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The continuum function on singular cardinals
Stejskalová, Šárka ; Honzík, Radek (advisor) ; Verner, Jonathan (referee)
Bachelor thesis studies the behaviour of the continuum function on singular cardinals in theory ZFC. The work is divided into two parts. The focus of the first part is on the Silver's Theorem and it analyzes two different proofs of this Theorem, Silver's original proof and the second, purely combinatorial, proof by Baumgartner and Prikry. The second part is devoted to the Singular Cardinal Hypothesis, which influences the behaviour of the continuum function. In the thesis it is shown that, in the presence of large cardinals, Singular Cardinal Hypothesis is not provable in ZFC. Using Easton and Prikry forcing a model is found where the Singular Cardinal Hypothesis does not hold.
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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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Machine-Free Characterization of Polynomially Computable Functions
Profeld, Michal ; Švejdar, Vítězslav (advisor) ; Verner, Jonathan (referee)
This work is focused into constructing mathematical structure. This structure is closed under it's operations. Structure was developed to contain all functions of certain growth rate. To be More specific functi- ons with polynomial growth rate. We can say that our structure con- tains all functions that have growth rate slower or equal to polynomial growth rate and no other function. Development of our structure was influenced mostly by work of Samuel R. Buss [1] 1
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Koza and Prolog
Frauknecht, Jan ; Švarný, Petr (advisor) ; Verner, Jonathan (referee)
This paper introduces the artificial intelligence background of ge- netic programming and some properties of logical programming para- digm. However, the main task of this work is to create the genetic programming algorithm that operates with the logical programming paradigm. SWI-Prolog is used for the actual implementation of such a program. This implementation is in detail described. The testing of this implementation shows some possible path for the future work. 1
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Combinatorics of filters on the natural numbers
Chládek, Alexandr ; Verner, Jonathan (advisor) ; Grebík, Jan (referee)
The work is devoted to combinatorial properties of filters on natural num- bers as an introduction and motivation to the definability of the filters and its combinatorics. The work contains definitions of basic filter types: P- filter, Q-filter, Rapid filter; orders: Rudin-Kiesler, Rudin-Blass, Katětov and Tukey; filter constructions; basic definitions related to combinatorics on ω; introduction to basic descriptive set theory and topology and some specific results. 1
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Combinatorics of filters on the natural numbers
Chládek, Alexandr ; Verner, Jonathan (advisor) ; Grebík, Jan (referee)
The work is devoted to combinatorial properties of filters on natural num- bers as an introduction and motivation to the definability of the filters and its combinatorics. Basic filter types: P-filter, Q-filter, Rapid filter; orders: Rudin-Kiesler, Rudin-Blass, Katětov and Tukey; filter constructions; basic definitions related to combinatorics on ω; introduction to basic descriptive set theory and topology and some specific results. 1
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Combinatorics of filters on the natural numbers
Chládek, Alexandr ; Verner, Jonathan (advisor) ; Grebík, Jan (referee)
The work is intended to combinatorial properties of filters on natural numbers as an introduction and motivation to the problematics between definability of the filters and its combinatorics. Basic filter types: P-filter, Q-filter, Rapid filter; orders: Rudin-Kiesler, Rudin-Blass, Katětov and Tukey; filter kon- structions; basic definitions related to combinatorics on ω; introduction to basic descriptive set theory and topology and some specific results. 1
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Machine-Free Characterization of Polynomially Computable Functions
Profeld, Michal ; Švejdar, Vítězslav (advisor) ; Verner, Jonathan (referee)
This thesis focuses on machine-free definition of polynomial functions. The main goal is to not only make the readers familiar with this de- finition, but also to introduce them to the other pivotal terms of this thesis. The other pivotal terms are: basic functions, function composi- tion, recursive schemes a polynomial conditions. Throughout the thesis the readers will be introduced, among other things, to derivation of the most used polynomially bounded functions, like multiplication, addi- tion, or other arithmetic functions. 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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