Academic literature on the topic 'Iterated forcing'

AbstractWe define the notion of Souslin forcing, and we prove that some properties are preserved under iteration. We define a weaker form of Martin's axiom, namely , and using the results on Souslin forcing we show that is consistent with the existence of a Souslin tree and with the splitting number s = ℵ1. We prove that proves the additivity of measure. Also we introduce the notion of proper Souslin forcing, and we prove that this property is preserved under countable support iterated forcing. We use these results to show that ZFC + there is an inaccessible cardinal is equiconsistent with ZFC + the Borel conjecture + -measurability.

The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Veličković. We introduce a stronger form of resurrection axioms (the iterated resurrection axioms [Formula: see text] for a class of forcings [Formula: see text] and a given ordinal [Formula: see text]), and show that [Formula: see text] implies generic absoluteness for the first-order theory of [Formula: see text] with respect to forcings in [Formula: see text] preserving the axiom, where [Formula: see text] is a cardinal which depends on [Formula: see text] ([Formula: see text] if [Formula: see text] is any among the classes of countably closed, proper, semiproper, stationary set preserving forcings). We also prove that the consistency strength of these axioms is below that of a Mahlo cardinal for most forcing classes, and below that of a stationary limit of supercompact cardinals for the class of stationary set preserving posets. Moreover, we outline that simultaneous generic absoluteness for [Formula: see text] with respect to [Formula: see text] and for [Formula: see text] with respect to [Formula: see text] with [Formula: see text] is in principle possible, and we present several natural models of the Morse–Kelley set theory where this phenomenon occurs (even for all [Formula: see text] simultaneously). Finally, we compare the iterated resurrection axioms (and the generic absoluteness results we can draw from them) with a variety of other forcing axioms, and also with the generic absoluteness results by Woodin and the second author.

AbstractWe shall show the consistency of CH+⌝(+) and CH+(+)+there are no club guessing sequences on ω1. We shall also prove that ◊+ does not imply the existence of a strong club guessing sequence on ω1.

If U is a normal measure on κ then we can add indiscernibles for U either by Prikry forcing [P] or by taking an iterated ultrapower which will add a sequence of indiscernibles for over M. These constructions are equivalent: the set C of indiscernibles for added by the iterated ultrapower is Prikry generic for [Mat]. Prikry forcing has been extended for sequences of measures of length by Magidor [Mag], and his method readily extends to . In this case the measure U is replaced by a sequence of measures and the set C of indiscernibles is replaced by a system of indiscernibles for : is a function such that (κ, β) is a set of indiscernibles for (κ, β) for each . The equivalence between forcing and iterated ultra-powers still holds true for such sequences: there is an interated ultrapower j: V → M (which is defined in detail later in this paper) such that the system of indiscernibles for j() constructed by j is Magidor generic over M.The construction of the system of indiscernibles works equally well for o(κ) ≧ κ+. Radin has defined a variant of Prikry forcing which also works for o(κ) > κ+ ([R]; see also [Mi82] where Radin forcing is applied specifically to sequences of measures, rather than to hypermeasures as in Radin's paper), but Radin's forcing is weaker than Magidor's extension of Prikry forcing in the sense that the system of indiscernibles generated by the interated ultrapower is not Radin generic for j(), but only the set . That is, an indiscernible does not belong to a specific measure, but only to the whole sequence of measures on the cardinal κ.

AbstractWe prove that if I is a partially ordered set in a countable transitive model of ZFC then can be extended by a generic sequence of reals ai, i ∈ I, such that is preserved and every ai is Sacks generic over [〈aj: j < i〉]. The structure of the degrees of -constructibility of reals in the extension is investigated.As applications of the methods involved, we define a cardinal invariant to distinguish product and iterated Sacks extensions, and give a short proof of a theorem (by Budinas) that in ω2-iterated Sacks extension of L the Burgess selection principle for analytic equivalence relations holds.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1993.
On t.p., "[omega]" appears as the lower case Greek letter.
Includes bibliographical references (leaves 38-39 ).
by Dimitrios V. Tzimas.
Ph.D.

