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arXiv · 1703.05493

Definable Continuous Induction on Ordered Abelian Groups

Jafar S. Eivazloo
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Abstract

As mathematical induction is applied to prove statements on natural numbers, {\it continuous induction} (or, {\it real induction}) is a tool to prove some statements in real analysis.(Although, this comparison is somehow an overstatement.) Here, we first consider it on densely ordered abelian groups to prove Heine-Borel theorem (every closed and bounded interval is compact with respect to order topology) in those structures. Then, using the recently introduced notion of pseudo finite sets, we introduce a first order definable version of {\it continuous induction} in the language of ordered groups and we use it to prove a definable version of Heine-Borel theorem on densely ordered abelian groups.

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Jafar S. Eivazloo. 2017-03-16. Definable Continuous Induction on Ordered Abelian Groups. https://arxiv.org/abs/1703.05493

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