Let M be a countable recursively saturated model of Th(), and let GAut(M), considered as a topological group. We examine connections between initial segments of M and subgroups of G. In particular, for each of the following classes of subgroups HG, we give characterizations of the class of terms of the topological group structure of H as a subgroup of G. (a) for some (b) for some (c) for some (d) for some (Here, M(a) denotes the smallest M containing a, (...) , , and .). (shrink)

We give a survey of automorphisms of countable recursively saturated models of Peano Arithmetic.

We give an examination of the automorphism group Aut of a countable recursively saturated model M of PA. The main result is a characterisation of strong elementary initial segments of M as the initial segments consisting of fixed points of automorphisms of M. As a corollary we prove that, for any consistent completion T of PA, there are recursively saturated countable models M1, M2 of T, such that Aut[ncong]Aut, as topological groups with a natural topology. Other results include a classification (...) of the normal subgroups of Aut of the form [lcub]g: g [uharr] A = idA[rcub], for sets A M, and a highly homogeneous representation of Aut as a subgroup of Aut. (shrink)

We give some information about new proofs of the incompleteness theorems, found in 1990s. Some of them do not require the diagonal lemma as a method of construction of an independent statement.

We give some information about the action of Aut on M, where M is a countable arithmetically saturated model of Peano Arithmetic. We concentrate on analogues of moving gaps and covering gaps inside M.

We give new proofs of both incompleteness theorems. We do not use the diagonalization lemma, but work with some quickly growing functions instead.

We transform the proof of the second incompleteness theorem given in [3] to a proof-theoretic version, avoiding the use of the arithmetized completeness theorem. We give also new proofs of old results: The Arithmetical Hierarchy Theorem and Tarski's Theorem on undefinability of truth; the proofs in which the construction of a sentence by means of diagonalization lemma is not needed.

Continuing the earlier research from [T. Bigorajska, H. Kotlarski, Partitioning α-large sets: some lower bounds, Trans. Amer. Math. Soc. 358 4981–5001] we show that for the price of multiplying the number of parts by 3 we may construct partitions all of whose homogeneous sets are much smaller than in [T. Bigorajska, H. Kotlarski, Partitioning α-large sets: some lower bounds, Trans. Amer. Math. Soc. 358 4981–5001]. We also show that the Paris–Harrington independent statement remains unprovable if the number of (...) colors is restricted to 2, in fact, the statement is unprovable in IΣb. Other results concern some lower bounds for partitions of pairs. (shrink)

If $M \models PA (= Peano Arithmetic)$ , we set $A^M = \{N \subset_e M: N \models PA\}$ and study this family.

For a primitive recursive consistent and strong enough theory T we construct an independent statement which has some clear metamathematical meaning.

We study cofinal extensions of models of arithmetic, in particular we show that some properties near to expandability are preserved under cofinal extensions.

In this paper we study the model theory of extensions of models of first-order Peano Arithmetic by means of the arithmetized completeness theorem applied to a definable complete extension of PA in the original model. This leads us to many interesting model theoretic properties equivalent to reflection principles and ω-consistency, and these properties together with the associated first-order schemes extending PA are studied.

Let M be a countable recursively saturated model of PA and H an open subgroup of G = Aut. We prove that I = sup {b ∈ M : ∀u < bfu = u and J = inf{b ∈ MH} may be invariant, i. e. fixed by all automorphisms of M.

We show that if M is a countable recursively saturated model of True Arithmetic, then G = Aut has nonmaximal open subgroups with unique extension to a maximal subgroup of Aut.

