ZF and the axiom of choice in some paraconsistent set theories

Libert, Thierry

doi:10.12775/llp.2003.005

Cited by 6 publications

(5 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…24 Now P cl (X) × P cl (X), d max is also a complete metric space with Thus, seeing that F(X) ⊂ P cl (X) × P cl (X), it will be a complete metric space provided we show that F(X) is closed:…”

Section: Working With Compact Complete Metric Spacesmentioning

confidence: 98%

“…The models we are going to present were first introduced and manufactured in a different way in [22], and then further studied in [24]. It seems quite clear that the construction of such models should involve other techniques.…”

Section: The Issuementioning

confidence: 99%

“…The natural paraconsistent first-order theory to which all these properties give rise is axiomatized in [24]. There it was conjectured that, with a relevant formulation of an axiom of infinity, that theory could interpret ZF.…”

Section: The Modelmentioning

confidence: 99%

See 2 more Smart Citations

Models for a paraconsistent set theory

Libert

2005

Journal of Applied Logic

Self Cite

View full text Add to dashboard Cite

In this paper the existence of natural models for a paraconsistent version of naive set theory is discussed. These stand apart from the previous attempts due to the presence of some non-monotonic ingredients in the comprehension scheme they fulfill. Particularly, it is proved here that allowing the equality relation in formulae defining sets, within an extensional universe, compels the use of nonmonotonic operators. By reviewing the preceding attempts, we show how our models can naturally be obtained as fixed points of some functor acting on a suitable category (stressing the use of fixed-point arguments in obtaining such alternative semantics).

show abstract

Section: Working With Compact Complete Metric Spacesmentioning

confidence: 98%

Section: The Issuementioning

confidence: 99%

See 1 more Smart Citation

Models for a paraconsistent set theory

Libert

2005

Journal of Applied Logic

Self Cite

View full text Add to dashboard Cite

show abstract

“…There are many approaches to paraconsistent set theory: for LP set theory, see (Restall, 1992;Martinez, 2021); for non-transitive approaches, see (Ripley, 2015;Istre, 2017); for others, see (Libert, 2003), (Carnielli and Coniglio, 2016, ch. 8), (Batens, 2020).…”

Section: } Obeying Some Limited Form Of Inductionmentioning

confidence: 99%

True, Untrue, Valid, Invalid, Provable, Unprovable

Weber

2024

LLP

View full text Add to dashboard Cite

There are many approaches to paraconsistency, ranging from the very moderate to the more radical. In this paper I explore and extend the more radical end of the spectrum, where there are truth-value gluts. In particular I will look at paraconsistent metatheory – the machinery of truth, validity, and proof  as developed in a glut-friendly paraconsistent setting. The aim is to evaluate the philosophical and technical tenability of such an approach. I will show that there are very significant technical challenges to face on this sort of radical approach, but that there is good philosophical support for facing these challenges.

show abstract

“…In this theory, the axiom of choice is discussed. The second technique leads to the proof the model which is constructed by any classical set theory universe (for example, a model of ZF) by using corresponding mutual simulation techniques [8]. In 2005, T. Libert pointed out in Models for Paraconsistent Set Theory, "From the Russell paradox we know Cantor's naive set theory based on the first-order axiomatization of the classical logic is inconsistent, while the classical solution to contradictions is to get rid of 'contradictions set'".…”

mentioning

confidence: 99%

Research on Set Theory Based on Paraconsistent Logic

Shi¹

2020

IJP

View full text Add to dashboard Cite

Different from ZF axiomatic set theory, the paraconsistent set theory has changed the basic logic of set theory and selected paraconsistent logic which can accommodate or deal with contradictions, it effectively avoids the whole theory falling into a non-trivial dilemma when there are contradictions in set theory. In this paper, we first review the history and current situation of the praconsistent set theory; then, we give three kinds of paraconsistent logic which can be used to construct the praconsistent set theory among many kinds of paraconsistent logics. And then, we analyze the differences of methods of the paraconsistent set theory with strong or weak structure of paraconsistent logic and get different paraconsistent set theory. Finally, we verify that paraconsistent set theory is a new method to solve the paradox of set theor. The development of paraconsistent set theory can solve the difficulties in the development of set theory in a unique way, which is not only the extension of the application of paraconsistent logic, but also the new form and new trend of the development of set theory.

show abstract

ZF and the axiom of choice in some paraconsistent set theories

Cited by 6 publications

References 9 publications

Models for a paraconsistent set theory

Models for a paraconsistent set theory

True, Untrue, Valid, Invalid, Provable, Unprovable

Research on Set Theory Based on Paraconsistent Logic

Contact Info

Product

Resources

About