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Matemaattis-Luonnontieteellinen Tiedekunta Matematiikan Ja Tilastotieteen Laitos Tekijä  Författare  Author Juho Leppänen

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Matemaattis-Luonnontieteellinen Tiedekunta Matematiikan Ja Tilastotieteen Laitos Tekijä  Författare  Author Juho Leppänen View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Helsingin yliopiston digitaalinen arkisto HELSINGIN YLIOPISTO HELSINGFORS UNIVERSITET UNIVERSITY OF HELSINKI Tiedekunta/Osasto Fakultet/Sektion Faculty Laitos Institution Department Matemaattis-luonnontieteellinen tiedekunta Matematiikan ja tilastotieteen laitos Tekijä Författare Author Juho Leppänen Työn nimi Arbetets titel Title Solovay's Theorem Oppiaine Läroämne Subject Matematiikka Työn laji Arbetets art Level Aika Datum Month and year Sivumäärä Sidoantal Number of pages Pro gradu -tutkielma Joulukuu 2014 38 s. Tiivistelmä Referat Abstract Tutkielmassa esitellään Robert Solovayn artikkelin A model of set-theory in which every set of reals is Lebesgue measurable (Annals of Mathematics, Second Series, 92, 1970) reaalijoukkojen Lebesgue-mitallisuutta koskevia tuloksia. Lähtien liikkeelle ZFC:n numeroituvasta ja transitiivises- ta mallista jossa lisäksi oletetaan olevan saavuttamaton kardinaali Solovay konstruoi artikke- lissa ZF+DC:n mallin, missä jokainen reaalilukujen osajoukko on Lebesgue-mitallinen. Edellä DC viittaa riippuvan valinnan aksioomaan, joka implikoi numeroituvan valinnan aksiooman, ja on siten riittävä todistamaan kaikki tavanomaiset positiiviset tulokset Lebesguen mitasta. Tunnetusti ZFC todistaa ei-mitallisen joukon olemassaolon, mutta Solovayn tuloksen perusteella edes ZF + DC ei hylkää lausetta, jonka mukaan kaikki reaalijoukot olisivat Lebesgue-mitallisia. Solovayn mallin re- aalijoukoilla on edellä mainitun lisäksi myös muita tärkeitä säännöllisyysominaisuuksia, joista tässä tutkielmassa tarkastellaan täydellisen joukon ominaisuutta: mallissa jokainen reaalijoukko on joko numeroituva tai sisältää täydellisen joukon. Solovayn mallin konstruktio on sovellus Cohenin pakotusmenetelmästä, ja tutkielmassa keskitytään tarkastelemaan lähinnä kahta tapaa tuottaa pakotukseen liittyviä geneerisiä jatkeita. Ensimmäinen näistä on Lévy-romautus, missä lähtömallin saavuttamaton kardinaali pakotetaan olemaan genee- risen jatkeen seuraajakardinaali. Tutkielmassa todistettavien tulosten avulla nähdään menettelyn yhteys Solovayn mallin konstruktioon. Yhdessä tulopakotuksen perusteorian kanssa, tätä pakotus- menettelyä voidaan lisäksi soveltaa !1:n tai !2:n kombinatorisia ominaisuuksia koskeviin ongelmiin; esimerkkinä tästä osoitetaan, että kun lähtömallin saavuttamaton kardinaali Lévy-romautetaan ge- neerisen jatkeen kardinaaliksi !2, ei geneerisessä jatkeessa ole tällöin lainkaan Kurepa-puita. Toinen tarkasteltava pakotusmenettely on satunnaisen reaalin pakotus, joka mahdollistaa Cohenin pakotusmenetelmän soveltamisen mittateoreettisiin ongelmiin. Tämän pakotusmenettelyn tuotta- man geneerisen jatkeen määrää yksikäsitteinen reaaliluku nimeltään satunnainen reaali, ja näitä reaalilukuja koskevat tulokset ovat avainasemassa Solovayn mallin ominaisuuksia tutkittaessa. Kar- keasti ottaen reaaliluku on satunnainen, jos se välttää kaikki sellaiset nollamittaiset reaalien osajou- kot, joiden konstruktio on riittävän yksinkertainen. Satunnaisuuden käsite määritellään tarkasti hyödyntämällä koodia, jonka avulla pakotuksen lähtömallin ja oikean maailman Borel-joukkojen vä- lille saadaan useimmat topologiset ja mittateoreettiset ominaisuudet säilyttävä yhteys. Satunnaisten reaalien tutkimus tuottaa työkaluja, joita hyödyntämällä lopulta nähdään, että Lévy-romautuksen tuottamassa geneerisessä jatkeessa kaikki !-pituisista ordinaalijonoista määriteltävät reaalijoukot ovat Lebesgue-mitallisia. Tämä havainto on huomattava askel kohti Solovayn mallin konstruktiota. Toisaalta tulos on jo itsessään merkittävä, sillä esimerkiksi jokainen projektiivinen reaalien osajouk- ko on määriteltävissä !-jonosta ordinaaleja, joten seurauksena voidaan päätellä, että ZFC ei todista ei-mitallisen projektiivisen joukon olemassaoloa. Tutkielman lopuksi kommentoidaan myös lyhyesti Solovayn tuloksessa tehtävän (edellä mainitun) kardinaalioletuksen tarpeellisuutta. Avainsanat Nyckelord Keywords Joukko-oppi, pakotus, Lebesgue-mitallisuus Säilytyspaikka Förvaringsställe Where deposited Kumpulan tiedekirjasto Muita tietoja Övriga uppgifter Additional information Ohjaaja: Tapani Hyttinen Solovay’s Theorem Juho Leppänen October 2014 Contents 1 Notation and Preliminaries . 4 1.1 Models and Absoluteness . 4 1.2 Sets of Reals . 5 1.3 Measurability and the Perfect Set Property . 6 1.4 Relative Constructibility . 9 1.5 Forcing . 10 2 Products and the Lévy Collapse . 15 2.1 Products . 16 2.2 The Lévy Collapse . 17 2.3 Independence of Kurepa’s Hypothesis . 23 3 Random Reals . 26 4 Solovay’s Theorem . 31 4.1 Sets Definable from Sequences of Ordinals . 