On the (In)Dependence of the Peano Axioms for Natural Numbers

Bibliographic Data

ID21451209
AuthorsMárcia R Cerioli (0000-0002-9712-3140, Universidade Federal do Rio de Janeiro), Hugo Nobrega (0000-0001-7597-2208, Universidade Federal do Rio de Janeiro), Guilherme Silveira (0009-0004-8843-2015, Universidade Federal do Rio de Janeiro), Petrucio Viana (0000-0002-3517-6706, Universidade Federal Fluminense, corresponding author)
Year2022
Volume43
Issue1
Pages51-69
Publication date2022-01-02
Open AccessNo
TypeARTICLE
VenueHistory and Philosophy of Logic (JOURNAL)
Journal identifiersISSN: 0144-5340 • E-ISSN: 1464-5149
PublisherInforma UK Limited (PUBLISHER • GB)
DOI10.1080/01445340.2021.1971005
OpenAlexW3206911194
LanguageEN
References cited13

We investigate two notions of independence—(usual) independence and complete independence—applied to the Peano axioms for the sequence of natural numbers. We review the results that, although they are independent, the Peano axioms are not completely independent. The standard proof that the Peano axioms are not completely independent is algebraic, in the sense that it makes essential reference to the relationship between several mathematical structures that satisfy, or do not satisfy, these axioms. We then present an alternative logical proof, which makes no essential references to the relationship between mathematical structures. There is a completely independent set of axioms for the sequence of natural numbers, but it is based on primitives different from those originally adopted by Peano. Therefore, we present a new completely independent set of axioms based on the same set of primitives as the one originally adopted by Peano

Axiom · Axiomatic system · Discrete mathematics · Independence (probability theory) · Natural number · Peano axioms · Sequence (biology) · Set (abstract data type) · Set theory · Advanced Algebra and Logic · Computability, Logic, AI Algorithms · Computer Science · Logic, Reasoning, and Knowledge · Mathematics

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