{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/20320"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/20320","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Combinatorial principles in second-order theories of bounded arithmetic","abstract":"An attempt is made to study the mathematical strength of the weak second order theories of Bounded Arithmetic U$\\sbsp{2}{i}$ and V$\\sbsp{2}{i}$, i $\\geq$ 0, introduced by S. Buss. It is first shown that U$\\sbsp{2}{1}$ can $\\Sigma\\sbsp{1}{1,b}$-define the functions in the second class of Grzegorczyk, $\\varepsilon\\sp2$, or, equivalently, the class of $\\Sigma\\sbsp{1}{1,b}$-definable functions in U$\\sbsp{2}{1}$ is closed under bounded recursion.","abstract_html":"An attempt is made to study the mathematical strength of the weak second order theories of Bounded Arithmetic U$\\sbsp{2}{i}$ and V$\\sbsp{2}{i}$, i $\\geq$ 0, introduced by S. Buss. It is first shown that U$\\sbsp{2}{1}$ can $\\Sigma\\sbsp{1}{1,b}$-define the functions in the second class of Grzegorczyk, $\\varepsilon\\sp2$, or, equivalently, the class of $\\Sigma\\sbsp{1}{1,b}$-definable functions in U$\\sbsp{2}{1}$ is closed under bounded recursion.","abstract_has_math":true,"creators":["De Castro, Rodrigo"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Jockusch, Carl G., Jr."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T12:35:53Z","date_published":"2011-05-07T12:35:53Z","updated_at":"2026-07-22T22:25:15Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1992 De Castro, Rodrigo"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9236438","(UMI)AAI9236438"],"render_values":[{"text":"AAI9236438","href":null,"code":true},{"text":"(UMI)AAI9236438","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/20320","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Jockusch, Carl G., Jr."]},{"key":"dc:creator","label":"Author","values":["De Castro, Rodrigo"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T12:35:53Z","10000-01-01","1992"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"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","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1992 De Castro, Rodrigo"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9236438","(UMI)AAI9236438","http://hdl.handle.net/2142/20320"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["An attempt is made to study the mathematical strength of the weak second order theories of Bounded Arithmetic U$\\sbsp{2}{i}$ and V$\\sbsp{2}{i}$, i $\\geq$ 0, introduced by S. Buss. It is first shown that U$\\sbsp{2}{1}$ can $\\Sigma\\sbsp{1}{1,b}$-define the functions in the second class of Grzegorczyk, $\\varepsilon\\sp2$, or, equivalently, the class of $\\Sigma\\sbsp{1}{1,b}$-definable functions in U$\\sbsp{2}{1}$ is closed under bounded recursion.","It is shown next that U$\\sbsp{2}{1}$ proves the $\\Delta\\sbsp{1}{1,b}$-pigeonhole principle. Two general combinatorial principles, the $\\Delta\\sbsp{1}{1,b}$-partition principle and the $\\Delta\\sbsp{1}{1,b}$-equipartition principle, are obtained from it thereby demonstrating that the $\\Delta\\sbsp{1}{1,b}$-PHP embodies a strong notion of cardinality. The introduction of these principles is motivated with some examples, notably by showing that Euler's theorem is provable in U$\\sbsp{2}{1}$. The provability of Euler's theorem in weak first order fragments of Peano Arithmetic is an open problem.","\"A theory of polynomials is developed in U$\\sbsp{2}{1}$ and it is proved that U$\\sbsp{2}{1}$ + B $\\vdash$ \"\"existence of primitive roots\"\" where formula B asserts that a nontrivial polynomial of degree n can have at most n solutions modulo p if p is a prime. By an essential use of the $\\Delta\\sbsp{1}{1,b}$-PHP and the $\\Delta\\sbsp{1}{1,b}$-partition principle it is shown that V$\\sbsp{2}{1}\\/\\vdash$ B, hence V$\\sbsp{2}{1}\\/\\vdash$ \"\"existence of primitive roots\"\".\"","Made available in DSpace on 2011-05-07T12:35:53Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9236438.pdf: 3051390 bytes, checksum: 650f5f11d665e556c4113b9ffc71dcc7 (MD5) Previous issue date: 1992","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:43:06Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:18:49-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Combinatorial principles in second-order theories of bounded arithmetic"]}]}],"canonical_facts":{"dc:contributor":["Jockusch, Carl G., Jr."],"dc:creator":["De Castro, Rodrigo"],"dc:date":["2011-05-07T12:35:53Z","10000-01-01","1992"],"dc:description":["An attempt is made to study the mathematical strength of the weak second order theories of Bounded Arithmetic U$\\sbsp{2}{i}$ and V$\\sbsp{2}{i}$, i $\\geq$ 0, introduced by S. Buss. It is first shown that U$\\sbsp{2}{1}$ can $\\Sigma\\sbsp{1}{1,b}$-define the functions in the second class of Grzegorczyk, $\\varepsilon\\sp2$, or, equivalently, the class of $\\Sigma\\sbsp{1}{1,b}$-definable functions in U$\\sbsp{2}{1}$ is closed under bounded recursion.","It is shown next that U$\\sbsp{2}{1}$ proves the $\\Delta\\sbsp{1}{1,b}$-pigeonhole principle. Two general combinatorial principles, the $\\Delta\\sbsp{1}{1,b}$-partition principle and the $\\Delta\\sbsp{1}{1,b}$-equipartition principle, are obtained from it thereby demonstrating that the $\\Delta\\sbsp{1}{1,b}$-PHP embodies a strong notion of cardinality. The introduction of these principles is motivated with some examples, notably by showing that Euler's theorem is provable in U$\\sbsp{2}{1}$. The provability of Euler's theorem in weak first order fragments of Peano Arithmetic is an open problem.","\"A theory of polynomials is developed in U$\\sbsp{2}{1}$ and it is proved that U$\\sbsp{2}{1}$ + B $\\vdash$ \"\"existence of primitive roots\"\" where formula B asserts that a nontrivial polynomial of degree n can have at most n solutions modulo p if p is a prime. By an essential use of the $\\Delta\\sbsp{1}{1,b}$-PHP and the $\\Delta\\sbsp{1}{1,b}$-partition principle it is shown that V$\\sbsp{2}{1}\\/\\vdash$ B, hence V$\\sbsp{2}{1}\\/\\vdash$ \"\"existence of primitive roots\"\".\"","Made available in DSpace on 2011-05-07T12:35:53Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9236438.pdf: 3051390 bytes, checksum: 650f5f11d665e556c4113b9ffc71dcc7 (MD5) Previous issue date: 1992","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:43:06Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:18:49-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9236438","(UMI)AAI9236438","http://hdl.handle.net/2142/20320"],"dc:language":["eng"],"dc:rights":["Copyright 1992 De Castro, Rodrigo"],"dc:subject":["Mathematics"],"dc:title":["Combinatorial principles in second-order theories of bounded arithmetic"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:15Z"}