20.500.12556/RUL-108796 Toposni modeli principov teorije množic magistrsko delo Topos models of set-theoretic principles In this thesis we approach axioms of choice of different strength by considering topoi. First, we present Grothendieck topoi and afterwards their abstractly axiomatised counterparts called elementary topoi. For these we show, that they carry logic, through which we can express set-theoretic statements. We consider topos versions of axiom of choice, axiom of dependent choice and axiom of countable choice. For Grothendieck topoi (over atomic sites) we give natural conditions (without reference to the internal language of the topos) for validity of the internal choice principles. We show that the same implications as for the set-theoretic versions also hold for the topos versions. We present three concrete examples of Grothendieck topoi, which show that the converses of these implications are not valid and that the three considered axioms are independent from IZFA (intuitionistic ZF, where atoms are allowed). V delu pristopimo k obravnavi aksiomov izbire različnih moči preko toposov. Najprej predstavimo Grothendieckove topose, nato pa abstraktno aksiomatizirane elementarne topose. Za slednje pokažemo, da so na naravni način nosilci logike, preko katere lahko znotraj toposa izrazimo trditve iz teorije množic. Obravnavamo toposne različice aksioma izbire, aksioma odvisne izbire in aksioma števne izbire. Za Grothendieckove topose (nad atomskimi odri) navedemo naravne pogoje (brez uporabe notranjega jezika toposa) za veljavnost toposnih različic aksiomov izbire. Dokažemo, da tudi za toposne različice formulacij veljajo iste implikacije kot v teoriji množic. Predstavimo tudi tri konkretne primere Grothendieckovih toposov in tako dokažemo, da obratne implikacije ne veljajo ter da so vsi trije obravnavani aksiomi neodvisni od IZFA (intuitionistične ZF, kjer so dovoljeni atomi). topos sheaf site axiom of choice dependent choice countable choice independence topos snop oder aksiom izbire aksiom odvisne izbire aksiom števne izbire neodvisnost true false false Angleški jezik Angleški jezik Magistrsko delo/naloga 2019-07-26 07:45:03 2019-07-26 07:45:14 2024-05-30 11:04:28 0000-00-00 00:00:00 2019 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 510.6 96767 18690905 781.pdf 781.pdf 1 7F5310AF224C1072CA7D173E15AE4391 edabc504bba4d0cac72d3805eab46c5d83474ed509d0d89e56cafe8d9eafb235 ba88af6f-a1b6-11eb-a523-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=119914 Fakulteta za matematiko in fiziko 0 0 0