Some interesting problems

A ≤W B. (This refers to Wadge reducible.) Answer: The first question was answered by Hjorth [83] who showed that it is independent. 1.2 A subset A ⊂ ω ω is compactly-Γ iff for every compact K ⊂ ω ω we have that A ∩ K is in Γ. Is it consistent relative to ZFC that compactly-Σ 1 1 implies Σ 1 1? (see Miller-Kunen [111], Becker [11]) 1.3 (Miller [111]) Does ∆ 1 1 = compactly- ∆ 1 1 imply Σ 1 1 = compactly-Σ 1 1? 1.4 (Prikry see [62]) Can L ∩ ω ω be a nontrivial Σ 1 1 set? Can there be a nontrivial perfect set of constructible reals? Answer: No, for first question Velickovic-Woodin [192]. question Groszek-Slaman [71]. See also Gitik [67]

Notions of Relative Ubiquity for Invariant Sets of Relational Structures

Given a finite lexicon L of relational symbols and equality, one may view the collection of all L-structures on the set of natural numbers w as a space in several different ways. We consider it as: (i) the space of outcomes of certain infinite two-person games; (ii) a compact metric space; and (iii) a probability measure space. For each of these viewpoints, we can give a notion of relative ubiquity, or largeness, for invariant sets of structures on w. For example, in every sense of relative ubiquity considered here, the set of dense linear orderings on w is ubiquitous in the set of linear orderings on w

Towards establishing an epistemological position for library and information science

Bibliography: leaves 497-531.This study examines the need for and value of a theory of knowledge for library and information science that would account for the 'Ways in which given philosophical assumptions lead to certain modes of professional practice and styles of academic research. Since given theoretical standpoints influence the nature of library practice and tend to structure the way in which library and information science research is conducted, this investigation focuses on an analysis of the fundamental conceptions of knowledge, information, truth and reality in the context of the unique complex of functions of this profession. The main method applied in this study is a representative consultation and review of the literatures of library and information science, and of a few cognate or classical fields of study. A special focus is the examination and analysis of the writings of more than 40 selected library and information science theorists, as well as those of non-librarians. The inductively-derived results of this examination are reflected in analytical typologies. The holistic intellectual tradition that underlies the presumed continuities and commonalities in the typologies is developed as a framework for developing suitable criteria to establish and evaluate an appropriate epistemological position for library and information science. An epistemological position called holistic perspectivism is proposed as one which satisfies the postulated criteria. A graphic model of this position is explained as a means of demonstrating the application of holistic perspectivism in given areas of the knowledge-transfer role of library and information science

The problem of immanence in Kant and Deleuze

In The Problem of Immanence in Kant and Deleuze, I reassess Kant's project in the light of its origins in Leibnizian rationalism. In his early works Kant seeks to ground the principle of sufficient reason as a 'real' rather than a 'logical' principle; it is this project that shapes his 'critical' formulation of the problem of the 'synthetic apriori'. I claim that Kant's project of 'immanent critique' never quite escapes the continuing requirement for metaphysical and teleological grounds, and that in the Opus Posthumum we find Kant returning to his rationalist roots in order to find a new relation between self, world and God, the three Ideas of reason. In parallel to this story, I argue that in his major work Difference and Repetition, the French philosopher Gilles Deleuze effects a return to Leibnizian philosophy (in pursuit of a new account of sufficient reason) which allows him to resolve in retrospect certain problems that arose in the unfolding of Kant's philosophy. My account is conducted on both historical and philosophical levels. From the historical point of view, I suggest firstly that Deleuze's return to the problematic of 'immanence' should be seen as providing an alternative transformation of Kantianism to the better known trajectory of German idealism, one that is more faithful to Kant's project in its historical totality. Secondly, I demonstrate how Deleuze's interpretation is facilitated by insightful readings of more neglected thinkers of the post-Kantian period such as Maimon, Novalis and Holderlin. Philosophically, the weight of the thesis lies with the extensive development of two themes. Firstly Kant's theories of ideas and intuition are interpreted from a Deleuzian standpoint, in order to provide materials for a theory of nonconceptual difference. Secondly, a new perspective is taken on the question of the primacy of self-consciousness in Kantian philosophy

