The ontological import of adding proper classes

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Palavras-chave:

Set theory, Class theory, Formal ontology.

Resumo

In this article, we analyse the ontological import of adding classes to set theories. We assume that this increment is well represented by going from ZF system to NBG. We thus consider the standard techniques of reducing one system to the other. Novak proved that from a
model of ZF we can build a model of NBG (and vice versa), while Shoenfield have shown that from a proof in NBG of a set-sentence we can generate a proof in ZF of the same formula. We argue that the first makes use of a too strong metatheory. Although meaningful, this symmetrical reduction does not equate the ontological content of the theories. The strong metatheory levels the two theories. Moreover,
we will modernize Shoenfield’s proof, emphasizing its relation to Herbrand’s theorem and that it can only be seen as a partial type of reduction. In contrast with symmetrical reductions, we believe that asymmetrical relations are powerful tools for comparing ontological content. In virtue of this, we prove that there is no interpretation of NBG in ZF, while NBG trivially interprets ZF. This challenges the standard view that the two systems have the same ontological content.

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Biografia do Autor

Alfredo Roque de Oliveira Freire Filho, University of Campinas

PhD in Philosophy, Department of Philosophy, University of Campinas.

Rodrigo de Alvarenga Freire, University of Brasilia

PhD in Mathematics from the University of São Paulo. Associate Professor at the University of Brasilia.

Referências

[1] Freire, A. R. Estudo Comparado do Comprometimento Ontológico das Teorias de Classes e Conjuntos, PhD thesis, Campinas, 2019.

[2] Freire, A. R. On what counts as a translation. In: Logica Yearbook, pp. 61-76 2018.

[3] Freire, A. R. Translation non interpretable theories. Forthcoming in: South America Journal of Logic.

[4] Freire, R. On existence in set theory. In: Notre Dame Journal of Formal Logic, 53, 4, pp. 525– 547, 2012.

[5] Shoenfield, J. R. Mathematical logic. AddisonWestley Publ. Comp, 1967.

[6] Jech, T. Set Theory Springer Science & Business Media, 2013.

[7] Murawski, R. and Mickiewicz, A. John von Neumann and Hilbert’s School of Foundations of Mathematics. Studies in Logic, Grammar and Rhetoric, 2004.

[8] Novak, I.L. Models of consistency systems. Fundamenta Mathematica, pp. 87-110, 1950.

[9] Shoenfield, J.R. A relative consistency proof. The Journal of Symbolic Logic 19, 4, pp. 21-28, 1954.

[10] Hilbert, D. and Bernays, P. Grundlagen der mathematik.[Von] D. Hilbert Und P. Bernays, 1968.

[11] Freire, R. A. and Tausk, D. V. Inner models of set theory can’t satisfy V 6= L. Preprint, 2009.

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Publicado

2019-09-17

Como Citar

FREIRE FILHO, A. R. de O.; FREIRE, R. de A. The ontological import of adding proper classes. Manuscrito: Revista Internacional de Filosofia, Campinas, SP, v. 42, n. 2, p. 85–112, 2019. Disponível em: https://periodicos.sbu.unicamp.br/ojs/index.php/manuscrito/article/view/8656681. Acesso em: 4 out. 2022.

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