chapter 2 of book 5 of his famous work, Sidrat muntaha al-afkar fī malakūt al-falak al-dawwār. In this article, we provide a detailed analysis of the accuracy of his solar and lunar observations. (shrink)

We investigated the causes and impact of procrastination on “slippery deadlines,” where the due date is ill-defined and can be autonomously extended, using the unique applied setting of grievance arbitration across two studies. In Study One, using 3 years of observed performance data derived from Canadian arbitration cases and a survey of leading arbitrators, we examined the effect of individual differences, self-regulatory skills, workloads and task characteristics on time delay. Observed delay here is a critical criterion, where justice is emphasized (...) to be swift and sure. Multilevel Modeling established trait procrastination as a substantive predictor of observed delay, equivalent to the environmental contributors of expediting the arbitration procedure or grievance complexity. Also, despite substantive negative consequence of delay for both arbitrators and their clients, arbitrators who scored one standard deviation above the mean in procrastination took approximately 83 days to write their decisions compared to the 26 days for arbitrators one standard deviation below the mean. In Study Two, we conducted a replication and extension survey with a much larger group of American arbitrators. Consistent with Temporal Motivation Theory, trait procrastination was largely explained by expectancy, value, and sensitivity to time related traits and skills, which together accounted for majority of the variance in trait procrastination, leaving little left for other explanations. For example, perfectionism connection to procrastination appears to be distal, being largely mediated by each of TMT’s core variables. Finally, procrastination was largely synonymous with a deadline pacing style, indicating that observed delay can be used as a proxy for procrastination as long as little or no prior work was done. In all, our results indicate that procrastination is rampant in the workplace and has seriously detrimental effects. (shrink)

Posner [6] has shown, by a nonuniform proof, that every ▵ 0 2 degree has a complement below 0'. We show that a 1-generic complement for each ▵ 0 2 set of degree between 0 and 0' can be found uniformly. Moreover, the methods just as easily can be used to produce a complement whose jump has the degree of any real recursively enumerable in and above $\varnothing'$ . In the second half of the paper, we show that the complementation (...) of the degrees below 0' does not extend to all recursively enumerable degrees. Namely, there is a pair of recursively enumerable degrees a above b such that no degree strictly below a joins b above a. (This result is independently due to S. B. Cooper.) We end with some open problems. (shrink)

We present equiconsistency results at the level of subcompact cardinals. Assuming SBHδ, a special case of the Strategic Branches Hypothesis, we prove that if δ is a Woodin cardinal and both □ and □δ fail, then δ is subcompact in a class inner model. If in addition □ fails, we prove that δ is Π12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_1^2}$$\end{document} subcompact in a class inner model. These results are optimal, and lead to equiconsistencies. As a corollary (...) we also see that assuming the existence of a Woodin cardinal δ so that SBHδ holds, the Proper Forcing Axiom implies the existence of a class inner model with a Π12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_1^2}$$\end{document} subcompact cardinal. Our methods generalize to higher levels of the large cardinal hierarchy, that involve long extenders, and large cardinal axioms up to δ is δ+ supercompact for all n < ω. We state some results at this level, and indicate how they are proved. (shrink)

Let L[E] be an iterable tame extender model. We analyze to which extent L[E] knows fragments of its own iteration strategy. Specifically, we prove that inside L[E], for every cardinal K which is not a limit of Woodin cardinals there is some cutpoint t K > a>ω1 are cardinals, then ◊$_{K.\lambda }^* $ holds true, and if in addition λ is regular, then ◊$_{K.\lambda }^* $ holds true.

We show that every sufficiently iterable countable mouse has a unique iteration strategy whose associated iteration maps are lexicographically minimal. This enables us to extend the results of [3] on the good behavior of the standard parameter from tame mice to arbitrary mice.

We introduce two new iteration games: the game , which is a strengthening of the weak iteration game, and the game , which is somewhat stronger than but weaker than the full iteration game of length ω1. For a countable M elementarily embeddable in some Vη, with two players I and II, we can show that II wins and that I does not win.

In this paper we shall answer some questions in the set theory of L, the universe of all sets constructible from the reals. In order to do so, we shall assume ADL, the hypothesis that all 2-person games of perfect information on ω whose payoff set is in L are determined. This is by now standard practice. ZFC itself decides few questions in the set theory of L, and for reasons we cannot discuss here, ZFC + ADL yields the most (...) interesting “completion” of the ZFC-theory of L.ADL implies that L satisfies “every wellordered set of reals is countable”, so that the axiom of choice fails in L. Nevertheless, there is a natural inner model of L, namely HODL, which satisfies ZFC.. The superscript “L” indicates, here and below, that the notion in question is to be interpreted in L.) HODL is reasonably close to the full L, in ways we shall make precise in § 1. The most important of the questions we shall answer concern HODL: what is its first order theory, and in particular, does it satisfy GCH?These questions first drew attention in the 70's and early 80's. (shrink)

The Repugnant Conclusion served an important purpose in catalyzing and inspiring the pioneering stage of population ethics research. We believe, however, that the Repugnant Conclusion now receives too much focus. Avoiding the Repugnant Conclusion should no longer be the central goal driving population ethics research, despite its importance to the fundamental accomplishments of the existing literature.

