The Extender Algebra and Vagaries of Σ 21 Absoluteness Philipp

نویسندگان

  • Philipp Doebler
  • Ralf Schindler
چکیده

We review the construction of the extender algebra, a Boolean algebra which is due to Woodin, with δ-many generators. The resulting genericity iteration is applied to prove a new Σ1-absoluteness theorem for c.c.c. forcings with ordinal parameters. Additionally we introduce and discuss sets that extend to a class with unique condensation. We analyse the sets that extend to classes with unique condensation in detail and construct non-trivial examples. We show that these sets can trivialize granted large cardinals. Given a name τ for a set that extends to a class with unique condensation, we construct a genericity iteration such that all interpretations of τ are generic over the final model. We show that if τ is in a reasonable forcing extension, then such a genericity iteration behaves like genericity iterations for reals. We apply this Lemma to show two absoluteness results. We give an overview first: this paper mostly deals with the extender algebra, a Boolean algebra which was discovered by W. H. Woodin and the absoluteness results one can obtain using the extender algebra. Given a (fine-structural) sufficiently iterable countable model M that contains a Woodin cardinal δ, one can construct the extender algebra Wδ inM. This construction then has the following application due to Woodin: given some x ⊂ ω, one can find an iteration map j : M → M∗ such that x is generic over M∗ for j(Wδ). This iteration is known as a genericity iteration. In Theorem 3.1 we will construct in detail a genericity iteration. It should be stressed that x was arbitrary to begin with and even more than that: if P is a notion of forcing,M is highly iterable and τ is a forcing-name for a real in V P, then there is an iterate M∗ of M, such that regardless of the choice of G ⊂ P generic over V we have that τ is generic over M∗; this result is also due to Woodin. So all possible interpretations of the name τ are generic over M∗. We will carry out the construction of the extender algebra for a fine-structural model with a Woodin cardinal δ in Section 2; for reals this has also been done in detail by Steel in [Ste], but we also review the construction of the extender algebra, due to Woodin, for subsets of ω1. The construction for reals starts with ω-many generators, the construction for subsets of ω1 starts with δ-many generators. For the version with δ-many generators sources besides this paper are scarce: see [Far] for the coarse case and a slightly different construction of the extender algebra, or see [SS09] for another application of the extender algebra with δ-many generators. These genericity iterations are applied to prove Woodin’s Σ1-absoluteness Theorem 4.1 (see also the discussion in the following section) and a new Σ1-absoluteness theorem for c.c.c. forcings with ordinal parameters, Theorem 5.3. ∗Both authors gratefully acknowledge support by DFG grant no. SCHI 484/3-1. This article contains material from the first author’s Ph.D. thesis.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

The Extender Algebra and Vagaries of Σ 21 Absoluteness

We review the construction of the extender algebra, a Boolean algebra which is due to Woodin, with δ-many generators. The resulting genericity iteration is applied to prove a new Σ1-absoluteness theorem for c.c.c. forcings with ordinal parameters. We also introduce and discuss sets of ordinals that extend to a class with unique condensation. This paper mostly deals with the extender algebra, a ...

متن کامل

On (σ, τ)-module extension Banach algebras

Let A be a Banach algebra and X be a Banach A-bimodule. In this paper, we define a new product on $Aoplus X$ and generalize the module extension Banach algebras. We  obtain characterizations of Arens regularity, commutativity, semisimplity, and study the ideal structure and derivations of this new Banach algebra.

متن کامل

Generic Σ 1 3 Absoluteness

In this article we study the strength of Σ 1 3 absoluteness (with real parameters) in various types of generic extensions, correcting and improving some results from [2]. We shall also make some comments relating this work to the bounded forcing axioms BMM, BPFA and BSPFA. The statement " Σ 1 3 absoluteness holds for ccc forcing " means that if a Σ 1 3 formula with real parameters has a solutio...

متن کامل

Generalized states on EQ-algebras

In this paper, we introduce a notion of generalized states from an EQ-algebra E1 to another EQ-algebra E2, which is a generalization of internal states (or state operators) on an EQ-algebra E. Also we give a type of special generalized state from an EQ-algebra E1 to E1, called generalized internal states (or GI-state). Then we give some examples and basic properties of generalized (internal) st...

متن کامل

Generic absoluteness under projective forcing∗

We study the preservation of the property of L(R) being a Solovay model under projective ccc forcing extensions. We compute the exact consistency strength of the generic absoluteness of L(R) under forcing with projective ccc partial orderings and, as an application, we build models in which Martin’s Axiom holds for Σ ∼ 1 n partial orderings, but it fails for the Σ ∼ 1 n+1.

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2010