# Minimal elementary end extensions

Minimal elementary end extensions Suppose that $${\mathcal M}\models \mathsf{PA}$$ M ⊧ PA and $${\mathfrak X} \subseteq {\mathcal P}(M)$$ X ⊆ P ( M ) . If $${\mathcal M}$$ M has a finitely generated elementary end extension $${\mathcal N}\succ _\mathsf{end} {\mathcal M}$$ N ≻ end M such that $$\{X \cap M : X \in {{\mathrm{Def}}}({\mathcal N})\} = {\mathfrak X}$$ { X ∩ M : X ∈ Def ( N ) } = X , then there is such an $${\mathcal N}$$ N that is, in addition, a minimal extension of $${\mathcal M}$$ M iff every subset of M that is $$\Pi _1^0$$ Π 1 0 -definable in $$({\mathcal M}, {\mathfrak X})$$ ( M , X ) is the countable union of $$\Sigma _1^0$$ Σ 1 0 -definable sets. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

# Minimal elementary end extensions

, Volume 56 (6) – May 15, 2017
13 pages

/lp/springer_journal/minimal-elementary-end-extensions-6JVwDWiibJ
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Mathematical Logic and Foundations; Mathematics, general; Algebra
ISSN
0933-5846
eISSN
1432-0665
D.O.I.
10.1007/s00153-017-0556-5
Publisher site
See Article on Publisher Site

### Abstract

Suppose that $${\mathcal M}\models \mathsf{PA}$$ M ⊧ PA and $${\mathfrak X} \subseteq {\mathcal P}(M)$$ X ⊆ P ( M ) . If $${\mathcal M}$$ M has a finitely generated elementary end extension $${\mathcal N}\succ _\mathsf{end} {\mathcal M}$$ N ≻ end M such that $$\{X \cap M : X \in {{\mathrm{Def}}}({\mathcal N})\} = {\mathfrak X}$$ { X ∩ M : X ∈ Def ( N ) } = X , then there is such an $${\mathcal N}$$ N that is, in addition, a minimal extension of $${\mathcal M}$$ M iff every subset of M that is $$\Pi _1^0$$ Π 1 0 -definable in $$({\mathcal M}, {\mathfrak X})$$ ( M , X ) is the countable union of $$\Sigma _1^0$$ Σ 1 0 -definable sets.

### Journal

Archive for Mathematical LogicSpringer Journals

Published: May 15, 2017

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