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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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Reflection principles and large cardinals
Mrva, Mikuláš ; Honzík, Radek (advisor) ; Verner, Jonathan (referee)
This thesis aims to examine relations between so called "Reflection Princi- ples" and Large cardinals. Lévy has shown that Reflection Theorem is a sound theorem of ZFC and it is equivalent to Replacement Scheme 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 existence of Large Cardinals. This thesis will establish Inaccessible, Mahlo and Indescribable cardinals and their definition via reflection. A natural limit of Large Cardi- nals obtained via reflection are cardinals inconsistent with L. The thesis will offer an intuitive explanation of why this is the case. 1
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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 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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