Material set theory

Elementhood Relation Class Axiom

The Elementhood Relation Class Axiom is a possible axiom of material set theory with classes: nbg

(∃ℭ𝔩𝔰⁡𝑋)(∀𝑢)(∀𝔐⁡𝑣)[(𝑢,𝑣)∈𝑋⟺𝑢∈𝑣]

which is to say, there exists a Class, unique by extensionality, representing elementhood as a Relation class.¹ It is necessary to fix a construction of the ordered pair within the theory.

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tidy | en | SemBr

Footnotes

1. 2015. Introduction to Mathematical Logic, §4.1, p. 236 ↩