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The real line in elementary submodels of set theory

Published online by Cambridge University Press:  12 March 2014

Kenneth Kunen  and
Franklin D. Tall
Kenneth Kunen
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706, USA, E-mail: kunen@math.wisc.edu
Franklin D. Tall
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3., Canada, E-mail: tall@math.toronto.edu
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Extract

The use of elementary submodels has become a standard tool in set-theoretic topology and infinitary combinatorics. Thus, in studying some combinatorial objects, one embeds them in a set, M, which is an elementary submodel of the universe, V (that is, (M; Є) ≺ (V; Є)). Applying the downward Löwenheim-Skolem Theorem, one can bound the cardinality of M. This tool enables one to capture various complicated closure arguments within the simple “≺”.

However, in this paper, as in the paper [JT], we study the tool for its own sake. [JT] discussed various general properties of topological spaces in elementary submodels. In this paper, we specialize this consideration to the space of real numbers, ℝ. Our models M are not in general transitive. We will always have ℝ Є M, but not usually ℝ ⊆ M. We plan to study properties of the ℝ ⋂ M's. In particular, as M varies, we wish to study whether any two of these ℝ ⋂ M's are isomorphic as topological spaces, linear orders, or fields.

As usual, it takes some sleight-of-hand to formalize these notions within the standard axioms of set theory (ZFC), since within ZFC, one cannot actually define the notion (M;Є) ≺ (V;Є). Instead, one proves theorems about M such that (M;Є) ≺ (H(θ);Є), where θ is a “large enough” cardinal; here, H(θ) is the collection of all sets whose transitive closure has size less than θ.

Keywords

Elementary submodelreal lineorder-isomorphic
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 2 , June 2000 , pp. 683 - 691
Copyright
Copyright © Association for Symbolic Logic 2000

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References

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