Topos models of set-theoretic principles

master's thesis

Severin Mejak (Author), Alex Simpson (Mentor)

Abstract

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).

Keywords

topos;sheaf;site;axiom of choice;dependent choice;countable choice;independence;

Data

Language:	English
Year of publishing:	2019
Typology:	2.09 - Master's Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[S. Mejak]
UDC:	510.6
COBISS:	18690905
Views:	1029
Downloads:	262
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Slovenian
Secondary title:	Topos models of set-theoretic principles
Secondary abstract:	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).
Secondary keywords:	topos;snopi;oder;aksiom izbire;aksiom odvisne izbire;aksiom števne izbire;neodvisnost;
Type (COBISS):	Master's thesis/paper
Study programme:	0
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 2. stopnja
Pages:	IX, 82 str.
ID:	11194158