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Reflection principles and large cardinals
Mrva, Mikuláš ; Honzík, Radek (advisor) ; Verner, Jonathan (referee)
This thesis aims to examine the relation between the so called Reflection Principles and Large Cardinals. Lévy has shown that the Reflection Theorem is a sound theorem of ZFC and it is equivalent to the Replacement Schema and the Axiom of Infinity. From this point of view, Reflection theorem can be seen a specific version of an Axiom of Infinity. This paper aims to examine the Reflection Principle and its generalisations with respect to the existence of Large Cardinals. This thesis will establish the Inaccessible, Mahlo and Indescribable cardinals and show how can those be defined via reflection. A natural limit of Large Cardinals obtained via reflection are cardinals inconsistent with L. This thesis will offer an intuitive explanation of why this holds. 1
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The constructive universe L
Ketner, Michal ; Honzík, Radek (advisor) ; Přenosil, Adam (referee)
The theme explores the universe of constructive set L as it was defined by Godel. The work compares two methods of construction L set: one through the formalization of satisfaction relationand the other one with several (finitely many) called rudimentary functions that generate L. The work continues with verification of the implications Con(ZF)→Con(ZFC + CH). The goal is to give a comprehensive view of the construction L and verification of 's relative consistency CH. Powered by TCPDF (www.tcpdf.org)
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Tree property at more cardinals
Stejskalová, Šárka ; Honzík, Radek (advisor) ; Zdomskyy, Lyubomyr (referee)
In this thesis we study the Aronszajn and special Aronszajn trees, their existence and nonexistence. We introduce the most common definition of special Aronszajn tree and some of its generalizations and we examine the relations between them. Next we study the notions of the tree property and the weak tree property at a given regular cardinal κ. The tree property means that there are no Aronszajn trees at κ and the weak tree property means that there are no special Aronszajn trees at κ. We define and compare two forcings, the Mitchell forcing and the Grigorieff forcing, and we use them to obtain a model in which the (weak) tree property holds at a given cardinal. At the end, we show how to use the Mitchell forcing to construct a model in which the (weak) tree property holds at more than one cardinal. 1
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Cohen forcing and its properties
Bydžovský, Jan ; Honzík, Radek (advisor) ; Verner, Jonathan (referee)
This bachelor thesis studies properties of Cohen Forcing and its relation to the unprovability of Continuum Hypothesis and Generalised Continuum Hypothesis. The thesis is divided into four parts. In the first part the technique of forcing based on partial orders is introduced. The second part introduces a notion of Cohen forcing, shows properties of cardinal arithmetic sufficient to preservation of cardinals by Cohen forcing and focuses mainly on generic sets added by concrete variations of Cohen Forcing. Finally some of the properties of Cohen reals are shown in this part. The third part reconstruct a proof of unprovability of Continuum Hypothesis and shows a use of Cohen Forcing in relation to the statements about the Generalised Continuum Hypothesis. The last part discusses briefly a non-minimality of generic filters on Cohen forcing and introduce a notion of Sacks forcing in order to show an existence of forcing notion whose generic filters are minimal. Keywords Cohen forcing, CH, GCH, Cohen reals.
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The continuum function on regular cardinals in the presence of large cardinals
Blicha, Martin ; Honzík, Radek (advisor) ; Verner, Jonathan (referee)
This thesis examines the interactions between the continuum function and large cardinals. It is know, by a result of Easton, that the continuum function on regular cardinals has great freedom in ZFC. However, large cardinals lay additional constraints to possible behaviour of the continuum function. We focus on weakly compact and measurable cardinal to point out the differences in interactions with the continuum function between various types of large cardinals. We also study the case of indescribable cardinals for the comparison, and the results lead us to conclude that it is not easy to pinpoint the reason for these differences. 1
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Logical background of forcing
Glivická, Jana ; Honzík, Radek (advisor) ; Chodounský, David (referee)
This thesis examines the method of forcing in set theory and focuses on aspects that are set aside in the usual presentations or applications of forcing. It is shown that forcing can be formalized in Peano arithmetic (PA) and that consis- tency results obtained by forcing are provable in PA. Two ways are presented of overcoming the assumption of the existence of a countable transitive model. The thesis also studies forcing as a method giving rise to interpretations between theories. A notion of bi-interpretability is defined and a method of forcing over a non-standard model of ZFC is developed in order to argue that ZFC and ZF are not bi-interpretable. 1
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Generalized Boolean models and classical predicate logic
Lávička, Tomáš ; Honzík, Radek (advisor) ; Haniková, Zuzana (referee)
This bachelor thesis is dealing with complete Boolean algebras and its use in semantics of first-order predicate logic. This thesis has two main goals, at first it is to show that every Boolean al- gebra can be extended to a complete Boolean algebra such that the former Boolean algebra is its dense subalgebra. This theorem is proved using topological construction. Afterwards, in the sec- ond part, we define semantics for first-order predicate logic with respect to complete Boolean algebras, which includes introduc- tion of the Boolean-valued model. Then we prove completeness theorem with respect to all complete Boolean algebras. The the- orem is proven using ultrafilters on Boolean algebras. Keywords: Boolean algebras, complete Boolean algebras, clas- sical logic.
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Ultrapower construction in set theory
Holík, Lukáš ; Honzík, Radek (advisor) ; Verner, Jonathan (referee)
The presented work contains the history of origin of measure, its connection with measurable cardinals and summary of all elementary definitions and no- tions needed for the generalization of ultrapower construction in model theory for proper classes. One of the parts of the presented theory is the proof of el- ementary properties needed for the application of ultrapower construction to measurable cardinals. Using all previous results we prove the Theorem of Dana Scott about the connection between existence of a measurable cardinal and the size of the universe.
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Skolem paradox in set theory
Liepoldová, Tereza ; Honzík, Radek (advisor) ; Hykšová, Magdalena (referee)
This works aims to map the development of the Löwenheim-Skolems the- orem from Ernst Schröder to Thoralf Skolem using original mathematical notation. It describes its consequence in the form of the Skolem paradox and its influence on set theory and associated issues concerning orders of logic. 1
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