This book presents a detailed treatment of ordinal combinatorics of large sets tailored for independence results. It uses model theoretic and combinatorial methods to obtain results in proof theory, such as incompleteness theorems or a description of the provably total functions of a theory. In the first chapter, the authors first discusses ordinal combinatorics of finite sets in the style of Ketonen and Solovay. This provides a background for an analysis of subsystems of Peano Arithmetic as well as for combinatorial (...) independence results. Next, the volume examines a variety of proofs of Gödel's incompleteness theorems. The presented proofs differ strongly in nature. They show various aspects of incompleteness phenomena. In additon, coverage introduces some classical methods like the arithmetized completeness theorem, satisfaction predicates or partial satisfaction classes. It also applies them in many contexts. The fourth chapter defines the method of indicators for obtaining independence results. It shows what amount of transfinite induction we have in fragments of Peano arithmetic. Then, it uses combinatorics of large sets of the first chapter to show independence results. The last chapter considers nonstandard satisfaction classes. It presents some of the classical theorems related to them. In particular, it covers the results by S. Smith on definability in the language with a satisfaction class and on models without a satisfaction class. Overall, the book's content lies on the border between combinatorics, proof theory, and model theory of arithmetic. It offers readers a distinctive approach towards independence results by model-theoretic methods. (shrink)

Continuing the earlier research in [14] we give some more information about nonmaximal open subgroups of G = Aut with unique maximal extension, where ℳ is a countable recursively saturated model of True Arithmetic.

We prove that for each β, γ < ε0 there existsα < ε0 such that whenever A ⊆ ω is α ‐large and G: A → β is such that (∀a ∈ A)(psn(G (a)) ≤ a), then there exists a γ ‐large C ⊆ A on which G is nondecreasing. Moreover, we give upper bounds for α for small ordinals β ≤ ω (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim).

We study some variations of the so-called Hardy hierarchy of quickly growing functions, known from the literature, and obtain analogues of Ratajczyk's approximation lemma for them.

We show that the stabilizer of an element a of a countable recursively saturated model of arithmetic M is a maximal subgroup of Aut iff the type of a is selective. This is a point of departure for a more detailed study of the relationship between pointwise and setwise stabilizers of certain subsets of M and the types of elements in those subsets. We also show that a complete type of PA is 2-indiscernible iff it is minimal in the sense (...) of Gaifman. (shrink)

Historically, Ehrenfest’s theorem is the first one which shows that classical physics can emerge from quantum physics as a kind of approximation. We recall the theorem in its original form, and we highlight its generalizations to the relativistic Dirac particle and to a particle with spin and izospin. We argue that apparent classicality of the macroscopic world can probably be explained within the framework of standard quantum mechanics.

Our article is an overview of a selection of findings in physics relating to the issue of time—we do not present in it any “time theory” of our own. After making some general remarks on the issue of time, we present historical outline and a brief description of the current state of time interval measurements. Subsequently, we go on to discuss certain (relating to the concept of time) consequences of both theories of relativity: special and general. Here, time is a (...) geometrical component of space-time continuum. Following section is dedicated to time in the so-called Hamiltonian formulations of the theory of particles, where it appears as a parameter of evolution. The last section contains remarks referring to certain attempts of going beyond the recognized physical theories relating to the question of time. (shrink)

Our article is an overview of a selection of findings in physics relating to the issue of time—we do not present in it any “time theory” of our own. After making some general remarks on the issue of time, we present historical outline and a brief description of the current state of time interval measurements. Subsequently, we go on to discuss certain (relating to the concept of time) consequences of both theories of relativity: special and general. Here, time is a (...) geometrical component of space-time continuum. Following section is dedicated to time in the so-called Hamiltonian formulations of the theory of particles, where it appears as a parameter of evolution. The last section contains remarks referring to certain attempts of going beyond the recognized physical theories relating to the question of time. (shrink)

This article deals with basic changes in Polish society over past 20 years. On the basis of sociological surveys, the author attempts to answer the following questions: to what extent has systemic change brought about significant changes in social stratification? In which dimensions did it take place? What are the consequences of these changes for individuals and the social system? How does Polish society differ from others? As for social stratification, the only clear and unambiguous tendencies were growing inequality and (...) more marked effects of educational credentials on incomes. Contrary to expectations, barriers to mobility have not become more rigid, nor is there any clear evidence of social origins mattering more for educational achievement. Transformations in social structure have been accompanied by rising discontent with material conditions, the government, and the system as a whole. Interestingly, this wave runs counter to positive evaluations in the private sphere, displayed in relatively high self-esteem and satisfaction with friendships and family life. These optimistic attitudes contrasted with disapproval of the political class, and with the workings of democracy. The degree of legitimization of the political system in Poland appears to be the lowest among 21 European societies. When compared with the capitalist West, this constitutes one of the obstacles to modernization along Western lines. (shrink)