Les principaux résultats de cette thèse sont liés au forcing, mais notre présentation bénéficie de sa mise en relation avec un autre domaine de la logique: la théorie des modèles des logiques infinitaires. Une idée clé de notre travail, qui était plus ou moins implicite dans les recherches de nombreux auteurs, est que le forcing joue un rôle en logique infinitaire similaire à celui joué par le théorème de compacité en logique du premier ordre. Plus précisément, de la même manière que le théorème de compacité est l'outil clé pour produire des modèles de théories du premier ordre, le forcing peut être l'outil clé pour produire les modèles des théories infinitaires. La première partie de cette thèse explore la relation entre les logiques infinitaires et les modèles à valeurs booléennes. Une propriété de consistance est une famille d'ensembles de formules non contradictoires, fermée sous certaines opérations logiques naturelles. Les propriétés de consistance reproduisent dans le contexte des logiques infinitaires la technique donnée par la méthode de résolution pour produire des modèles d'une formule du premier ordre; elles sont l'outil standard pour produire des modèles de formules infinitaires non contradictoires. Le premier résultat majeur que nous établissons dans cette thèse est le Théorème d'Existence des Modèles Booléens, affirmant que toute formule dans un ensemble qui est dans une propriété de consistance possède un modèle à valeurs booléennes avec la propriété de "mixing", et renforce le résultat original de Mansfield. Le Théorème d'Existence des Modèles Booléens nous permet de prouver trois résultats supplémentaires dans la théorie des modèles des logiques infinitaires munis de la sémantique des modèles à valeurs booléennes avec la propriété de ``mixing": un théorème de complétude par rapport à un calcul de type Gentzen, un théorème d'interpolation et un théorème d'omission des types. Cependant, nous croyons que le résultat central de cette partie de la thèse est le Théorème de Compacité Conservative. Dans la poursuite d'une généralisation de la compacité du premier ordre pour les logiques infinitaires, nous introduisons le concept de "renforcement conservatif" et de "conservativité finie". Nous soutenons que la généralisation appropriée de la consistance finie (relative à la sémantique de Tarski pour la logique du premier ordre) est la conservativité finie (relative à la sémantique donnée par les modèles à valeurs booléennes). À notre avis, ces résultats nous permettent de soutenir que: Les modèles à valeurs booléennes avec la propriété de "mixing" fournissent une sémantique naturelle pour les logiques infinies. Dans la seconde partie de la thèse, nous nous appuyons sur les résultats de la première partie pour aborder la question suivante: pour quelle famille de formules infinitaires peut-on forcer l'existence d'un modèle de Tarski sans détruire les sous-ensembles stationnaires? Kasum et Velickovic ont introduit une caractérisation des formules pour lesquelles un modèle de Tarski peut être forcé par un forcing préservant les ensembles stationnaires (AS-goodness). Leur travail s'appuie sur le résultat révolutionnaire d'Asperò et Schindler. Nous définissons la propriété ASK - une variante de l'AS-goodness - que nous utilisons également de la même manière que Kasum et Velickovic. Il est démontré que pour toute formule ayant la propriété ASK, on peut forcer l'existence d'un modèle de Tarski d'une manière qui préserve les ensembles stationnaires. La preuve de ce résultat s'appuie sur la perspective de la théorie des modèles de forcing présentée dans la première partie de la thèse, tout en introduisant une nouvelle notion de forcing itéré. Cette présentation du forcing itéré est étroitement liée au Théorème de Compacité Conservateur, soulignant à nouveau l'analogie entre les paires (forcing, logiques infinitaires) et (compacité, logique du premier ordre)
The main results of this thesis are related to forcing, but our presentation benefits from relating them to another domain of logic: the model theory of infinitary logics. In the 1950s, after the basic framework of first-order model theory had been established, Carol Karp, followed by Makkai, Keisler and Mansfield among others, developed the area of logic known as "infinitary logics". One key idea from our work, which was more or less implicit in the research of many, is that forcing plays a role in infinitary logic similar to the role compactness plays in first-order logic. Specifically, much alike compactness is the key tool to produce models of first-order theories, forcing can be the key tool to produce the interesting models of infinitary theories. The first part of this thesis explores the relationship between infinitary logics and Boolean valued models. Leveraging on the translation of forcing in the Boolean valued models terminology, this part lays the foundations connecting infinitary logics to forcing. A consistency property is a family of sets of non-contradictory sentences closed under certain natural logical operations. Consistency properties are the standard tools to produce models of non-contradictory infinitary sentences. The first major result we establish in the thesis is the Boolean Model Existence Theorem, asserting that any sentence which belongs to some set which is in some consistency property has a Boolean valued model with the mixing property, and strengthens Mansfield's original result. The Boolean Model Existence Theorem allows us to prove three additional results in the model theory of Boolean valued models for the semantics induced by Boolean valued models with the mixing property: a completeness theorem, an interpolation theorem, and an omitting types theorem. These can be shown to be generalizations of the corresponding results for first order logic in view of the fact that a first order sentence has a Tarski model if and only if it has a Boolean valued model. However we believe that the central result of this part of the thesis is the Conservative Compactness Theorem. In pursuit of a generalization of first-order compactness for infinitary logics, we introduce the concepts of conservative strengthening and of finite conservativity. We argue that the appropriate generalization of finite consistency (relative to Tarski semantics for first order logic) is finite conservativity (relative to the semantics given by Boolean valued models). The Conservative Compactness Theorem states that any finitely conservative family of sentences admits a Boolean valued model with the mixing property. In our opinion these results support the claim: Boolean-valued models with the mixing property provide a natural semantics for infinitary logics. In the second part of the thesis we leverage on the results of the first part to address the following question: For what family of infinitary formulae can we force the existence of a Tarski model for them without destroying stationary sets? Kasum and Velickovic introduced a characterization of which sentences can be forced by a stationary set preserving forcing (AS-goodness). Their work builds on the groundbreaking result of Asperò and Schindler. We define the ASK property -a variant of AS-goodness- which we also employ to the same effect of Kasum and Velickovic. It is shown that for any formula with the ASK-property, one can force the existence of a Tarski model in a stationary set preserving way. The proof of this result builds on the model theoretic perspective of forcing presented in the first part of the thesis, and does so introducing a new notion of iterated forcing. This presentation of iterated forcing is strictly intertwined with the Conservative Compactness Theorem, thereby emphasizing again the analogy between the pairs (forcing, infinitary logics) and (compactness, first-order logic)

This paper considers numerical simulations of two-dimensional viscous flow past oscillating cylinders using an efficient, oscillation-free, Cartesian grid based Immersed Boundary Method (IBM). The direct forcing approach originally developed by Uhlmann [1] for fixed and moving boundaries is employed. The IBM utilizes an improved smoothing technique for the discrete delta function and a solid-domain forcing strategy. A strong-coupling scheme is employed in which both, fluid and structure, are treated as linked components of a single dynamical system and all governing equations are iterated until convergence within the same time step. The accuracy, validity and efficiency of the utilized IBM are demonstrated by a series of validation cases including Vortex Induced Vibration (VIV) of an elastically mounted single cylinder and VIV assessment of a pair of cylinders in tandem arrangement. The method provides a good estimation of the single cylinder vortex lock in regime and fairly accurate predictions of the wake interference effect on the onset of resonance in flows involving multiple cylinders in an in-line arrangement.