Prošireni modeli teorije skupova

Abstract

Glavni je cilj rada dokazati nezavisnost aksioma kontruktibilnosti, hipoteze kontinuuma i aksioma izbora od teorije $ZF$. Za dokaze tih tvrdnji ne koristimo poznatu Cohenovu metodu forcinga, već nešto intuitivniju teoriju proširenih modela. U prvom poglavlju uvodimo osnovne pojmove matematičke logike, teorije skupova, teorije Booleovih algebri te topologije. Relativna opsežnost prvog poglavlja je uvjetovana potrebama u daljnjem tekstu, a smatramo da je bolje neke stvari naglasiti odmah na početku, pa se u nastavku na njih samo pozivati. Ipak, to poglavlje sadrži neke same po sebi interesantne rezultate (primjerice Stoneov teorem). Drugo poglavlje, u kojem se uvodi pojam proširenog modela, glavni je dio teksta. Zatim dokazujemo neka osnovna svojstva takve strukture, da bismo na kraju dokazali da u takvim modelima vrijede svi aksiomi teorije $ZFC$. Treće poglavlje sadrži prve dokaze nezavisnosti. Nakon kratkog uvođenja nekih pojmova forcinga, u nastavku dokazujemo nezavisnost aksioma konstruktibilnosti od teorije $ZF$, a kao krunu poglavlja dokazujemo nezavisnost hipoteze kontinuuma od teorije $ZF$. Kako bismo to dokazali, izabiremo posebnu Booleovu algebru pomoću koje dobivamo prošireni model u kojem vrijedi negacija hipoteze kontinuuma. U zadnjem, četvrtom poglavlju, dokazujemo nezavisnost aksioma izbora od teorije $ZF$ i time završavamo dokaze nezavisnosti. Interesantno je da u dokazu nezavisnosti aksioma izbora koristimo metodu sličnu onoj koju su razvili Fraenkel i Mostowski desetljećima prije forcinga.The main goal of this thesis is to prove the independence of the axiom of constructibility, continuum hypothesis and the axiom of choice from $ZF$ theory. In the first chapter, we introduce basic notions of mathematical logic, set theory, theory of Boolean algebras, and topology. Relatively long first chapter is due to the needs in later parts of the text, and we think that it is better to emphasize certain things at the beginning, in order to refer to them later. Nevertheless, this chapter is not without interesting results (Stone’s theorem, for example). The second chapter is the main part of the text, in which we introduce Boolean-valued models. We then prove some basic properties of such structures, and at the end, we prove that every axiom of $ZF$ theory is true in every Boolean-valued model. Main topic of the third chapter are the independence results. After a short introduction of the forcing notion, we prove the independence of axiom of constructibility from $ZF$ theory, and then we prove the most important result of this chapter: independence of continuum hypothesis from $ZF$ theory. In order to prove that, we choose a concrete Boolean algebra, and then construct Boolean-valued model with it, in which continuum hypothesis fails. In the last, fourth chapter, we prove independence of axiom of choice from $ZF$ theory, thus concluding our work on independence proofs. It is interesting to note that in this chapter we use somewhat similar technique to the one which was discovered and ramified by Fraenkel and Mostowski, decades before the invention of the method of forcing