Every countable model of set theory embeds into its own constructible universe

The main theorem of this article is that every countable model of set theory M, including every well-founded model, is isomorphic to a submodel of its own constructible universe. In other words, there is an embedding $j:M\to L^M$ that is elementary for quantifier-free assertions. The proof uses universal digraph combinatorics, including an acyclic version of the countable random digraph, which I call the countable random Q-graded digraph, and higher analogues arising as uncountable Fraisse limits, leading to the hypnagogic digraph, a set-homogeneous, class-universal, surreal-numbers-graded acyclic class digraph, closely connected with the surreal numbers. The proof shows that $L^M$ contains a submodel that is a universal acyclic digraph of rank $Ord^M$. The method of proof also establishes that the countable models of set theory are linearly pre-ordered by embeddability: for any two countable models of set theory, one of them is isomorphic to a submodel of the other. Indeed, they are pre-well-ordered by embedability in order-type exactly $\omega_1+1$. Specifically, the countable well-founded models are ordered by embeddability in accordance with the heights of their ordinals; every shorter model embeds into every taller model; every model of set theory $M$ is universal for all countable well-founded binary relations of rank at most $Ord^M$; and every ill-founded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the same proof method shows that if $M$ is any nonstandard model of PA, then every countable model of set theory---in particular, every model of ZFC---is isomorphic to a submodel of the hereditarily finite sets $HF^M$ of $M$. Indeed, $HF^M$ is universal for all countable acyclic binary relations.Comment: 25 pages, 2 figures. Questions and commentary can be made at http://jdh.hamkins.org/every-model-embeds-into-own-constructible-universe. (v2 adds a reference and makes minor corrections) (v3 includes further changes, and removes the previous theorem 15, which was incorrect.

Facing the Unknown Unknowns of Data Analysis

Empirical claims are inevitably associated with uncertainty, and a major goal of data analysis is therefore to quantify that uncertainty. Recent work has revealed that most uncertainty may lie not in what is usually reported (e.g., p value, confidence interval, or Bayes factor) but in what is left unreported (e.g., how the experiment was designed, whether the conclusion is robust under plausible alternative analysis protocols, and how credible the authors believe their hypothesis to be). This suggests that the rigorous evaluation of an empirical claim involves an assessment of the entire empirical cycle and that scientific progress benefits from radical transparency in planning, data management, inference, and reporting. We summarize recent methodological developments in this area and conclude that the focus on a single statistical analysis is myopic. Sound statistical analysis is important, but social scientists may gain more insight by taking a broad view on uncertainty and by working to reduce the “unknown unknowns” that still plague reporting practice.</p

Is it time for studying real-life debiasing? Evaluation of the effectiveness of an analogical intervention technique.

The aim of this study was to initiate the exploration of debiasing methods applicable in real-life settings for achieving lasting improvement in decision making competence regarding multiple decision biases. Here, we tested the potentials of the analogical encoding method for decision debiasing. The advantage of this method is that it can foster the transfer from learning abstract principles to improving behavioral performance. For the purpose of the study, we devised an analogical debiasing technique for 10 biases (covariation detection, insensitivity to sample size, base rate neglect, regression to the mean, outcome bias, sunk cost fallacy, framing effect, anchoring bias, overconfidence bias, planning fallacy) and assessed the susceptibility of the participants (N = 154) to these biases before and 4 weeks after the training. We also compared the effect of the analogical training to the effect of 'awareness training' and a 'no-training' control group. Results suggested improved performance of the analogical training group only on tasks where the violations of statistical principles are measured. The interpretation of these findings require further investigation, yet it is possible that analogical training may be the most effective in the case of learning abstract concepts, such as statistical principles, which are otherwise difficult to master. The study encourages a systematic research of debiasing trainings and the development of intervention assessment methods to measure the endurance of behavior change in decision debiasing

Tenzing and the importance of tool development for research efficiency

The way science is done is changing. While some tools are facilitating this change, others lag behind. The resulting mismatch between tools and researchers' workflows can be inefficient and delay the progress of research. As an example, information about the people associated with a published journal article was traditionally handled manually and unsystematically. However, as large-scale collaboration, sometimes referred to as “team science,” is now common, a more structured and easy-to-automate approach to managing meta-data is required. In this paper we describe how the latest version of tenzing (A.O. Holcombe et al., Documenting contributions to scholarly articles using CRediT and tenzing, PLOS One 15(12) (2020)) helps researchers collect and structure contributor information efficiently and without frustration. Using tenzing as an example, we discuss the importance of efficient tools in reforming science and our experience with tool development as researchers.</p

The lattice Schwarzian KdV equation and its symmetries

In this paper we present a set of results on the symmetries of the lattice Schwarzian Korteweg-de Vries (lSKdV) equation. We construct the Lie point symmetries and, using its associated spectral problem, an infinite sequence of generalized symmetries and master symmetries. We finally show that we can use master symmetries of the lSKdV equation to construct non-autonomous non-integrable generalized symmetries.Comment: 11 pages, no figures. Submitted to Jour. Phys. A, Special Issue SIDE VI