Notions of Independence: Examples and Properties

• Bruno Bandeira UFRJ http://orcid.org/0000-0003-0406-0775
• Márcia R. Cerioli UFRJ https://orcid.org/0000-0002-9712-3140
• Petrucio Viana UFF https://orcid.org/0000-0002-3517-6706

DOI: https://doi.org/10.5753/wbl.2021.15775

Resumo

We analyze the notions of independence of a set of postulates, proposed by G. Peano, E. H. Moore, A. Church and F. Harary, together with a presumably new notion inspired by H. M. Sheffer. For each of these notions, we present a proper definition and give illustrative examples. We give a complete picture of the implications between these notions of independence. Then, we make some additional observations that shed some light on these notions and some concepts related to them.

Palavras-chave: Independence of axioms, Primality, Irredundance, Complete independence, Absolute independence

Referências

Cerioli, M. R., Nobrega, H., Silveira, G., and Viana, P. (2021). On the (in)dependence of the peano axioms for natural numbers. Accepted for publication in The History and Philosophy of Logic.

Church, A. (1925). On irredundant sets of postulates. Transactions of the American Mathematical Society, 27(3):318–328.

Harary, F. (1961). A very independent axiom system. The American Mathematical Monthly, 68(2):159–162.

Moore, E. H. (1910). Introduction to a form of general analysis, volume 2. Yale University Press.

Peano, G. (1889). Arithmetices principia: Nova methodo exposita. Fratres Bocca.

Peano, G. (1891). Sul concetto di numero. Rivista Matematica, 1:256–267.

Sheffer, H. M. (1916). Mutually prime postulates. Bulletin of the American Mathematical Society, 22:287.