Skip to main content
×
Home
    • Aa
    • Aa

Inner Models and Large Cardinals

  • Ronald Jensen (a1)
Abstract

In this paper, we sketch the development of two important themes of modern set theory, both of which can be regarded as growing out of work of Kurt Gödel. We begin with a review of some basic concepts and conventions of set theory.

§0. The ordinal numbers were Georg Cantor's deepest contribution to mathematics. After the natural numbers 0, 1, …, n, … comes the first infinite ordinal number ω, followed by ω + 1, ω + 2, …, ω + ω, … and so forth. ω is the first limit ordinal as it is neither 0 nor a successor ordinal. We follow the von Neumann convention, according to which each ordinal number α is identified with the set {ν ∣ ν α} of its predecessors. The ∈ relation on ordinals thus coincides with <. We have 0 = ∅ and α + 1 = α ∪ {α}. According to the usual set-theoretic conventions, ω is identified with the first infinite cardinal ℵ0, similarly for the first uncountable ordinal number ω1 and the first uncountable cardinal number ℵ1, etc. We thus arrive at the following picture:

The von Neumann hierarchy divides the class V of all sets into a hierarchy of sets V α indexed by the ordinal numbers. The recursive definition reads: (where } is the power set of x);

V λ = ∪ v Vv for limit ordinals λ. We can represent this hierarchy by the following picture.

Copyright
References
Hide All
[1] DevonK., Constructability, Springer-Verlag, 1984.
[2] DrakeF. R., Set theory: An introduction to large cardinals, North-Holland, 1974.
[3] JensenR., Measures of order 0, unpublished manuscript.
[4] KunenK., Set theory: An introduction to independence proofs, North-Holland, 1980.
[5] MartinD. A. and SteelJ. R., A proof of projective determinacy, Journal of the American Mathematical Society, vol. 2 (1989), pp. 71125.
[6] MartinD. A. and SteelJ. R., Iteration trees, Journal of the American Mathematical Society, vol. 7 (1994), pp. 173.
[7] MitchellW. and SteelJ. R., Fine structure and iteration trees, Lecture notes in logic 3, Springer-Verlag, 1994.
[8] SteelJ. R., The core model iterability problem, unpublished manuscript.
[9] SteelJ. R., Inner models with many Woodin cardinals, Annals of Pure and Applied Logic, vol. 65 (1993), pp. 185209.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 2 *
Loading metrics...

Abstract views

Total abstract views: 88 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd October 2017. This data will be updated every 24 hours.