### Lambda theory: Introduction of a constant for "nothing" into set theory, a model of consistency and most noticable conclusions

#### Abstract

The purpose of this article is to present several immediate consequences of the introduction of a new constant called Lambda in order to represent the object “nothing” or “void” into a standard set theory. The use of Lambda will appear natural thanks to its role of condition of possibility of sets.

On a conceptual level, the use of Lambda leads to a legitimation of the empty set and to a redefinition of the notion of set. It lets also clearly appear the distinction between the empty set, the nothing and the ur-elements.

On a technical level, we introduce the notion of pre-element and we suggest a formal definition of the nothing distinct of that of the null-class. Among other results, we get a relative resolution of the anomaly of the intersection of a family free of sets and the pos- sibility of building the empty set from “nothing”. The theory is presented with equi-consistency results (model and interpretation).

On both conceptual and technical levels, the introduction of Lambda leads to a resolution of the Russell’s puzzle of the null- class.

On a conceptual level, the use of Lambda leads to a legitimation of the empty set and to a redefinition of the notion of set. It lets also clearly appear the distinction between the empty set, the nothing and the ur-elements.

On a technical level, we introduce the notion of pre-element and we suggest a formal definition of the nothing distinct of that of the null-class. Among other results, we get a relative resolution of the anomaly of the intersection of a family free of sets and the pos- sibility of building the empty set from “nothing”. The theory is presented with equi-consistency results (model and interpretation).

On both conceptual and technical levels, the introduction of Lambda leads to a resolution of the Russell’s puzzle of the null- class.

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