{"id":"W3186095916","doi":"10.1016/j.apal.2023.103317","title":"An undecidable extension of Morley's theorem on the number of countable models","year":2023,"lang":"en","type":"article","venue":"Annals of Pure and Applied Logic","topic":"Advanced Topology and Set Theory","field":"Mathematics","cited_by":1,"is_retracted":false,"has_abstract":false,"ca_institutions":"University of Toronto; University of Victoria","funders":"Natural Sciences and Engineering Research Council of Canada","keywords":"Undecidable problem; Countable set; Mathematics; Discrete mathematics; Continuum hypothesis; Extension (predicate logic); Model theory; Equivalence (formal languages); Decidability; Combinatorics; Computer science","routes":{"ca_aff":true,"ca_fund":true,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"codex-gemma-dda1882f352a","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.0007099166,0.0001056712,0.0002990119,0.00003828197,0.00006798845,0.000002906569,0.0001510598,0.0001024942,0.0001333211],"category_scores_gemma":[0.00004679238,0.00006457102,0.00004706625,0.0001626207,0.000243726,0.00005119183,0.00004398932,0.00009983085,0.00000907376],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.000001607775,"about_ca_system_score_gemma":0.000011671,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.00000477809,"about_ca_topic_score_gemma":0.000002640802,"domain_scores_codex":[0.9992431,0.0000544634,0.0002398299,0.0001539141,0.0001432057,0.0001654965],"domain_scores_gemma":[0.9987614,0.0006055014,0.0001757685,0.0003514849,0.0000776704,0.00002813616],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.0001121087,0.0001392429,0.00002430203,0.00005610677,0.00002621136,0.000001120965,0.0003583495,0.001223389,0.006262804,0.9899676,0.001411154,0.0004176464],"study_design_scores_gemma":[0.0001399448,0.00009164542,0.0000567661,0.00002709544,0.00001549131,0.000001736891,0.001514457,0.0004560535,0.04039847,0.9571838,0.00005015358,0.00006436126],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.9685668,0.00006995529,0.002240326,0.0003578448,0.00002266347,0.0002164023,0.0000231016,0.00003271529,0.02847017],"genre_scores_gemma":[0.9984816,0.0001798455,0.0006535467,0.0003460633,0.00001660255,0.00001211964,0.000005829919,0.00001193367,0.0002924979],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.03413567,"threshold_uncertainty_score":0.263313,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.1478129913710508,"score_gpt":0.3812278666988754,"score_spread":0.2334148753278246,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}