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]

Mengenlehre

[no abstract available

The power function

The axioms of ZFC provide very little information about the possible values of the power function (i.e. the map K---->2ᴷ). In this dissertation, we examine various theorems concerning the behaviour of the power function inside the formal system ZFC , and we :;hall be p:trticul:trly interested in results which provide eonstraints on the possible values of the power function. Thus most of the results presented here will be consistency results. A theorem of Easton (Theorem 2.3.1) shows that, when restricted to regular cardinals, the power function may take on any reasonable value, and thus a considerable part of this thesis is concerned with the power function on singular cardinals. We also examine the influence of various strong axioms of infinity, and their generalization to smaller cardinals, on the possible behaviour of the power function

The power function

The axioms of ZFC provide very little information about the possible values of the power function (i.e. the map K---->2ᴷ). In this dissertation, we examine various theorems concerning the behaviour of the power function inside the formal system ZFC , and we :;hall be p:trticul:trly interested in results which provide eonstraints on the possible values of the power function. Thus most of the results presented here will be consistency results. A theorem of Easton (Theorem 2.3.1) shows that, when restricted to regular cardinals, the power function may take on any reasonable value, and thus a considerable part of this thesis is concerned with the power function on singular cardinals. We also examine the influence of various strong axioms of infinity, and their generalization to smaller cardinals, on the possible behaviour of the power function

A framework for forcing constructions at successors of singular cardinals

We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal κ of uncountable cofinality, while κ^+ enjoys various combinatorial properties. As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal κ of uncountable cofinality where SCH fails and such that there is a collection of size less than 2^{κ^+} of graphs on κ^+ such that any graph on κ^+ embeds into one of the graphs in the collection

Generalized Domination.

This thesis develops the theory of the everywhere domination relation between functions from one infinite cardinal to another. When the domain of the functions is the cardinal of the continuum and the range is the set of natural numbers, we may restrict our attention to nicely definable functions from R to N. When we consider a class of such functions which contains all Baire class one functions, it becomes possible to encode information into these functions which can be decoded from any dominator. Specifically, we show that there is a generalized Galois-Tukey connection from the appropriate domination relation to a classical ordering studied in recursion theory. The proof techniques are developed to prove new implications regarding the distributivity of complete Boolean algebras. Next, we investigate a more technical relation relevant to the study of Borel equivalence relations on R with countable equivalence classes. We show than an analogous generalized Galois-Tukey connection exists between this relation and another ordering studied in recursion theory.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113539/1/danhath_1.pd

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

Shelah's pcf theory and its applications

This is a survey paper giving a self-contained account of Shelah&apos;s theory of the pcf function pcf(a) = {cf(PI a/D, < D): D is an ultrafilter on a}, where a is a set of regular cardinals such that \a\ < min(a). We also give several applications of the theory to cardinal arithmetic, the existence of Jonsson algebras, and partition calculus.PT: J; CR: DEVLIN KJ, 1973, ANN MATH LOGIC, V5, P291 EASTON WB, 1970, ANN MATH LOGIC, V1, P139 ERDOS P, 1984, COMBINATORIAL SET TH GALVIN F, 1975, ANN MATH, V101, P491 GITIK M, SINGULAR CARDINALS P JECH T, CONJECTURE TARSKI PR JECH T, IN PRESS TRIBUTE P E JECH TJ, 1978, SET THEORY KUNEN K, 1980, STUDIES LOGIC F MATH, V102 RUBIN M, 1987, ANN PURE APPL LOGIC, V33, P43 SHELAH S, ALEPH OMEGA PLUS ONE SHELAH S, CARDINAL ARITHMETIC SHELAH S, IN PRESS ARCH MATH L SHELAH S, MORE PCF SHELAH S, 1978, ISRAEL J MATH, V30, P57 SHELAH S, 1980, J SYMBOLIC LOGIC, V45, P56 SHELAH S, 1980, STUD LOGIC FDN MATH, V95, P373 SHELAH S, 1982, LECTURE NOTES MATH, V940 SHELAH S, 1986, NOTRE DAME J FORM L, V27, P263 SHELAH S, 1987, ISRAEL J MATH, V59, P299 SHELAH S, 1988, ISRAEL J MATH, V62, P213 SHELAH S, 1988, ISRAEL J MATH, V62, P355 SILVER J, 1974, P INT C MATH, V1, P265 TODORCEVIC S, 1986, COMPOS MATH, V57, P357 TODORCEVIC S, 1987, ACTA MATH-DJURSHOLM, V159, P261 TODORCEVIC S, 1989, CONT MATH, V84; NR: 26; TC: 27; J9: ANN PURE APPL LOGIC; PG: 48; GA: EU720Source type: Electronic(1