The real line in elementary submodels of set theory. (English) Zbl 0960.03033

From the text: 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, \((M;\in)\prec (V;\in))\). 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 “\(\prec\)”.
However, in this paper, as in the paper “The topology of elementary submodels” [L. R. Junqueira and F. D. Tall, Topology Appl. 82, No. 1-3, 239-266 (1998; Zbl 0903.54002)], we study the tool for its own sake. Junqueira and Tall discussed various general properties of topological spaces in elementary submodels. In this paper, we specialize this consideration to the space of real numbers, \(\mathbb{R}\). Our models \(M\) are not in general transitive. We will always have \(\mathbb{R}\in M\), but not usually \(\mathbb{R}\subseteq M\). We plan to study properties of the \(\mathbb{R}\cap M\)’s. In particular, as \(M\) varies, we wish to study whether any two of these \(\mathbb{R}\cap M\)’s are isomorphic as topological spaces, linear orders, or fields.


03C62 Models of arithmetic and set theory
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
03E55 Large cardinals
03C55 Set-theoretic model theory
06A05 Total orders
12L99 Connections between field theory and logic


Zbl 0903.54002
Full Text: DOI


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