5 research outputs found

    Some interesting problems

    Get PDF
    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-ÎŁ 1 1 implies ÎŁ 1 1? (see Miller-Kunen [111], Becker [11]) 1.3 (Miller [111]) Does ∆ 1 1 = compactly- ∆ 1 1 imply ÎŁ 1 1 = compactly-ÎŁ 1 1? 1.4 (Prikry see [62]) Can L ∩ ω ω be a nontrivial ÎŁ 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

    Get PDF
    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

    Acta Scientiarum Mathematicarum : Tomus 51. Fasc. 1-2.

    Get PDF

    Playing Games on Sets and Models

    Get PDF
    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 sitÀkin enemmÀn. Millaisia niitÀ on? Miten ne voi luokitella? ErÀs lÀhestymistapa on luokitella ne monimutkaisuudensa mukaan. Jos joukko on helposti kuvailtavissa (avoimet, suljetut ja puoliavoimeet vÀlit, Cantorin joukko, irrationaaliluvut jne..), niin se on monimutkaisuushierarkiassa matalalla tasolla ja jos sen kuvaileminen on vaikea (jatkuvien funktioiden kuvajoukot, epÀmitalliset joukot,..) on se korkealla. Iso osa matemaattisia ongelmia voidaan palauttaa kysymykseen "Kuuluuko x joukkoon A?". Joissakin onnekkaissa tapauksissa, tÀmÀ ongelma palautuu tilanteeseen, jossa A on reaalilukujen osajoukko ja x on reaaliluku. Silloin y.o. kysymykseen vastaaminen riippuu siitÀ, kuinka korkealla monimutkaisuushierarkiassa A on... Vai onko?! Jos on, niin tÀllÀ tavalla voidaan analysoida matemaattisten ongelmien vaativuutta (jo ennen kuin niitÀ lÀhdetÀÀn ratkaisemaan!). TÀtÀ teoriaa kutsutaan deskriptiiviseksi (kuvailevaksi) joukko-opiksi. EntÀ jos matemaattinen ongelma ei palaudukkaan muotoon "Kuuluuko x joukkoon A?", missÀ A on reaalilukujen joukko? VÀitöskirjassa sama asetelma on siirretty pois reaaliluvuista ja reaalilukujen tilalla on niiden yleistyksiÀ: siinÀ missÀ reaaliluvut voidaan ilmaista numeroituvina binÀÀrijonoina, voidaan meidÀn objektit kuvata ylinumeroituvina binÀÀrijonoina. VÀitöskirjan keskeinen aihe on kehittÀÀ yllÀ mainittua monimutkaisuushierarkian teoriaa nÀille yleistetyille reaaliluvuille, jotta voitaisiin tutkia tiettyjen matemaattisten ongelmien (lÀhinnÀ malliteorian alalta) vaativuutta silloinkin, kun y.o. A ei voi olla reaalilukujen osajoukko
    corecore