{"abstracts":[{"sha1":"0587ebe9a42f97ad4e7bab351d45a3039bd58a9f","content":"In this paper we exploit the structural properties of standard and\nnon-standard models of set theory to produce models of set theory admitting\nautomorphisms that are well-behaved along an initial segment of their ordinals.\nNFU is Ronald Jensen's modifcation of Quine's `New Foundations' set\ntheory that allows non-sets into the domain of discourse. The axioms\nAxCount, AxCount_≤ and AxCount_≥ each\nextend NFU by placing restrictions on the cardinality of a finite\nset of singletons relative to the cardinality of its union. Using the results\nabout automorphisms of models of set theory we separate the consistency\nstrengths of these three extensions of NFU. We show that\nNFU + AxCount proves the consistency of NFU +\nAxCount_≤, and NFU + AxCount_≤ proves the\nconsistency of NFU + AxCount_≥.","mimetype":"text/plain","lang":"en"},{"sha1":"b421215fbe82eee515e9025b5602c9fed0a4594f","content":"In this paper we exploit the structural properties of standard and\nnon-standard models of set theory to produce models of set theory admitting\nautomorphisms that are well-behaved along an initial segment of their ordinals.\n$\\mathrm{NFU}$ is Ronald Jensen's modifcation of Quine's `New Foundations' set\ntheory that allows non-sets into the domain of discourse. The axioms\n$\\mathrm{AxCount}$, $\\mathrm{AxCount}_\\leq$ and $\\mathrm{AxCount}_\\geq$ each\nextend $\\mathrm{NFU}$ by placing restrictions on the cardinality of a finite\nset of singletons relative to the cardinality of its union. Using the results\nabout automorphisms of models of set theory we separate the consistency\nstrengths of these three extensions of $\\mathrm{NFU}$. We show that\n$\\mathrm{NFU} + \\mathrm{AxCount}$ proves the consistency of $\\mathrm{NFU} +\n\\mathrm{AxCount}_\\leq$, and $\\mathrm{NFU} + \\mathrm{AxCount}_\\leq$ proves the\nconsistency of $\\mathrm{NFU} + \\mathrm{AxCount}_\\geq$.","mimetype":"application/x-latex","lang":"en"}],"refs":[],"contribs":[{"index":0,"raw_name":"Zachiri McKenzie","role":"author"}],"license_slug":"ARXIV-1.0","language":"en","version":"v2","ext_ids":{"arxiv":"1309.4631v2"},"release_year":2013,"release_date":"2013-10-19","release_stage":"submitted","release_type":"article","webcaptures":[],"filesets":[],"files":[{"release_ids":["dgthjaijonfffnbdwns2glpu5q"],"mimetype":"application/pdf","urls":[{"url":"https://arxiv.org/pdf/1309.4631v2.pdf","rel":"repository"},{"url":"https://web.archive.org/web/20200824185544/https://arxiv.org/pdf/1309.4631v2.pdf","rel":"webarchive"}],"sha256":"9a03bb88c4cdc28209d56a0bb50c65ec743b91a5e958434d1fbeaeb44749cea5","sha1":"a274cbd26042d71e098394b1f1fd76cfee0539aa","md5":"81ba827b8c2c4f18327be4353059baf1","size":368052,"revision":"486bf0c4-a74e-4608-9bf5-2e8360a97043","ident":"a6q6r7gtazh43geuwdimiktu7q","state":"active"}],"work_id":"aww46slt5failex7fscfclivdm","title":"Automorphisms of models of set theory and extensions of NFU","state":"active","ident":"dgthjaijonfffnbdwns2glpu5q","revision":"e20ea222-7161-49b0-ae10-a9a89471cc71","extra":{"arxiv":{"base_id":"1309.4631","categories":["math.LO"],"comments":"41 pages"}}}