Cardinal Arithmetic: From Silver’s Theorem to Shelah’s PCF Theory

Treballs Finals del Màster de Lògica Pura i Aplicada, Facultat de Filosofia, Universitat de Barcelona, Curs: 2019-2020, Tutor: Joan Bagaria PigrauThe main goal of this master’s thesis is to give a detailed description of the major ZFC advances in cardinal arithmetic from Silver’s Theorem to Shelah’s pcf theory and his bound on 2אω. In our attempt to make this thesis as self-contained as possible, we have devoted the first chapter to review the most elementary concepts of set theory, which include all the classical results from the first period of developement of cardinal arithmetic, from 1870 to 1930, due to Cantor, Hausdorff, König, and Tarski

Definable Combinatorics of Graphs and Equivalence Relations

Let D = (X, D) be a Borel directed graph on a standard Borel space X and let χB(D) be its Borel chromatic number. If F0, …, Fn-1: X → X are Borel functions, let DF0, …, Fn-1 be the directed graph that they generate. It is an open problem if χB(DF0, …, Fn-1) ∈ {1, …, 2n + 1, ℵ0}. Palamourdas verified the foregoing for commuting functions with no fixed points. We show here that for commuting functions with the property that there is a path from each x ∈ X to a fixed point of some Fj, there exists an increasing filtration X = ⋃m &lt; ω Xm such that χB(DF0, …, Fn-1↾ Xm) ≤ 2n for each m. We also prove that if n = 2 in the previous case, then χB(DF0, F1) ≤ 4. It follows that the approximate measure chromatic number χapM(D) ≤ 2n + 1 when the functions commute. If X is a set, E is an equivalence relation on X, and n ∈ ω, then define [X]nE = {(x0, ..., xn - 1) ∈ nX: (∀i,j)(i ≠ j → ¬(xi E xj))}. For n ∈ ω, a set X has the n-Jónsson property if and only if for every function f: [X]n= → X, there exists some Y ⊆ X with X and Y in bijection so that f[[Y]n=] ≠ X. A set X has the Jónsson property if and only for every function f : (⋃n ∈ ω [X]n=) → X, there exists some Y ⊆ X with X and Y in bijection so that f[⋃n ∈ ω [Y]n=] ≠ X. Let n ∈ ω, X be a Polish space, and E be an equivalence relation on X. E has the n-Mycielski property if and only if for all comeager C ⊆ nX, there is some Borel A ⊆ X so that E ≤B E ↾ A and [A]nE ⊆ C. The following equivalence relations will be considered: E0 is defined on ω2 by x E0 y if and only if (∃n)(∀k &gt; n)(x(k) = y(k)). E1 is defined on ω(ω2) by x E1 y if and only if (∃n)(∀k &gt; n)(x(k) = y(k)). E2 is defined on ω2 by x E2 y if and only if ∑{1⁄(n + 1): x(n) ≠ y(n)} &lt; ∞. E3 is defined on ω(ω2) by x E3 y if and only if (∀n)(x(n) E0 y(n)). Holshouser and Jackson have shown that ℝ is Jónsson under AD. The present research will show that E0 does not have the 3-Mycielski property and that E1, E2, and E3 do not have the 2-Mycielski property. Under ZF + AD, ω2/E0 does not have the 3-Jónsson property. Let G = (X, G) be a graph and define for b ≥ 1 its b-fold chromatic number χ(b)(G) as the minimum size of Y such that there is a function c from X into b-sets of Y with c(x) ∩ c(y) = ∅ if x G y. Then its fractional chromatic number is χf(G) = infb χ(b)(G)⁄b if the quotients are finite. If X is Polish and G is a Borel graph, we can also define its fractional Borel chromatic number χfB(G) by restricting to only Borel functions. We similarly define this for Baire measurable and μ-measurable functions for a Borel measure μ. We show that for each countable graph G, one may construct an acyclic Borel graph G' on a Polish space such that χfBM(G') = χf(G) and χBM(G') = χ(G), and similarly for χfμ and χμ. We also prove that the implication χf(G) = 2 ⇒ χ(G) = 2 is false in the Borel setting.</p

Leiðarvísir. Its Genre and Sources, with Particular Reference to the Description of Rome