Decisions concerning price development are best visible in view of their direct impact on the market activities of the company and the general level of profitability. In every activity, the success is measured by an excess of sales revenue over the costs of used resources. In an ideal case, a determined price provides the highest margin resulting from an analysis of the sales volume, takes into consideration the customer's evaluation of the product/service, and allows to react to competitive threats and (...) to fend off the competition's attacks. There is no single universal way to develop prices for various enterprise types, regardless of the conditions of demand, the specific character of the sector they operate in, or the competitive situation, even if we assume that the enterprise has only one goal of activity. That's why, Author recommends three price determination methods. The ways of price determination presented in this article are just the first step on the way to reach the optimal prices. The prices determined in such way, as well as product or service prices already functioning in the market, are subject to modification depending on the assumed goals, taking into consideration the character of demand for a given commodity and the behaviour of the competition. (shrink)

We start from the assumption that the main problem, and at the same time the goal, is socio-economic development. We also assume that this pursuit is universal, development is the ambition of entities at all levels of aggregation, development is a desire of each country, as well as of business entities and individuals. From the other side, the governments create conditions for entities at the microeconomic level which generate additional value, that is, contribute to the creation of this development. Peculiar (...) to them they are developmental paths. That’s why, the main hypothesis of this article, as stating that the choice of economic policy financed by excessive public debt does not, in the long run, serve well real socio-economic development. At the beginning of the 21st century, the vast majority of these countries had serious problems with public debt. These problems already had their history and were so serious that some efforts to discipline the forum of this group appeared. It is, in a sense, natural in the sense that integration itself is heading to a high degree of coordination and to the unification of all economic policies. The picture of public finances of the European Union does not look good despite the fact that this group can boast a long history of making efforts to control and monitor the budgets of its member states. Moreover, one can even talk about a certain intensification of this process, especially with respect to the euro area countries, but we still can not say that satisfactory results have been achieved. (shrink)

The article deals with the perception of the Pope John Paul II and Poland in the two speeches of the Presidents of the United States of America. The George W. Bush’s speech given on the occasion of the dedication of the Pope John Paul II Cultural Center in Washington DC on March 22, 2001 and the Donald Trump’s speech delivered in front of the Warsaw Uprising Monument on the Krasinski Square in Warsaw on July 8, 2017 were used as the (...) sources for the study. The former speech is a laudation in honor of the Pope from Poland, whose biography was considered by the 43rd President of the United States as one of the most interesting in human history. Among the most notable episodes of the pontificate, George W. Bush listed the 1979 pilgrimage to his homeland, the 1995 visit to Manila in Philippines, the unprecedented visits to a synagogue and a Muslim country, the forgiveness shown to Ali Ağca – the would-be assassin, and the visits to the United States during which the Pope from Poland reminded the people of human dignity and the virtues of a just society. The latter speech is a laudation on the spirit of the Polish nation. Donald Trump emphasized the steadfastness of the Poles in the struggle to preserve their cultural identity despite the loss of their homeland. He recalled the period of partition and extermination of the Polish nation by the German Nazis and the Soviet Communists during World War II. The 45th President of the United States reminded the world about famous Poles: Nicolaus Copernicus, Frederic Chopin, Casimir Pulaski, Thaddeus Kosciuszko and Saint John Paul II. He also spoke about the lasting ties between the Americans and the Poles and about the common values cherished on both sides of the Atlantic. On the example of social transformations in Poland and Europe initiated by John Paul II’s pilgrimage to Poland in 1979, Donald Trump emphasized the influence of faith in God on political and social transformations in the world. He also expressed hope for the survival of Western civilization based on faith in God and the preservation of family and freedom. (shrink)