This study examines Babylonian records of the new moon interval NA (sunset to moonset on the day of first lunar visibility) and the connection of this interval to the length of the moon. I show that the NA intervals in the Normal Star Almanacs were computed using the goal-year method and were then used in turn to predict the lengths of each month of the year. I further argue that these predicted month lengths, adjusted occasionally on the basis of observation (...) in cases where the moon’s visibility was considered marginal, formed the basis of the Late Babylonian calendar. (shrink)

In this paper we publish three fragments of a cuneiform tablet that, when complete, contained the dates and zodiacal positions of Saturn’s synodic phenomena for roughly 60 years. The text is unique in containing comparisons of computed data with observations. Through an analysis of the preserved data we propose that the dates and positions were computed by an otherwise unknown two-zone System A-type scheme and show that the computed data in the tablet can be dated to the fourth century BC. (...) This early date and the comparisons with observations suggest that the text was produced during the period of active development of the planetary systems. (shrink)

We produce counterexamples to the unique and cofinal branches hypotheses, assuming (slightly less than) the existence of a cardinal which is strong past a Woodin cardinal.

In this paper I present an alternative reading and interpretation of the cuneiform tablet BM 76829. I suggest that the obverse of the tablet contains a simple astrological scheme linking the sign of the zodiac in which a child is born to the maximum length of life, and that the reverse contains a copy of a scheme relating parts of the body to the signs of the zodiac.

Otto Neugebauer’s early academic career was marked by a series of transitions. His interests shifted from physics to mathematics, and finally to the history of ancient mathematics and exact sciences. Yet even from his early years in Graz, Neugebauer was strongly attracted to the mathematical culture of Göttingen. When he arrived there in 1922, he quickly established a strong personal friendship with Richard Courant, the newly appointed Director of the Mathematics Institute. Neugebauer and Courant worked together closely up until 1933, (...) when the Nazi government decimated the Göttingen scientific community. In this essay, Neugebauer’s historical work and his vision for a new approach to the study of the exact sciences are viewed through the prism of these events. By so doing, one can easily appreciate how Neugebauer’s scholarship reflects the ideals he and Courant shared as leading representatives of the Göttingen mathematical tradition. (shrink)

Babylonian methods for predicting planetary phenomena using the so-called goal-year periods are well known. Texts known as Goal-Year Texts contain collections of the observational data needed to make predictions for a given year. The predictions were then recorded in Normal Star Almanacs and Almanacs. Large numbers of Goal-Year Texts, Normal Star Almanacs and Almanacs are attested from the early third century BC onward. A small number of texts dating from before the third century present procedures for using the goal-year periods (...) to predict planetary phenomena. In addition, two texts, one dating to the late sixth century BC and the other to the late fifth century BC, contain planetary data which was probably predicted using these methods. In this article, I discuss a further example of a tablet dating from before the third century BC which contains planetary data predicted using the goal-year periods. I show that the planetary phenomena contained in this tablet can be dated to the twelfth year of the reign of Artaxerxes III (347/6 BC) and that they were predicted using goal-year periods without the application of the kind of corrections which were used in the third century BC texts in order to produce more accurate predictions. (shrink)

Significant progress has been made in understanding Mesopotamian astronomy and astrology since the decipherment of cuneiform tablets containing astronomical and astrological texts in the late nineteenth century. However, until now few attempts have been made to write a detailed history of the Mesopotamian astral sciences as opposed to detailed studies of particular texts and types of astronomy or astrology. My aim in this paper is to present some ideas of how such a history should be written and in particular to (...) consider if and how visual evidence can be brought alongside textual evidence in the writing of this history. (shrink)

The normally staid topics of judicial ethics and the standards for judicial recusal have become the focus of political debates, editorials and letter writing campaigns. Most of the recent focus falls on conservative justices of the US Supreme Court and in particular on their anticipated participation in what is expected to be an important ruling on the constitutionality of the heath care reforms championed by President Obama and the Democratic Party. But the issue is not simply about partisan politics. It's (...) also about the structure of the US federal judiciary and the need to think about the impact of two-career couples on the judicial recusal rules. (shrink)