{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/8591"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/8591","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Decidable prime models","abstract":"A set S of types over a theory T is strongly free if for all subsets X [strict subset] S, there is a countable model of T which realizes X and omits S\\X. Throughout, all theories are assumed complete and consistent unless otherwise stated. Theorem 1 If all strongly free sets of types over a recursive theory T are finite, then T has a decidable prime model. Definition 2 A model is decidable if it is isomorphic to a model whose elementary diagram is recursive (technically speaking, this just means the model has a decidable presentation. Throughout this paper, however, we will just say the model is decidable} A classical result in model theory is that any theory with less than 2No many countable models must have a prime model. Our theorem gives an effective extension of this result: Corollary 3 If a countable theory T has less than 2No many countable models, then there is a prime model of T decidable in T.","abstract_html":"A set S of types over a theory T is strongly free if for all subsets X [strict subset] S, there is a countable model of T which realizes X and omits S\\X. Throughout, all theories are assumed complete and consistent unless otherwise stated. Theorem 1 If all strongly free sets of types over a recursive theory T are finite, then T has a decidable prime model. Definition 2 A model is decidable if it is isomorphic to a model whose elementary diagram is recursive (technically speaking, this just means the model has a decidable presentation. Throughout this paper, however, we will just say the model is decidable} A classical result in model theory is that any theory with less than 2No many countable models must have a prime model. Our theorem gives an effective extension of this result: Corollary 3 If a countable theory T has less than 2No many countable models, then there is a prime model of T decidable in T.","abstract_has_math":false,"creators":["Young, Jessica Millar, 1973-"],"institution":"Massachusetts Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Dept. of Mathematics.","school":null,"contributors":[],"advisors":["Gerald E. Sachs."],"committee_chairs":[],"committee_members":[],"year":2001,"date_issued":"2001","date_published":"2001","updated_at":"2026-07-22T22:22:30Z","subjects":["Mathematics."],"languages":["eng"],"rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"rights_urls":["http://dspace.mit.edu/handle/1721.1/7582"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1721.1/8591","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Gerald E. Sachs."]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. Dept. of Mathematics."]},{"key":"dc:contributor.other","label":"Dc Contributor Other","values":["Massachusetts Institute of Technology. Dept. of Mathematics."]},{"key":"dc:creator","label":"Author","values":["Young, Jessica Millar, 1973-"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2005-08-23T21:32:21Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2005-08-23T21:32:21Z"]},{"key":"dc:date.issued","label":"Date","values":["2001"]},{"key":"dc:publisher","label":"Institution","values":["Massachusetts Institute of Technology"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics."]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://dspace.mit.edu/handle/1721.1/7582"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1721.1/8591"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2001.","Includes bibliographical references (leaf 31)."]},{"key":"dc:description.abstract","label":"Abstract","values":["A set S of types over a theory T is strongly free if for all subsets X [strict subset] S, there is a countable model of T which realizes X and omits S\\X. Throughout, all theories are assumed complete and consistent unless otherwise stated. Theorem 1 If all strongly free sets of types over a recursive theory T are finite, then T has a decidable prime model. Definition 2 A model is decidable if it is isomorphic to a model whose elementary diagram is recursive (technically speaking, this just means the model has a decidable presentation. Throughout this paper, however, we will just say the model is decidable} A classical result in model theory is that any theory with less than 2No many countable models must have a prime model. Our theorem gives an effective extension of this result: Corollary 3 If a countable theory T has less than 2No many countable models, then there is a prime model of T decidable in T."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Decidable prime models"]}]}],"canonical_facts":{"dc:contributor.advisor":["Gerald E. Sachs."],"dc:contributor.department":["Massachusetts Institute of Technology. Dept. of Mathematics."],"dc:contributor.other":["Massachusetts Institute of Technology. Dept. of Mathematics."],"dc:creator":["Young, Jessica Millar, 1973-"],"dc:date.accessioned":["2005-08-23T21:32:21Z"],"dc:date.available":["2005-08-23T21:32:21Z"],"dc:date.issued":["2001"],"dc:description":["Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2001.","Includes bibliographical references (leaf 31)."],"dc:description.abstract":["A set S of types over a theory T is strongly free if for all subsets X [strict subset] S, there is a countable model of T which realizes X and omits S\\X. Throughout, all theories are assumed complete and consistent unless otherwise stated. Theorem 1 If all strongly free sets of types over a recursive theory T are finite, then T has a decidable prime model. Definition 2 A model is decidable if it is isomorphic to a model whose elementary diagram is recursive (technically speaking, this just means the model has a decidable presentation. Throughout this paper, however, we will just say the model is decidable} A classical result in model theory is that any theory with less than 2No many countable models must have a prime model. Our theorem gives an effective extension of this result: Corollary 3 If a countable theory T has less than 2No many countable models, then there is a prime model of T decidable in T."],"dc:description.degree":["Ph.D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/1721.1/8591"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Mathematics."],"dc:title":["Decidable prime models"],"dc:type":["Thesis"]},"updated_at":"2026-07-22T22:22:30Z"}