This thesis examines Leiðarvísir, a medieval itinerary from Iceland to the Holy Land. The itinerary indicates stops and distances, but is also rich in significant information on the places along the route. Leiðarvísir has been attributed to the twelfth-century Abbot Nikulás of the Benedictine abbey of Munkaþverá in Iceland and the text has been considered to be a travel account based on the direct experiences gained by the abbot during a journey. An analysis, however, of all the datable termini present in the itinerary demonstrates that the attribution of the whole itinerary to Nikulás cannot be maintained. Having taken into consideration the most relevant criteria and definitions required to categorise a text as ‘travel writing’ and as a ‘travel account’, this thesis will then show that Leiðarvísir does not share any of the distinguishing genre features typical of a travel account, and that it should rather be classified as an impersonal guide. Finally, the thesis focuses on the description of Rome in Leiðarvísir, putting it in the context of other medieval descriptions of Rome. Not only does this contextualization make evident that the description of Rome is largely based on written sources, but it also proves that some of its details are incompatible with a twelfth-century dating and with its attribution to Nikulás. It emerges that Leiðarvísir is a work composed by an erudite scholar using written sources, and that it was probably successively enriched and updated with relevant information by one or more later scribes

Minor Papal Penitentiaries of Dacia, their Lives and Careers in Context (1263 â 1408)

From 1263 â 1391, fifteen mendicants are mentioned as minor penitentiaries for the province Dacia in the Apostolic Penitentiary. Nine of them ended their careers as bishops in the Baltic and Scandinavia. The hypothesis in this article is that the Scandinavian papal penitentiaries were intermediates between a geographic periphery of Christendom and the papacy, and contributed to a further centralisation of the universal church. For this purpose it gives a comprehensive biography of the Scandinavian papal penitentiaries in context

Sixteenth Century Morisco Devotional Manuscripts in their Mediterranean Contexts

This thesis examines the religiosities of sixteenth century Morisco communities. By exploring the language, composition and contents of their devotional manuscripts, I demonstrate an alternative perspective from the current historiography. While the predominant narrative tends to focus upon the Moriscos as 'crypto-Muslims' and 'minorities', this was not the case for all Moriscos. The manuscripts examined here demonstrate the presence of communities less interested in dissimulation and far more focused on 'normative' Islamic beliefs and practices. Also, these works reveal religious interests, shared not only by the Moriscos, but also other confessional communities in the wider early modern Mediterranean. I argue that the Moriscos should be understood as complex and dynamic communities and participants in the spaces in which they lived. The thesis begins with an historical overview of Morisco communities in sixteenth century Spain. I demonstrate that far from being on the peripheries of their world(s), Morisco communities were at the centre of emerging and conflicting notions of 'Spanishness' during this period. Similarly, the language and composition of their written works evidence Morisco participation in broader social and intellectual trends. The thesis next turns to a sample of their extant sixteenth century devotional manuscripts in order to explore what the contents reveal about the textual interests of these communities. Rather than finding dissimulation and hybridity at the forefront of these works we instead see the largest textual emphasis upon Islamic beliefs and practices in their 'normative' conditions. The thesis then explores the way in which these contents are presented and demonstrates that the overarching textual focus of these works concerns the sacralisation of time with a programme of structured devotion. Once again, in exhibiting this textual interest, these Moriscos show themselves to be vibrant contributors to a Mediterranean-wide shared interest in structured devotions and sacred time. While many scholars continue to propose that the Moriscos were a 'tragic minority', these works attest to the presence of other more 'positive' narratives. In the numerous ways in which the manuscripts facilitate the sacralisation of daily life they normalise the 'everyday' rather than the 'crisis'. The thesis concludes by discussing the implications of this more complex understanding of Morisco communities in Mediterranean studies and Islamic studies more generally

Images in seals of chapters and bishops in the medieval dioceses of Norway and Iceland

