The Ontological Import of Adding Proper Classes

Freire, Alfredo Roque; Freire, Rodrigo de Alvarenga

doi:10.1590/0100-6045.2019.v42n2.rf

Cited by 2 publications

(2 citation statements)

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“…Theories PA 2 , ZFC − , and GBc − have been known to be equiconsistent for a while, see e.g., [20,30,31] for PA 2 vs. ZFC − , and [32][33][34] for ZFC − vs. GBc − . Such objects as ω 1 and HC are legitimate classes in GBc − , and such are all ZFCsets that play any role in the proof of Theorem 1 above, with one notable exception.…”

Section: Proof Of Theorem 2 and Commentsmentioning

confidence: 99%

A Model in Which Well-Orderings of the Reals First Appear at a Given Projective Level, Part II

Kanovei

Lyubetsky

2023

Mathematics

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We consider the problem of the existence of well-orderings of the reals, definable at a certain level of the projective hierarchy. This research is motivated by the modern development of descriptive set theory. Given n≥3, a finite support product of forcing notions similar to Jensen’s minimal-13-real forcing is applied to define a model of set theory in which there exists a good Δ1n of the reals, but there are no Δ1n−1s of the reals (not necessarily good). We conclude that the existence of a good well-ordering of the reals at a certain level n≥3 of the projective hierarchy is strictly weaker than the existence of a such well-ordering at the previous level n−1. This is our first main result. We also demonstrate that this independence theorem can be obtained on the basis of the consistency of ZFC‾ (that is, a version of ZFC without the Power Set axiom) plus `there exists the power set of ω’, which is a much weaker assumption than the consistency of ZFC usually assumed in such independence results obtained by the forcing method. This is our second main result. Further reduction to the consistency of second-order Peano arithmetic PA2 is discussed. These are new results in such a generality (with n≥3 arbitrary), and valuable improvements upon earlier results. We expect that these results will lead to further advances in descriptive set theory of projective classes.

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Section: Proof Of Theorem 2 and Commentsmentioning

confidence: 99%

A Model in Which Well-Orderings of the Reals First Appear at a Given Projective Level, Part II

Kanovei

Lyubetsky

2023

Mathematics

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“…This research was supported by grant 2017/21020-0, São Paulo Research Foundation (FAPESP). The research project grew out of the first author's Ph.D. dissertation [5], with related philosophical work in [4,6].…”

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confidence: 99%

Bi-Interpretation in Weak Set Theories

Freire¹,

Hamkins²

2020

J. symb. log.

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In contrast to the robust mutual interpretability phenomenon in set theory, Ali Enayat proved that bi-interpretation is absent: distinct theories extending ZF are never bi-interpretable and models of ZF are bi-interpretable only when they are isomorphic. Nevertheless, for natural weaker set theories, we prove, including Zermelo-Fraenkel set theory ZFCwithout power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of ZFCthat are bi-interpretable, but not isomorphic-even H 1 , ∈ and H 2 , ∈ can be bi-interpretable-and there are distinct bi-interpretable theories extending ZFC -. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails.

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The Ontological Import of Adding Proper Classes

Abstract: 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

Cited by 2 publications

References 4 publications

A Model in Which Well-Orderings of the Reals First Appear at a Given Projective Level, Part II

A Model in Which Well-Orderings of the Reals First Appear at a Given Projective Level, Part II

Bi-Interpretation in Weak Set Theories

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