The real line in elementary submodels of set theory
2000; Cambridge University Press; Volume: 65; Issue: 2 Linguagem: Inglês
10.2307/2586561
ISSN1943-5886
AutoresKenneth Kunen, Franklin D. Tall,
Tópico(s)History and Theory of Mathematics
ResumoThe 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.
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