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Study of Arithmetical Structures and Theories with Regard to Representative and Descriptive Analysis
Glivický, Petr
of doctoral thesis Study of Arithmetical Structures and Theories with Regard to Representative and Descriptive Analysis Petr Glivický We are motivated by a problem of understanding relations between local and global properties of an operation o in a structure of the form B, o , with regard to an application for the study of models B, · of Peano arithmetic, where B is a model of Presburger arithmetic. We are particularly interested in a dependency problem, which we formulate as the problem of describing the dependency closure iclO (E) = {d ∈ Bn ; (∀o, o ∈ O)(o E = o E ⇒ o(d) = o (d))}, where B is a structure, O a set of n-ary operations on B, and E ⊆ Bn. We show, that this problem can be reduced to a definability question in certain expansion of B. In particular, if B is a saturated model of Presburger arithmetic, and O is the set of all (saturated) Peano products on B, we prove that, for a ∈ B, iclO ({a}×B) is the smallest possible, i.e. it contains just those pairs (d0, d1) ∈ B2 for which at least one of di equals p(a), for some polynomial p ∈ Q[x]. We show that the presented problematics is closely connected to the descriptive analysis of linear theories. That are theories, models of which are - up to a change of the language - certain discretely ordered modules over specific discretely ordered...
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Russell's analysis of Peano Arithmetic
Jankovská, Lenka ; Holeček, Tomáš (advisor) ; Arazim, Pavel (referee)
The aim of this thesis is Russell's analysis of Peano arithmetic. This analysis was presented by Russell and A. N. Whitehead in the book Prin- cipia Mathematica and then in Russell's book Introduction to Mathema- tical Philosophy which provided this approach in a more accessible form. The thesis focuses on Russell's critique of original Peano axioms and his effort to use only logical definitions instead of axioms. Another goal of the thesis is Russell's theory of classes and substitution of classes by propositional functions. Furthermore, the type theory for propositional functions is introduced and explained. All is converted into present-day logical notation. Moreover, the non-standard models of Russell's Peano arithmetic are studied. Finally, there are two particular arithmetic exam- ples illustrating the purposes of the thesis. Key words Bertrand Russell, Peano Arithmetic, classes, propositional funkction
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Study of Arithmetical Structures and Theories with Regard to Representative and Descriptive Analysis
Glivický, Petr ; Mlček, Josef (advisor) ; Vopěnka, Petr (referee) ; Zlatoš, Pavol (referee)
of doctoral thesis Study of Arithmetical Structures and Theories with Regard to Representative and Descriptive Analysis Petr Glivický We are motivated by a problem of understanding relations between local and global properties of an operation o in a structure of the form B, o , with regard to an application for the study of models B, · of Peano arithmetic, where B is a model of Presburger arithmetic. We are particularly interested in a dependency problem, which we formulate as the problem of describing the dependency closure iclO (E) = {d ∈ Bn ; (∀o, o ∈ O)(o E = o E ⇒ o(d) = o (d))}, where B is a structure, O a set of n-ary operations on B, and E ⊆ Bn. We show, that this problem can be reduced to a definability question in certain expansion of B. In particular, if B is a saturated model of Presburger arithmetic, and O is the set of all (saturated) Peano products on B, we prove that, for a ∈ B, iclO ({a}×B) is the smallest possible, i.e. it contains just those pairs (d0, d1) ∈ B2 for which at least one of di equals p(a), for some polynomial p ∈ Q[x]. We show that the presented problematics is closely connected to the descriptive analysis of linear theories. That are theories, models of which are - up to a change of the language - certain discretely ordered modules over specific discretely ordered...
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Study of Arithmetical Structures and Theories with Regard to Representative and Descriptive Analysis
Glivický, Petr
of doctoral thesis Study of Arithmetical Structures and Theories with Regard to Representative and Descriptive Analysis Petr Glivický We are motivated by a problem of understanding relations between local and global properties of an operation o in a structure of the form B, o , with regard to an application for the study of models B, · of Peano arithmetic, where B is a model of Presburger arithmetic. We are particularly interested in a dependency problem, which we formulate as the problem of describing the dependency closure iclO (E) = {d ∈ Bn ; (∀o, o ∈ O)(o E = o E ⇒ o(d) = o (d))}, where B is a structure, O a set of n-ary operations on B, and E ⊆ Bn. We show, that this problem can be reduced to a definability question in certain expansion of B. In particular, if B is a saturated model of Presburger arithmetic, and O is the set of all (saturated) Peano products on B, we prove that, for a ∈ B, iclO ({a}×B) is the smallest possible, i.e. it contains just those pairs (d0, d1) ∈ B2 for which at least one of di equals p(a), for some polynomial p ∈ Q[x]. We show that the presented problematics is closely connected to the descriptive analysis of linear theories. That are theories, models of which are - up to a change of the language - certain discretely ordered modules over specific discretely ordered...
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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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Gentzen's Consistency Proof
Horská, Anna ; Švejdar, Vítězslav (advisor) ; Velebil, Jiří (referee)
This paper contains detailed description of two consistency proofs, which state that in the system called Peano arithmetic no contradiction can be obtained. The proofs were first published in 1936 and 1938 by the German mathematician Gerhard Gentzen. For the purpose of this paper, the proofs were read and studied from the original articles called "Die Widerspruchsfreiheit der reinen Zahlentheorie" and "Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie". The first mentioned proof is interesting from the historical point of view. Gentzen used a natural deduction sequent calculus and ordinal numbers in an unusual form he invented. The second proof is similar to the consistency proof, which is commonly known as a consistency proof for Peano arithmetic nowadays.
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