439 research outputs found
Class forcing, the forcing theorem and Boolean completions
The forcing theorem is the most fundamental result about set forcing, stating
that the forcing relation for any set forcing is definable and that the truth
lemma holds, that is everything that holds in a generic extension is forced by
a condition in the relevant generic filter. We show that both the definability
(and, in fact, even the amenability) of the forcing relation and the truth
lemma can fail for class forcing. In addition to these negative results, we
show that the forcing theorem is equivalent to the existence of a (certain kind
of) Boolean completion, and we introduce a weak combinatorial property
(approachability by projections) that implies the forcing theorem to hold.
Finally, we show that unlike for set forcing, Boolean completions need not be
unique for class forcing
Class forcing and second-order arithmetic
We provide a framework in a generalization of Gödel-Bernays set theory for performing class forcing. The forcing theorem states that the forcing relation is a (definable) class in the ground model (definability lemma) and that every statement that holds in a class-generic extension is forced by a condition in the generic filter (truth lemma). We prove both positive and negative results concerning the forcing theorem. On the one hand, we show that the definability lemma for one atomic formula implies the forcing theorem for all formulae in the language of set theory to hold. Furthermore, we introduce several properties which entail the forcing theorem. On the other hand, we give both counterexamples to the definability lemma and the truth lemma. In set forcing, the forcing theorem can be proved for all forcing notions by constructing a unique Boolean completion. We show that in class forcing the existence of a Boolean completion is essentially equivalent to the forcing theorem and, moreover, Boolean completions need not be unique. The notion of pretameness was introduced to characterize those forcing notions which preserve the axiom scheme of replacement. We present several new characterizations of pretameness in terms of the forcing theorem, the preservation of separation, the existence of nice names for sets of ordinals and several other properties. Moreover, for each of the aforementioned properties we provide a corresponding characterization of the Ord-chain condition. Finally, we prove two equiconsistency results which compare models of ZFC (with large cardinal properties) and models of second-order arithmetic with topological regularity properties (and determinacy hypotheses). We apply our previous results on class forcing to show that many important arboreal forcing notions preserve the boldface Pi_1^1-perfect set property over models of second-order arithmetic and also give an example of a forcing notion which implies the boldface Pi_1^1-perfect set property to fail in the generic extension.Wir führen Klassenforcing im axiomatischen Rahmen einer Verallgemeinerung von Gödel- Bernays-Mengenlehre ein. Das Forcing-Theorem besagt, dass die Forcingrelation eine (definierbare) Klasse im Grundmodell ist (Definierbarkeitslemma), und dass jede Aussage in einer generischen Erweiterung von einer Bedingung im generischen Filter erzwungen wird (Wahrheitslemma). Wir beweisen sowohl positive als auch negative Resultate über das Forcing-Theorem. Einerseits zeigen wir, dass das Definierbarkeitslemma für eine einzige atomare Formel reicht, um das Forcing-Theorem für alle Formeln in der Sprache der Mengenlehre zeigen. Außerdem stellen wir mehrere kombinatorische Eigenschaften von Klassenforcings vor, welche das Forcing-Theorem implizieren. Andrerseits präsentieren wir Gegenbeispiele für das Definierbarkeitslemma sowie für das Wahrheitslemma im Kontext von Klassenforcing. Im Mengenforcing ist das Forcing-Theorem eine Konsequenz der Existenz einer eindeutigen Booleschen Vervollständigung. Wir zeigen, dass im Klassenforcing die Existenz einer Booleschen Vervollständigung im Wesentlichen äquivalent zum Forcing-Theorem ist, und dass Boolesche Vervollständiungen im Allgemeinen nicht eindeutig sind. Pretameness ist eine Eigenschaft von Klassenforcings, welche definiert wurde um die Erhaltung des Ersetzungsaxioms zu charakterisieren. Wir beweisen mehrere neue Charakterisierungen von Pretameness anhand des Forcing-Theorems, der Erhaltung des Aussonderungsaxioms, der Existenz von Nice Names für Mengen von Ordinalzahlen sowie weiteren Eigenschaften von Klassenforcings. Des Weiteren verwenden wir alle diese Eigenschaften um die Ord-Kettenbedingung zu charakterisieren. Zu guter Letzt geben wir zwei Äquikonsistenzresultate an, welche Modelle von ZFC (mit grossen Kardinalzahlen) und Modelle der zweistufigen Arithmetik mit topologischer Regularität (und Determiniertheit) vergleichen. Wir wenden unsere Resultate über Klassenforcing an um nachzuweisen, dass zahlreiche wichtige Beispiele von Baumforcings die Π11-perfekte-Teilmengeneigenschaft über Modelle der zweistufigen Arithmetik erhalten. Andrerseits erläutern wir ein Beispiel eines Klassenforcings, welches die Π11-perfekte- Teilmengeneigenschaft in generischen Erweiterungen zerstört
Characterizations of pretameness and the Ord-cc
It is well known that pretameness implies the forcing theorem, and that
pretameness is characterized by the preservation of the axioms of
, that is without the power set axiom, or
equivalently, by the preservation of the axiom scheme of replacement, for class
forcing over models of . We show that pretameness in fact has
