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[{"key": "dc.contributor.advisor", "value": "Kurittu, Lassi", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Kosonen, Ossi", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2016-11-24T15:15:13Z", "language": "", "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2016-11-24T15:15:13Z", "language": "", "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2016", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1643489", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/52001", "language": "", "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Ossi Kosonen, G\u00f6delin ep\u00e4t\u00e4ydellisyyslauseet, G\u00f6del's incompleteness , matematiikan pro gradu -tutkielma, 57 sivua, Jyv\u00e4skyl\u00e4n yliopisto, Matematiikan ja tilastotieteen laitos, syksy 2016.\r\nT\u00e4m\u00e4n tutkielman tarkoituksena on todistaa G\u00f6delin kaksi ep\u00e4t\u00e4ydellisyyslausetta RA-kieless\u00e4. It\u00e4valtalais-amerikkalainen Kurt G\u00f6del todisti nime\u00e4ns\u00e4 kantavat lauseet artikkelissaan vuonna 1931. G\u00f6del ei itse varsinaisesti k\u00e4ytt\u00e4nyt RA-kielt\u00e4 lauseiden alkuper\u00e4isiss\u00e4 todistuksissa, mutta t\u00e4ss\u00e4 tutkielmassa RA-kieli on valittu formaaliksi kieleksi, koska se perustuu predikaattikielten pohjalle. RA-kielen tarkoitus on formalisoida mahdollisimman hyvin aritmetiikka, joka k\u00e4yt\u00e4nn\u00f6ss\u00e4 onnistuu mallintamalla luonnollisten lukujen joukko sek\u00e4 sen laskutoimitukset t\u00e4lle kielelle.\r\nEnsimm\u00e4inen ep\u00e4t\u00e4ydellisyyslause sanoo, ett\u00e4 RA-kieless\u00e4 on olemassa validi kaava, joka ei ole teoreema. Toisen ep\u00e4t\u00e4ydellisyyslauseen mukaan joukko N sek\u00e4 sit\u00e4 imitoiva RA-kieli on tarpeeksi kehittynyt matemaattinen j\u00e4rjestelm\u00e4, ettei sen ristiriidattomuutta pystyt\u00e4 todistamaan syntaktisesti. T\u00e4llaisen j\u00e4rjestelm\u00e4n ristiriidattomuutta ei voida siis osoittaa p\u00e4\u00e4ttelyjonoa konstruoimalla.\r\nTutkielman luvussa 1 esitell\u00e4\u00e4n ensin primitiivirekursion avulla primitiivirekursiiviset funktiot ja joukot. Yleisesti voidaan sanoa, ett\u00e4 primitiivirekursiivisilla funktioilla ja joukoilla tarkoitetaan sellaisia matemaattisia objekteja, joiden laskettavuus pysyy \u00e4\u00e4rellisill\u00e4 toimenpiteill\u00e4 hallussa. K\u00e4yt\u00e4nn\u00f6ss\u00e4 t\u00e4m\u00e4 n\u00e4kyy esimerkiksi rajoitetun minimalisaation m\u00e4\u00e4ritelm\u00e4ss\u00e4, jossa minimi\u00e4 etsit\u00e4\u00e4n jostain \u00e4\u00e4rellisest\u00e4 joukosta. T\u00e4m\u00e4n j\u00e4lkeen kappaleessa 1.2 konstruoidaan RA-kielen syntaksi sek\u00e4 semantiikka l\u00e4hes vastaavalla tavalla kuin predikaattikieliss\u00e4.\r\nLuvussa 2 esitell\u00e4\u00e4n G\u00f6delin t\u00f6iden keskeisimm\u00e4t saavutukset. Ensin esitet\u00e4\u00e4n alkulukuesityksen yksik\u00e4sitteisyyteen perustuva G\u00f6del-numerointi, jolla jokainen RA-kielen kaava sek\u00e4 kaavajono saadaan vastaamaan yksik\u00e4sitteist\u00e4 luonnollista lukua. T\u00e4m\u00e4n avulla osoitetaan suurpiirteisesti, kuinka p\u00e4\u00e4ttelyn primitiivirekursiivisuus voidaan todistaa. P\u00e4\u00e4ttelyn primitiivirekursiivisuus on t\u00e4rkein, mutta my\u00f6s vaikein aputulos, jota tarvitaan ep\u00e4t\u00e4ydellisyyslauseiden todistamisessa.\r\nLuvun 2 lopuksi ep\u00e4t\u00e4ydellisyyslauseiden todistukset pyrit\u00e4\u00e4n esitt\u00e4m\u00e4\u00e4n mahdollisimman eksaktisti, kuitenkin samalla selitt\u00e4en kansantajuisesti etenkin niiden merkityksest\u00e4, sek\u00e4 mit\u00e4 ep\u00e4t\u00e4ydellisyyslauseet kertovat matemaattisista j\u00e4rjestelmist\u00e4. Viimeisess\u00e4 luvussa 3 esitell\u00e4\u00e4n G\u00f6delin tulosten seurauksia sek\u00e4 piirret\u00e4\u00e4n mahdollisia uusia suuntaviivoja matemaattisen logiikan tutkimuksessa.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Ossi Kosonen (osankoso) on 2016-11-24 15:15:13.330125. Form: Pro gradu -lomake (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake). JyX data: [jyx_publishing-allowed (fi) =True]", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija.group@korppi.jyu.fi) on 2016-11-24T15:15:13Z\r\nNo. of bitstreams: 2\r\nURN:NBN:fi:jyu-201611244750.pdf: 475771 bytes, checksum: dfded6ead2e4fea8b8189622ef2ded1f (MD5)\r\nlicense.html: 4788 bytes, checksum: e6b1e53bf981fd62508a0abcb29b42f0 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2016-11-24T15:15:13Z (GMT). No. of bitstreams: 2\r\nURN:NBN:fi:jyu-201611244750.pdf: 475771 bytes, checksum: dfded6ead2e4fea8b8189622ef2ded1f (MD5)\r\nlicense.html: 4788 bytes, checksum: e6b1e53bf981fd62508a0abcb29b42f0 (MD5)\r\n Previous issue date: 2016", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (57 sivua)", "language": null, "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "G\u00f6delin ep\u00e4t\u00e4ydellisyyslauseet", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "G\u00f6delin ep\u00e4t\u00e4ydellisyyslauseet", "language": null, "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-201611244750", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.date.updated", "value": "2016-11-24T15:15:14Z", "language": "", "element": "date", "qualifier": "updated", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": "fi", "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": null, "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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