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)
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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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References found in this work BETA
Model Theory.Makkai, Michael; Chang, C. C. & Keisler, H. J.
Set Theory.Jech, T.
Principia Mathematica, to *56.Whitehead, Alfred North & Russell, Bertrand
The Principles of Mathematics.Russell, Bertrand & Langer, Susanne K.

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