Playing Games on Sets and Models

The most prominent objective of the thesis is the development of the generalized descriptive set theory, as we call it. There, we study the space of all functions from a fixed uncountable cardinal to itself, or to a finite set of size two. These correspond to generalized notions of the universal Baire space (functions from natural numbers to themselves with the product topology) and the Cantor space (functions from natural numbers to the {0,1}-set) respectively. We generalize the notion of Borel sets in three different ways and study the corresponding Borel structures with the aims of generalizing classical theorems of descriptive set theory or providing counter examples. In particular we are interested in equivalence relations on these spaces and their Borel reducibility to each other. The last chapter shows, using game-theoretic techniques, that the order of Borel equivalence relations under Borel reduciblity has very high complexity. The techniques in the above described set theoretical side of the thesis include forcing, general topological notions such as meager sets and combinatorial games of infinite length. By coding uncountable models to functions, we are able to apply the understanding of the generalized descriptive set theory to the model theory of uncountable models. The links between the theorems of model theory (including Shelah's classification theory) and the theorems in pure set theory are provided using game theoretic techniques from Ehrenfeucht-Fraïssé games in model theory to cub-games in set theory. The bottom line of the research declairs that the descriptive (set theoretic) complexity of an isomorphism relation of a first-order definable model class goes in synch with the stability theoretical complexity of the corresponding first-order theory. The first chapter of the thesis has slightly different focus and is purely concerned with a certain modification of the well known Ehrenfeucht-Fraïssé games. There we (me and my supervisor Tapani Hyttinen) answer some natural questions about that game mainly concerning determinacy and its relation to the standard EF-gameReaalilukuja on paljon: ylinumeroituvasti. Reaalilukujen osajoukkoja on yksinkertaisen laskutoimituksen mukaan sitäkin enemmän. Millaisia niitä on? Miten ne voi luokitella? Eräs lähestymistapa on luokitella ne monimutkaisuudensa mukaan. Jos joukko on helposti kuvailtavissa (avoimet, suljetut ja puoliavoimeet välit, Cantorin joukko, irrationaaliluvut jne..), niin se on monimutkaisuushierarkiassa matalalla tasolla ja jos sen kuvaileminen on vaikea (jatkuvien funktioiden kuvajoukot, epämitalliset joukot,..) on se korkealla. Iso osa matemaattisia ongelmia voidaan palauttaa kysymykseen "Kuuluuko x joukkoon A?". Joissakin onnekkaissa tapauksissa, tämä ongelma palautuu tilanteeseen, jossa A on reaalilukujen osajoukko ja x on reaaliluku. Silloin y.o. kysymykseen vastaaminen riippuu siitä, kuinka korkealla monimutkaisuushierarkiassa A on... Vai onko?! Jos on, niin tällä tavalla voidaan analysoida matemaattisten ongelmien vaativuutta (jo ennen kuin niitä lähdetään ratkaisemaan!). Tätä teoriaa kutsutaan deskriptiiviseksi (kuvailevaksi) joukko-opiksi. Entä jos matemaattinen ongelma ei palaudukkaan muotoon "Kuuluuko x joukkoon A?", missä A on reaalilukujen joukko? Väitöskirjassa sama asetelma on siirretty pois reaaliluvuista ja reaalilukujen tilalla on niiden yleistyksiä: siinä missä reaaliluvut voidaan ilmaista numeroituvina binäärijonoina, voidaan meidän objektit kuvata ylinumeroituvina binäärijonoina. Väitöskirjan keskeinen aihe on kehittää yllä mainittua monimutkaisuushierarkian teoriaa näille yleistetyille reaaliluvuille, jotta voitaisiin tutkia tiettyjen matemaattisten ongelmien (lähinnä malliteorian alalta) vaativuutta silloinkin, kun y.o. A ei voi olla reaalilukujen osajoukko

Borel Complexity of the Isomorphism Relation for O-minimal Theories

In 1988, Mayer published a strong form of Vaught's Conjecture for o-minimal theories. She showed Vaught's Conjecture holds, and characterized the number of countable models of an o-minimal theory T if T has fewer than continuum many countable models. Friedman and Stanley have shown that several elementary classes are Borel complete. In this thesis we address the class of countable models of an o-minimal theory T when T has continuum many countable models. Our main result gives a model theoretic dichotomy describing the Borel complexity of isomorphism on the class of countable models of T. The first case is if T has no simple types, isomorphism is Borel on the class of countable models of T. In the second case, T has a simple type over a finite set A, and there is a finite set B containing A such that the class of countable models of the completion of T over B is Borel complete