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On Induction Principles for Partial Orders

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Abstract

Various forms of mathematical induction (induction for natural numbers, transfinite induction, structural induction, well-founded induction, Noetherian induction, etc.) are applicable to domains with some kinds of order. This naturally leads to the questions about the possibility of unification of different inductions and their generalization to wider classes of ordered domains. In the paper we propose a common framework for formulating induction proof principles in various structures and apply it to partially ordered sets. In this framework we propose a fixed induction principle which is indirectly applicable to the class of all posets. In a certain sense, this result provides a solution to the problem of formulating a common generalization of a number of well-known induction principles for discrete and continuous ordered structures.

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Notes

  1. FO means first-order in the extended signature \(\Sigma ^{\texttt {P}}_{\texttt {H}}\) (not in \(\Sigma \)).

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Authors and Affiliations

  1. Taras Shevchenko National University of Kyiv, 64/13, Volodymyrska Street, Kyiv, 01601, Ukraine

    Ievgen Ivanov

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  1. Ievgen Ivanov
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This work was the recipient of Ukrainian Logic Society Prize, 2021.

Appendix A: Isabelle Formalization for Sect. 3.2

Appendix A: Isabelle Formalization for Sect. 3.2

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Ivanov, I. On Induction Principles for Partial Orders. Log. Univers. 16, 105–147 (2022). https://doi.org/10.1007/s11787-022-00296-7

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