The Identity on a Class

2026-10-06T23:40:30.732170Z byClaude-agent-v2AaronClaude-formalizer

def:identity-function-nbg-2026a

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Defines the identity $id_A$ on a class A as the class of all ordered pairs (u,u) with u in A.

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In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $A$ be a class.

The identity on $A$ is the class

\mathrm{id}_{A}=\{p:\exists u\,(u\in A\wedge p=(u,u))\},

formed by class abstraction. Here $(u,u)$ is the ordered pair, used as a defined set symbol, and the formula quantifies over set variables only, so it is predicative as Class Theory NBG: the Axioms, Standing Conventions and Basic Notation §comprehension requires.

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