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)

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)

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

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

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 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.

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.

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 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 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)

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.

We take I.Q. tests but not Compassion Aptitude tests. Yet mind and emotions need to be seen as two different parts of the same spectrum if the battered human psyche is to be mended. Skolimowski sees the opportunity for a new order to be brought about if the Western mind can return to, then reintegrate, the spiritual.

The geography of religion is one of the younger areas of religious studies. The subject of her interests is the reciprocal ties that stand between religion and the geographical environment. On the one hand, the influence of the geographical environment on the formation of religious imaginations is investigated, and on the other hand, feedback, that is, what kind of religion does the change in the geographical environment. In addition, this area of ​​religious studies is engaged in the distribution of individual (...) religions, demographic and statistical issues, as well as analysis of the topography of holy places, problems of religious migration, pilgrimage, missions, religious tourism, etc. (shrink)

Celem artykułu jest analiza mobilności Antonia Santa Crocego – tytularnego arcybiskupa Seleucji i nuncjusza apostolskiego w Rzeczypospolitej Obojga Narodów w latach 1627–1630. Autorzy odtworzyli podróż dyplomaty papieskiego z Rzymu do Rzeczypospolitej odbytą w 1627 r., podróż powrotną nad Tyber z 1630 r. oraz dwa wojaże po Polsce – miesięczny wyjazd do Prus z 1627 r., podjęty w związku z trwającą wojną o ujście Wisły oraz polsko-szwedzkimi rokowaniami prowadzonymi przy udziale mediatorów niderlandzkich, a także – dwumiesięczny objazd Mazowsza, będący w rzeczywistości (...) ucieczką przed zarazą, jaka w drugiej połowie 1629 r. nawiedziła ziemie Korony. Podstawą opracowania jest korespondencja Santa Crocego z Sekretariatem Stanu Stolicy Apostolskiej, nad którym pieczę sprawował kardynał nepot Francesco Barberini. Wspomniana korespondencja przechowywana jest w Archivio Apostolico Vaticano oraz w Biblioteca Apostolica Vaticana. Znaczna jej część została wydana w serii Acta Nuntiaturae Polonae (t. XXIII, vol. 1–2). Interesującym źródłem o charakterze uzupełniającym jest _Diario di Roma (1608–1644) _autorstwa Giacinto Gigliego, w którym pamiętnikarz uwiecznił wjazd Santa Crocego do Wiecznego Miasta latem 1630 r. Ponadto autorzy wykorzystali literaturę przedmiotu poświęconą zagadnieniu podróży przedstawicieli papieskich, m.in. prace ks. Henryka Damiana Wojtyski, Teresy Chynczewskiej-Hennel czy Tadeusza Fitycha. Autorzy dokonali analizy rękopiśmiennych materiałów źródłowych pod kątem mobilności nuncjusza Santa Crocego, dzięki czemu udało się uzupełnić jego itinerarium. W publikacji zwrócono uwagę na opisy odwiedzanych przez niego miejsc oraz personalia osób, z którymi przedstawiciel papieski spotykał się podczas swych wojaży. Zarysowano również pewne wątki, które będą kontynuowane w przyszłych badaniach, a dotyczą sfery organizacyjnej nuncjatury w czasie, gdy przedstawiciel papieski przebywał poza dworem. Artykuł może stanowić cenne uzupełnienie do badań nad podróżami w czasach nowożytnych, jak również przyczynek do zagadnienia związanego z mobilnością dyplomatów w czasie realizowania ich misji. (shrink)

First published in 1967, Polish Analytical Philosophy presents the first comprehensive study of Polish analytical philosophy that has been written in Polish or English, traces the origin of the Polish analytical movement, it's development in the period between the World Wars, and its decline after the Second World War. The book shows that although inspired by the British movement and in close touch with the Vienna circle, Polish philosophy acquired its own distinctive character. Analytical philosophy in Poland was actively engaged (...) in logic and language but tended rather to be interested in constructing philosophical doctrines which attempted to resolve traditional problems of ontology and epistemology. The author contrasts leading representatives of analytical philosophy in Britain and in Poland, and his study will be a valuable addition to the literature of the analytical trend in Europe. (shrink)

Artykuł charakteryzuje zmianę francuskiej pluralistycznej demokracji zmierzająca do bipolaryzacji systemu partyjnego i podziału ideologiczno - programowego na prawicę i lewicę społeczną. Dwubiegunowo ść bvła pochodną prezydenckiego ustroju V Republiki przy odpowiedzialności premiera przed parlamentem. Ważne były także przemiany w strukturze społeczeństwa na dwie podstawowe klasy i siły społeczne, tj. pracodawców i pracowników najemnych. W kryterium stosunku do własności i zakresu ingerencji państwa w gospodarkę, orientacja prawicowa obejmuje gaullistowski Ruch na Rzecz Republiki i liberalną Unię Demokracji Francuskiej, natomiast do orientacji lewicowej (...) zalicza się nurt socjalizmu reformistycznego i ruch komunistyczny. Omówione zostaną także organizacje skrajne. (shrink)

Das Buch ist historisch-systematisch angelegt. In ihm stellt der Verfasser in interdisziplinarer - philosophischer, psychologischer und theologischer - Perspektive verstehend, sympathisierend, aber auch kritisch die Funktion des Christentums bei C. G. Jung dar. Fur ihn ist der Mensch von Natur aus religios und erlebt Gott in seiner Seele. Demzufolge liegt das Wesen des Christentums im Erleben, in dem der Glaubige von unbewussten Machten uberwaltigt wird. Religiose Erfahrung ist daher ein Erleben der Archetypen des kollektiven Unbewussten, was bedeutet, dass Gott und (...) das Unbewusste im religiosen Erlebnis nicht zu unterscheiden sind. Jung betrachtet Religionen als wahr, insofern sie die Krankheiten der Seele heilen. Sie therapieren deshalb, weil in ihnen die heilenden Krafte des Unbewussten besonders fruchtbar wirken. (shrink)