Bulletin of Symbolic Logic 1 (4):393-407 (1995)

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 Vα indexed by the ordinal numbers. The recursive definition reads: ;Vλ = ∪v<λVv for limit ordinals λ. We can represent this hierarchy by the following picture.
Keywords No keywords specified (fix it)
Categories (categorize this paper)
DOI 10.2307/421129
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

PhilArchive copy

Upload a copy of this paper     Check publisher's policy     Papers currently archived: 69,089
Through your library

References found in this work BETA

Set Theory: An Introduction to Large Cardinals.F. R. Drake & T. J. Jech - 1976 - British Journal for the Philosophy of Science 27 (2):187-191.
Set Theory. An Introduction to Large Cardinals.Azriel Levy - 1978 - Journal of Symbolic Logic 43 (2):384-384.
Inner Models with Many Woodin Cardinals.J. R. Steel - 1993 - Annals of Pure and Applied Logic 65 (2):185-209.

View all 6 references / Add more references

Citations of this work BETA

Set-Theoretic Pluralism and the Benacerraf Problem.Justin Clarke-Doane - 2020 - Philosophical Studies 177 (7):2013-2030.
The Ethics–Mathematics Analogy.Justin Clarke-Doane - 2020 - Philosophy Compass 15 (1).

View all 22 citations / Add more citations

Similar books and articles

Splitting Number at Uncountable Cardinals.Jindřich Zapletal - 1997 - Journal of Symbolic Logic 62 (1):35-42.
Gap Forcing: Generalizing the Lévy-Solovay Theorem.Joel David Hamkins - 1999 - Bulletin of Symbolic Logic 5 (2):264-272.
Abstract Logic and Set Theory. II. Large Cardinals.Jouko Väänänen - 1982 - Journal of Symbolic Logic 47 (2):335-346.
On Measurable Limits of Compact Cardinals.Arthur W. Apter - 1999 - Journal of Symbolic Logic 64 (4):1675-1688.
Nonexistence of Universal Orders in Many Cardinals.Menachem Kojman & Saharon Shelah - 1992 - Journal of Symbolic Logic 57 (3):875-891.
Proper Forcing and L(ℝ).Itay Neeman & Jindřich Zapletal - 2001 - Journal of Symbolic Logic 66 (2):801-810.
Chains of End Elementary Extensions of Models of Set Theory.Andrés Villaveces - 1998 - Journal of Symbolic Logic 63 (3):1116-1136.


Added to PP index

Total views
87 ( #132,404 of 2,499,037 )

Recent downloads (6 months)
1 ( #419,059 of 2,499,037 )

How can I increase my downloads?


My notes