Axiomatic WikiWWikiFFoundationReflexivity of identity
Foundation·F11
Everything is equal to itself.
Stated with the universal quantifier: everything, without exception, is equal to itself.
In words
Every x is equal to itself.
Part of the logical framework beneath ZFC (classical first-order logic with equality): taken as given rather than proven, it belongs to the language the set-theoretic axioms are stated in.
Remarks
The first axiom of identity, part of the logical framework beneath ZFC: ZFC is a theory formulated in first-order logic with equality, and so presupposes the identity axioms rather than restating them.
Used by
Propose an edit6 published revisions
- 7/12/2026 · Benjamin· Rename the quantifier key to "forall" for consistency with the other identity axioms.→what changed →
- 7/12/2026 · Benjamin· Add word-notation highlight markup linking "Every x" and "equal to itself" to the statement.→what changed →
- 7/11/2026 · Benjamin· Fix Mathlib link: add the #doc fragment doc-gen4's find endpoint requires (old link 404'd). Content unchanged.→what changed →
- 7/11/2026 · Benjamin· Cite the logic layer beneath the identity axioms→what changed →
- 7/11/2026 · Benjamin· Backfill: add plain-English prose and Mathlib docs link→what changed →
- 7/11/2026 · Benjamin· Initial foundations seed→what changed →