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Weak arithmetic theories and their models.
Glivický, Petr ; Mlček, Josef (advisor) ; Pajas, Petr (referee)
In the present thesis we study arithmetical theories in the language of arithmetic L extended by one binary functional symbol for exponentiation. For arbitrary theory in the language of arithmetic it is possible todefine its extension in this new language Le by adding axioms postulatingbasic properties of exponentiation. We consider two axiomatic systems for exponentiation - Exp1 and Exp2. Thus exponentiation is always defined axiomatically in the theories we deal with. We show that in such theories the Fermat's last theorem is unprovable no matter how strong the original theory is. In the thesis we develop a general method of construction of exponential function. This method subsists of "splitting some original exponential function in shorter segments and of rearranging them to form new exponential function which satisfies required properties. As an application of this method three independence results for stronger variants of negation of Fermat's last theorem are prooved. As a first result we construct model of theory Ar + Exp1 defined in the thesis in which the equation x + y = z has nonzero solution for cofinally many 's. The second result allows to expand an arbitrary model of I1 to model of theory Exp2 in which again Fermat's theorem is violated by cofinally many 's. The third result is a construction...
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Limit ultrapower and non-regular universe
Klimeš, Martin ; Mlček, Josef (advisor) ; Pajas, Petr (referee)
The limit ultrapower is generalized to complete distributive lattices equipped with a ultrafilter and a partition system. This construction provides a complete characterization of the internal universe in models of nonstandard set theory: we prove that bounded part of an elementary extension of a set universe is given by suitable partition ultrapower. Our special interest is in models where a weak form of standardization holds. The Rudin-Keisler preorder on ultrafilters is defined on partition systems on ultrafilters such that it corresponds to embeddings of related partition ultrapowers, whereas Rudin-Frolík ordering characterizes those embedddings which are standardizable. Finally, the problem whether set-many elements are always enough to generate the internal universe from its standard part is considered. It's shown that the existence of a narrow elementary extension which doesn't rise from adjunction of set-many elements implies an existence of highly nonregular ultrafilters, and thus is equivalent to a large-cardinal hypothesis.
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Boolean algebras and first order theories.
Cepák, Jiří ; Mlček, Josef (advisor) ; Pajas, Petr (referee)
We will study Lindenbaum algebras and algebras of definable subsets of selected first order theories: constants theory for a, Presburger, Robinson, Peano and standard arithmetic, successor theory, successor theory with zero, theory of dense linear orders without endpoints, theory of discrete linear orders, random graph theory and theory of algebraically closed fields. For finite algebras we will determine their cardinality, for countable algebras we will determine whether they are atomic or atomless and for some of them we will carry out classification up to isomorphism using algebras FA, ASA and CA. For this purpose we will prove several general theorems.
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Theories and algebras of formulas
Garlík, Michal ; Mlček, Josef (advisor) ; Glivický, Petr (referee)
In the present work we study first-order theories and their Lindenbaum alge- bras by analyzing the properties of the chain BnT n<ω, called B-chain, where BnT is the subalgebra of the Lindenbaum algebra given by formulas with up to n free variables. We enrich the structure of Lindenbaum algebra in order to cap- ture some differences between theories with term-by-term isomorphic B-chains. Several examples of theories and calculations of their B-chains are given. We also construct a model of Robinson arithmetic, whose n-th algebras of definable sets are isomorphic to the Cartesian product of the countable atomic saturated Boolean algebra and the countable atomless Boolean algebra. 1
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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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Differential Calculus of Functions of Several Variables
Ráž, Adam ; Mlček, Josef (advisor) ; Balcar, Bohuslav (referee)
The thesis follows on Petr Vopìnka's alternative theory of sets and semisets by extending notions of in nite closeness and monad for real spaces of several variables. It speci es and explains on examples the basic terminology of this theory, namely notions of sets, semisets and domains. It brings up two worlds | an ancient and a classical one | by which it shows a dual way of looking at real functions of several variables. That is used for examining local properties like continuity, limit or derivative of a function at a point. The peak of the thesis is an alternative formulation of the implicit function theorem and the inverse function theorem. The thesis also contains translation rules, which allow us to reformulate all these results from an alternative into a traditional formulation used in mathematical analysis.
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Differential Calculus of Functions of Several Variables
Ráž, Adam ; Mlček, Josef (advisor) ; Balcar, Bohuslav (referee)
The thesis follows on Petr Vopìnka's alternative theory of sets and semisets by extending notions of in nite closeness and monad for real spaces of several variables. It speci es and explains on examples the basic terminology of this theory, namely notions of sets, semisets and domains. It brings up two worlds | an ancient and a classical one | by which it shows a dual way of looking at real functions of several variables. That is used for examining local properties like continuity, limit or derivative of a function at a point. The peak of the thesis is an alternative formulation of the implicit function theorem and the inverse function theorem. The thesis also contains translation rules, which allow us to reformulate all these results from an alternative into a traditional formulation used in mathematical analysis.
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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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