##
**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.

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.

### MSC:

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 |

### Keywords:

order-isomorphism; elementary submodels; space of real numbers; topological spaces; linear orders; fields### Citations:

Zbl 0903.54002
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\textit{K. Kunen} and \textit{F. D. Tall}, J. Symb. Log. 65, No. 2, 683--691 (2000; Zbl 0960.03033)

Full Text:
DOI

### References:

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[2] | Fundamenta Mathematicae 79 pp 101– (1973) |

[3] | DOI: 10.1007/BF02764858 · Zbl 0674.54004 |

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.