The PhD thesis, Images in seals of chapters and bishops in the medieval dioceses of Norway and Iceland, is an innovative study on the imagery in the seals of chapters and bishops in the medieval dioceses of Norway and Iceland in the time frame of ca 1250-1537. The dioceses in question were those of Nidaros, Bergen, Hamar, Stavanger, Oslo, Skálholt and Hólar. The study aims partly to investigate the visual authority of the metropolitan see in Nidaros. Whether the iconography in the seals of the archbishops had an impact on the imagery in the seals of the suffragans. Another important aim is to explore architecture and architectural elements in the seals and estimate on what ground the topographical connection or accuracy in seal images rests. The first part of the work consists of an overview of both types of the seals in question and the later chapters provide an in-depth study on architecture and architectural elements in these two types of seals. Chapter 2. provides an overview of chapter seals which are different in essence from the seals of individuals. The seals are listed and described, diocese by diocese, starting with Nidaros. Chapter 3 presents the seals of bishops according to their iconographic types and is subdivided by them. The discussion in chapter 4 provides a ground for evaluation of church buildings in chapter seals in general by presenting important comparison examples where both seal and a medieval church are preserved. In chapter 5 individual examples from the dioceses in Norway and Iceland are presented. Because of a complete lack of preserved medieval church buildings in Iceland, the Icelandic chapter seals receive the most attention. Chapter 6. Deals with architectural elements in bishop´s seals. Suggestions are provided for what types of architectural elements would be appropriate in bishop´s seals and that is different from what is relevant for chapter seals. The seals with architectural elements are then compared to the built environment at their cathedrals and/or what is known about them. The conclusion of the thesis is multi-layered, providing both positive and negative answers to the research questions where topographical accuracy of church images in chapter seals is addressed and a plausible possibility in many instances.Doktorsritgerðin, Images in seals of chapters and bishops in the medieval dioceses of Norway and Iceland fjallar um myndefnið í tveimur tegundum kirkulegra miðaldainnsigla í norsku og íslensku biskupsdæmum miðalda. Annars vegar innsigli dómklerka og klaustursamfélaga og hins vegar einstaklinga, þ.e. biskupa. Samfélög dómklerka og klaustrafólks nefndust kapítuli og af því er orðið kapítulainnsigli dregið. Orðið kapítuli á einnig við um samkomustaðinn sjálfan, rýmið þar sem starfsemin fór fram. Tímarammi verksins er frá elstu varðveislu sem nær aftur á miðja 13. öld og fram til siðbreytingar. Þar er miðað við ártal kirkjuordinazíu Kristjáns III frá 1537 að teknu tilliti til breytileika eftir biskupsdæmum. Innsigli þessara hópa eru ólík í eðli sínu og því nauðsynlegt að fjalla um þau í aðskildum köflum. Kapítulainnsiglin höfðu langan líftíma, sum hver voru notuð öldum saman en aftur á móti voru biskupsinnsigli bundin við embættistíð hvers þeirra sem gat verið nokkuð breytileg. Stíll biskupsinnsigla breyttist því hraðar en kapítulainnsigla. Í köflum 2 og 3 er lagt fram yfirlit yfir varðveitt kapítula- og biskupsinnsigli í norsku og íslensku biskupsdæmunum eins og þau voru á miðöldum. Birtir það yfirlit ákv. mynstur menningartengsla sem eru mismunandi eftir því hvor flokkur innsigla á í hlut. Í köflum 4-6 er áhersla lögð á að skoða byggingar og byggingarþætti í þessum sömu flokkum innsigla. Grundvöllurinn sem liggur að baki því að kirkjumyndir voru í kapítulainnsiglum er kannaður og lögð fram skýr dæmi þar sem samanburður á innsiglismynd og traustri heimild um útlit kirkju kemur fram. Sérstök áhersla er lögð á að meta kirkjumyndir í íslensku kapítulainnsiglunum, einkum í ljósi þess að engar varðveittar miðaldakirkjur eru til í landinu. Myndefni þetta hefur því mikið gildi fyrir rannsóknir á íslenskum kirkjubyggingararfi.Rannsóknarsjóður Háskóla Ísland

Crusade preaching, c. 1095-1216: control, evolution and impact

This thesis investigates the papacy’s authority over the preaching of crusades between 1095 and 1216. It connects for the first time the administrative government of the Papal Monarchy with the data relating to known crusade participants to show the varied effectiveness of crusade preaching. The study is divided thematically into four key areas that show the different ways in which crusading was communicated: the role of successive popes as crusade preachers; the significance of epistolary communication of the popes’ crusade messages; the impact of the numerous proxy preachers who acted with or without the papacy’s authority; and the contribution of magnus rumor, which developed organically from the original crusade message but was not subject to direct papal influence. By inputting data relating to known crusaders into Geographical Information Software, it compares crusade recruits with papal letters and preaching events and proves that crusade preachers’ successes varied according to the time and regions in which they preached. The thesis shows that, despite a clear evolution in the methods of communication, the effectiveness of promoting a successful crusade recruitment campaign during a period in which successive popes sought to assert Petrine authority was constrained by external factors