{"paper":{"title":"The onto mapping property of Sierpinski","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Arnold W. Miller","submitted_at":"2014-08-12T20:58:53Z","abstract_excerpt":"Define\n  (*) There exists $(\\phi_n:\\omega_1\\to \\omega_1:n<\\omega)$ such that for every uncountable $I$ which is a subset of $\\omega_1$ there exists $n$ such that $\\phi_n$ maps $I$ onto $\\omega_1$.\n  This is roughly what Sierpinski in his book on the continuum hypothesis refers to as $P_3$ but I think he brings reals number line into it. I don't know French so I cannot say for sure what he says but I think he proves that (*) follows from the continuum hypothesis. We show that the existence of a Luzin set implies (*); and (*) implies that there exists a nonmeager set of reals of size $\\omega_1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.2851","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"}