# Aftermath Of The Nothing

In J.-Y. Beziau, A. Costa-Leite & I. M. L. D’Ottaviano (eds.),

*CLE, v.81, Aftermath of the Logical Paradise*. Rio de Janeiro, État de Rio de Janeiro, Brésil: pp. 93-124 (2017)**Abstract**

This article consists in two parts that are complementary and autonomous at the same time.
In the first one, we develop some surprising consequences of
the introduction of a new constant called Lambda in order to represent the object ``nothing"
or ``void" into a standard set theory. On a conceptual level, it allows to see sets in a new light and to give a legitimacy to the empty set. On a technical level, it leads to a relative resolution of the anomaly of the intersection of a family free of sets.
In the second part, we show the interest of introducing an operator of potentiality into a standard set theory. Among other results, this operator allows to prove the existence of a hierarchy of empty sets and to propose a solution to the puzzle of "ubiquity" of the empty set.
Both theories are presented with equi-consistency results (model and interpretation).
Here is a declaration of intent : in each case, the starting point is a conceptual questionning; the technical tools come in a second time\\[0.4cm]
\textbf{Keywords:} nothing, void, empty set, null-class, zero-order logic with quantifiers, potential, effective, empty set, ubiquity, hierarchy, equality, equality by the bottom, identity, identification.

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Archival date: 2019-01-02

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References found in this work BETA

Set Theory.Jech, T.

The Identity of Indiscernibles.Forrest, Peter

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