{"paper":{"title":"Incomparable $\\omega_1$-like models of set theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Gunter Fuchs, Joel David Hamkins, Victoria Gitman","submitted_at":"2015-01-05T22:09:48Z","abstract_excerpt":"We show that the analogues of the Hamkins embedding theorems, proved for the countable models of set theory, do not hold when extended to the uncountable realm of $\\omega_1$-like models of set theory. Specifically, under the $\\diamondsuit$ hypothesis and suitable consistency assumptions, we show that there is a family of $2^{\\omega_1}$ many $\\omega_1$-like models of ZFC, all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive $\\omega_1$-like model of ZFC that does not embed into its own constructible universe; and there can be an $\\omega_1$-lik"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.01022","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}