various other characterizations, for instance in terms of the forcing theorem,
the preservation of the axiom scheme of separation, the forcing equivalence of
partial orders and their dense suborders, and the existence of nice names for
sets of ordinals. These results show that pretameness is a strong dividing line
between well and badly behaved notions of class forcing, and that it is exactly
the right notion to consider in applications of class forcing. Furthermore, for
most properties under consideration, we also present a corresponding
characterization of the -chain condition
Category forcings, , and generic absoluteness for the theory of strong forcing axioms
We introduce a category whose objects are stationary set preserving complete
boolean algebras and whose arrows are complete homomorphisms with a stationary
set preserving quotient. We show that the cut of this category at a rank
initial segment of the universe of height a super compact which is a limit of
super compact cardinals is a stationary set preserving partial order which
forces and collapses its size to become the second uncountable
cardinal. Next we argue that any of the known methods to produce a model of
collapsing a superhuge cardinal to become the second uncountable
cardinal produces a model in which the cutoff of the category of stationary set
preserving forcings at any rank initial segment of the universe of large enough
height is forcing equivalent to a presaturated tower of normal filters. We let
denote this statement and we prove that the theory of
with parameters in is generically invariant
for stationary set preserving forcings that preserve . Finally we
argue that the work of Larson and Asper\'o shows that this is a next to optimal
generalization to the Chang model of Woodin's generic
absoluteness results for the Chang model . It remains open
whether and are equivalent axioms modulo large cardinals
and whether suffices to prove the same generic absoluteness results
for the Chang model .Comment: - to appear on the Journal of the American Mathemtical Societ
Indeterminateness and `The' Universe of Sets: Multiversism, Potentialism, and Pluralism
In this article, I survey some philosophical attitudes to talk concerning `the' universe of sets. I separate out four different strands of the debate, namely: (i) Universism, (ii) Multiversism, (iii) Potentialism, and (iv) Pluralism. I discuss standard arguments and counterarguments concerning the positions and some of the natural mathematical programmes that are suggested by the various views
Blow up and Blur constructions in Algebraic Logic
The idea in the title is to blow up a finite structure, replacing each
'colour or atom' by infinitely many, using blurs to represent the resulting
term algebra, but the blurs are not enough to blur the structure of the finite
structure in the complex algebra. Then, the latter cannot be representable due
to a {finite- infinite} contradiction. This structure can be a finite clique in
a graph or a finite relation algebra or a finite cylindric algebra. This theme
gives examples of weakly representable atom structures that are not strongly
representable. Many constructions existing in the literature are placed in a
rigorous way in such a framework, properly defined.
This is the essence too of construction of Monk like-algebras, one constructs
graphs with finite colouring (finitely many blurs), converging to one with
infinitely many, so that the original algebra is also blurred at the complex
algebra level, and the term algebra is completey representable, yielding a
representation of its completion the complex algebra.
A reverse of this process exists in the literature, it builds algebras with
infinite blurs converging to one with finite blurs. This idea due to Hirsch and
Hodkinson, uses probabilistic methods of Erdos to construct a sequence of
graphs with infinite chromatic number one that is 2 colourable. This
construction, which works for both relation and cylindric algebras, further
shows that the class of strongly representable atom structures is not
elementary.Comment: arXiv admin note: text overlap with arXiv:1304.114
Forcing and the Universe of Sets: Must we lose insight?
A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often forcing constructions that add subsets to models are cited as evidence in favour of the latter. This paper informs this debate by analysing ways the Universist might interpret this discourse that seems to necessitate the addition of subsets to . We argue that despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We analyse extant interpretations of such talk, and argue that while tradeoffs in naturality have to be made, they are not too severe
Perfect subsets of generalized Baire spaces and long games
We extend Solovay's theorem about definable subsets of the Baire space to the
generalized Baire space , where is an uncountable
cardinal with . In the first main theorem, we show
that that the perfect set property for all subsets of
that are definable from elements of is consistent
relative to the existence of an inaccessible cardinal above . In the
second main theorem, we introduce a Banach-Mazur type game of length
and show that the determinacy of this game, for all subsets of
that are definable from elements of
as winning conditions, is consistent relative to the
existence of an inaccessible cardinal above . We further obtain some
related results about definable functions on and
consequences of resurrection axioms for definable subsets of
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