An Elementary Approach to the Fine Structure of L

Author:

Friedman Sy D.,Koepke Peter

Abstract

We present here an approach to the fine structure of L based solely on elementary model theoretic ideas, and illustrate its use in a proof of Global Square in L. We thereby avoid the Lévy hierarchy of formulas and the subtleties of master codes and projecta, introduced by Jensen [3] in the original form of the theory. Our theory could appropriately be called ”Hyperfine Structure Theory”, as we make use of a hierarchy of structures and hull operations which refines the traditional Lα -or Jα-sequences with their Σn-hull operations.§1. Introduction. In 1938, K. Gödel defined the model L of set theory to show the relative consistency of Cantor's Continuum Hypothesis. L is defined as a unionof initial segments which satisfy: L0 = ∅, Lλ = ∪α<λLα for limit ordinals λ, and, crucially, Lα + 1 = the collection of 1st order definable subsets of Lα. Since every transitive model of set theory must be closed under 1st order definability, L turns out to be the smallest inner model of set theory. Thus it occupies the central place in the set theoretic spectrum of models.The proof of the continuum hypothesis in L is based on the very uniform hierarchical definition of the L-hierarchy. The Condensation Lemma states that if π : MLα is an elementary embedding, M transitive, then some ; the lemma can be proved by induction on α. If a real, i.e., a subset of ω, is definable over some Lα,then by a Löwenheim-Skolem argument it is definable over some countable M as above, and hence over some , < ω1. This allows one to list the reals in L in length ω1 and therefore proves the Continuum Hypothesis in L.

Publisher

Cambridge University Press (CUP)

Subject

Logic,Philosophy

Reference6 articles.

1. The fine structure of the constructible hierarchy

2. Silver Jack H. , How to eliminate the fine structure from the work of Jensen, handwritten manuscript, 197?

3. Richardson Thomas L. , Silver machine approach to the constructible universe, Ph.D. thesis , University of California, Berkeley, 1978.

4. Representing sets of ordinals as countable unions of sets in the core model

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