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Handbook of Set Theory pp 657–736Cite as

Σ* Fine Structure

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Abstract

Fine Structure is the name given to an analysis developed by Ronald Jensen in the late 60’s and early 70’s of Gödel’s universe L of constructible sets with which Gödel showed the consistency of the Axiom of Choice and Continuum Hypothesis with the usual axioms of set theory. As the name implies, the ramified nature of L that Gödel used, is given a microscopic treatment of how sets are constructed. Jensen sought to replace the Gödel levels L α with levels J α which are closed under certain rudimentary set functions and so are more useful for this purpose. This analysis turned out to be extremely fruitful and resulted in a wealth of information about the ordinal combinatorial structure of L, but moreover gave insight into a wealth of absolute facts about V the universe of all sets of mathematical discourse.

Gödel’s L has subsequently been much generalised in order to gain an understanding of V, and a whole spectrum of inner models are now used. Such inner models also require fine structure for their analysis (indeed even for their definition) and the Σ*-fine structure that Jensen developed is a very flexible tool covering a whole range of such inner models.

The chapter gives an account of this method, together with some history, as well as some applications to combinatorial principles such as Global-□.

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Authors and Affiliations

  1. Department of Mathematics, University of Bristol, Bristol, BS8 1TW, United Kingdom

    Philip D. Welch

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  1. Philip D. Welch
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Correspondence to Philip D. Welch .

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  1. School of Physical Sciences, University of California, Irvine, Irvine, 92697-3875, U.S.A.

    Matthew Foreman

  2. Dept. Manufacturing Engineering, Boston University, St. Mary's St. 15, Boston, 02215, U.S.A.

    Akihiro Kanamori

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Welch, P.D. (2010). Σ* Fine Structure. In: Foreman, M., Kanamori, A. (eds) Handbook of Set Theory. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5764-9_11

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