An Arithmetical-like Theory of Hereditarily Finite Sets
Resumo
This paper presents the (second-order) theory of hereditarily finite sets according to the usual pattern adopted in the presentation of the (second-order) theory of natural numbers. To this purpose, we consider three primitive concepts, together with four axioms, which are analogous to the usual Peano axioms. From them, we prove a homomorphism theorem, its converse, categoricity, and a kind of (semantical) completeness.
Palavras-chave:
second order arithmetic, hereditarily finite sets, homomorphism theorem, categoricity
Referências
Ackermann, W. (1937). Die widerspruchsfreiheit der allgemeinen mengenlehre. Mathe matische Annalen, 114(1):305–315.
Awodey, S. and Reck, E. H. (2002). Completeness and categoricity. part i: Nineteenth century axiomatics to twentieth-century metalogic. History and Philosophy of Logic, 23(1):1–30.
Cohen, P. J. (2008). Set theory and the continuum hypothesis. Courier Corporation.
Dedekind, R. (1965). Was sind und was sollen die zahlen? In Was sind und was sollen die Zahlen?. Stetigkeit und Irrationale Zahlen, pages 1–47. Springer.
Givant, S. and Tarski, A. (1977). Peano arithmetic and the zermelo-like theory of sets with finite ranks. Notices of the American Mathematical Society, 77:E51.
Kirby, L. (2009). Finitary set theory. Notre Dame Journal of Formal Logic, 50(3):227–244.
Lawvere, F. W. (1964). An elementary theory of the category of sets. Proceedings of the National academy of Sciences of the United States of America, 52(6):1506.
Lawvere, F. W. and McLarty, C. (2005). An elementary theory of the category of sets (long version) with commentary. Reprints in Theory and Applications of Categories, 11:1–35.
Peano, G. (1889). Arithmetices principia: Nova methodo exposita. Fratres Bocca.
Previale, F. (1994). Induction and foundation in the theory of hereditarily finite sets. Archive for Mathematical Logic, 33(3):213–241.
Smolka, G. and Stark, K. (2016). Hereditarily finite sets in constructive type theory. In International Conference on Interactive Theorem Proving, pages 374–390. Springer.
Swierczkowski, S. (2003). Finite sets and Godel’s incompleteness theorems ¨ .
Takahashi, M.-o. (1976). A foundation of finite mathematics. Publications of the Re search Institute for Mathematical Sciences, 12(3):577–70
Awodey, S. and Reck, E. H. (2002). Completeness and categoricity. part i: Nineteenth century axiomatics to twentieth-century metalogic. History and Philosophy of Logic, 23(1):1–30.
Cohen, P. J. (2008). Set theory and the continuum hypothesis. Courier Corporation.
Dedekind, R. (1965). Was sind und was sollen die zahlen? In Was sind und was sollen die Zahlen?. Stetigkeit und Irrationale Zahlen, pages 1–47. Springer.
Givant, S. and Tarski, A. (1977). Peano arithmetic and the zermelo-like theory of sets with finite ranks. Notices of the American Mathematical Society, 77:E51.
Kirby, L. (2009). Finitary set theory. Notre Dame Journal of Formal Logic, 50(3):227–244.
Lawvere, F. W. (1964). An elementary theory of the category of sets. Proceedings of the National academy of Sciences of the United States of America, 52(6):1506.
Lawvere, F. W. and McLarty, C. (2005). An elementary theory of the category of sets (long version) with commentary. Reprints in Theory and Applications of Categories, 11:1–35.
Peano, G. (1889). Arithmetices principia: Nova methodo exposita. Fratres Bocca.
Previale, F. (1994). Induction and foundation in the theory of hereditarily finite sets. Archive for Mathematical Logic, 33(3):213–241.
Smolka, G. and Stark, K. (2016). Hereditarily finite sets in constructive type theory. In International Conference on Interactive Theorem Proving, pages 374–390. Springer.
Swierczkowski, S. (2003). Finite sets and Godel’s incompleteness theorems ¨ .
Takahashi, M.-o. (1976). A foundation of finite mathematics. Publications of the Re search Institute for Mathematical Sciences, 12(3):577–70
Publicado
18/07/2021
Como Citar
CERIOLI, Márcia R.; KRAUSS, Vitor; VIANA, Petrucio.
An Arithmetical-like Theory of Hereditarily Finite Sets. In: WORKSHOP BRASILEIRO DE LÓGICA (WBL), 2. , 2021, Evento Online.
Anais [...].
Porto Alegre: Sociedade Brasileira de Computação,
2021
.
p. 17-24.
ISSN 2763-8731.
DOI: https://doi.org/10.5753/wbl.2021.15774.
