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Models of arithmetic and rich theories
Glivický, Petr ; Mlček, Josef (advisor) ; Vopěnka, Petr (referee)
In the present thesis we study the domain of Peano products (in a given model of the Presburger arithmetic (Pr)) as a potentially possible base for a construction of models of the Peano arithmetic (PA). This issue is a special case of the presentation problem which is closely connected to the concept of rich theories. We are especially concerned with one of the basic questions about Peano products domain, i.e. if there exist a pair of Peano products (· , ) such that these products coincide in some slice a: (x)(a · x = a x) and are different below a: (c, d < a)(c · d 6= c d). We reduce this problem to the question if the eliminating set of formulas of the linear arithmetic (LA) is a subset of the set of all existential formulas. We do not solve this problem completely, we only prove that all formulas (x)(z1, z2) , where is a conjunction of equations of terms, are equivalent to existential formulas. We also suggest that the quantifier elimination in the linear arithmetic is considerably more difficult than the elimination in Pr or in the module theory and that it is connected to the problem of description of finitely generated submonoids of Z. We introduce concepts (regular set, standard rationality, saw,... ) and methods which, as we believe, will be essential for an eventual solution of the problem.
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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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Models of arithmetic and rich theories
Glivický, Petr ; Vopěnka, Petr (referee) ; Mlček, Josef (advisor)
In the present thesis we study the domain of Peano products (in a given model of the Presburger arithmetic (Pr)) as a potentially possible base for a construction of models of the Peano arithmetic (PA). This issue is a special case of the presentation problem which is closely connected to the concept of rich theories. We are especially concerned with one of the basic questions about Peano products domain, i.e. if there exist a pair of Peano products (· , ) such that these products coincide in some slice a: (x)(a · x = a x) and are different below a: (c, d < a)(c · d 6= c d). We reduce this problem to the question if the eliminating set of formulas of the linear arithmetic (LA) is a subset of the set of all existential formulas. We do not solve this problem completely, we only prove that all formulas (x)(z1, z2) , where is a conjunction of equations of terms, are equivalent to existential formulas. We also suggest that the quantifier elimination in the linear arithmetic is considerably more difficult than the elimination in Pr or in the module theory and that it is connected to the problem of description of finitely generated submonoids of Z. We introduce concepts (regular set, standard rationality, saw,... ) and methods which, as we believe, will be essential for an eventual solution of the problem.
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Comparison of ontogenetic and phylogenetic development of understanding of phenomenon infinity in the geometrical context
Krátká, Magdalena ; Vopěnka, Petr (advisor) ; Novotná, Jarmila (referee) ; Potůček, Jiří (referee)
Nekonečno, ať to matematické, filozofické nebo teologické, fascinovalo a fascinuje lidstvo od počátků utváření vědeckého myšlení dodnes. Mnoho matematiků se nechalo omámit slastným pocitem toho, kdo rozřešil záhadu, když filozofovali nad problémy založenými na nekonečnu. Stejný pocit mohou zažít dnešní studenti, když znovu objevují překvapivé vlastnosti nekonečného. Podobné pocity jsem měla při studiu matematické analýzy, teorie množin nebo Zénónových aporií i já. Asi právě proto jsem se rozhodla zabývat se ve své disertační práci nekonečnem. Protože mám i zálibu v historii matematiky a měla jsem to štěstí pracovat pod vedením profesora Petra Vopěnky, bylo nasnadě zaměřit práci právě na srovnání ontogenetického a fylogenetického vývoje porozumění nekonečnu. Práce je rozdělena do 5 kapitol. V elektronické verzi je navíc doplněna o rejstřík pojmů. V první kapitole představuji argumenty pro i proti srovnávání fylogenetického a ontogenetického kognitivního vývoje. Druhá kapitola je exkurzem do historie matematiky se zaměřením na nekonečno, speciálně na výklady bodu, přímky a kontinua a jevů s tím souvisejících. Obsahuje postřehy o pojetí nekonečna od dob antické matematiky až po Cantorovu teorii množin. Tato kapitola mi jednak umožňuje vytypovat hlavní zdroje epistemologických překážek a dále konstruovat...
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