31 4.2 Proof of Solovay’s Theorem . 33 Introduction The classical construction of a Vitali set, originally discovered by Giuseppe Vitali in 1905, shows that there exists a set of reals that is not Lebesgue measurable. The construction, however, makes heavy use of the Axiom of Choice (AC), thus giving rise to the natural question whether the existence of non-measurable sets can be established solely on the basis of ZF1. The purpose of this thesis is to study the following answer to this question that was given by Robert Solovay in his 1970 article [Sol70]. THEOREM (Solovay). Suppose that there exists a model of ZFC with an inaccessible car- dinal. Then, there exists a model of ZF + DC in which every set of reals is Lebesgue measurable and has the perfect set property. The result is an application of Cohen’s method of forcing, and similarly to Cohen’s celebrated result on the independence of the continuum hypothesis, Solovay’s Theorem resolved a classical problem whose earliest formulations were known already at the beginning of the 20th century. Particularly remarkable was the level of the forcing ar- guments being used in the proof of the result, considering that Solovay had established his result already in 1964, only a year after Cohen’s formulation of forcing. The proof of the Theorem is mainly a study of two distinct notions of forcing that are nowadays known as the Lévy collapse and the random real forcing. The firstly men- tioned forces all ordinals below an inaccessible cardinal to become countable ordinals in the generic extension, so that the inaccessible then becomes the first uncountable ordinal !1 in the extension. The notion was devised by Azriel Lévy, who utilized it to obtain the relative consistency of ”every well-ordering of the reals that is definable from a countable sequence of ordinals is countable”, but only after Solovay’s appli- cation did the notion become a standard tool in forcing arguments. Since then, the method has proven to be useful in situations where some combinatorial principles of !1 or !2 are being considered, while assuming the existence of an inaccessible cardi- nal. An example is the model constructed by Jack Silver ([Sil71]) in which there are no Kurepa trees. The latter notion, random real forcing, was invented by Solovay for the purpose of applying Cohen’s method to solve measure theoretic problems. Most of the initial applications of forcing used partial orderings of functions in the spirit of Cohen’s original ideas, but Solovay stepped out of this framework by introducing his notion in which he considered the partial ordering of Borel sets with positive measure, ordered by set inclusion. A generic filter of such partial ordering will, in a sense, converge to a single real number called the random real, and these reals completely determine the generic extensions of this forcing. Studying properties of the random reals is a major part in the proof of Solovay’s result. A fact worth noting is that the formulation of Solovay’s Theorem which we gave above can be further strengthened and generalized: In [Sol70], Solovay considers one more regularity property of sets of reals, the property of Baire. Indeed, it can be shown 1ZF is Zermelo-Fraenkel set theory, ZFC is ZF with AC, and ZF + DC is ZF with the Principle of Dependent Choices. 2 that the model we construct for obtaining Solovay’s result satisfies also the statement ”every set of reals has the property of Baire”. The proof of this highly resembles the proof of the Lebesgue measurability portion of the Theorem, but some modifications are needed (see [Sol70] or [Jec02]). Furthermore, if R is replaced by a complete sep- arable metric space, and if moreover the Lebesgue measure is replaced by any σ-finite Borel-measure on X, the Theorem still holds (a proof of this is sketched at the end of [Sol70]). This paper consists of four sections. The first section briefly reviews essential back- ground needed for understanding the theory developed in later sections. This review includes basic results on models of set theory and Lebesgue measure, relative con- structibility and most notably forcing. The treatment of the topics reflects our ultimate desire for obtaining Solovay’s Theorem, and some of the results presented here are already crucial points in the very proof of the Theorem. Proofs are sometimes given for the sake of completeness or due to motivational reasons, but often times they are omitted as it is assumed that the reader is already familiar with most of the concepts touched upon in this section. The second section begins with a brief discussion on product forcing, and after this we swiftly proceed to consider forcing with the Lévy collapse partial ordering. We es- tablish some elementary properties of this forcing, and the section then culminates in the proof a special property which is later seen to be the key technical
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