nLab foundational axiom

Idea

A foundational axiom is an axiom that would be used in foundations.

Examples

foundational axioms

• basic constructions:

  • axiom of cartesian products
  • axiom of disjoint unions
  • axiom of the empty set
  • axiom of fullness
  • axiom of function sets
  • axiom of power sets
  • axiom of quotient sets

• material axioms:

  • axiom of extensionality
  • axiom of foundation
  • axiom of anti-foundation
  • Mostowski's axiom
  • axiom of pairing
  • axiom of transitive closure
  • axiom of union

• structural axioms:

  • axiom of materialization

• type theoretic axioms:

  • axioms of set truncation
    • uniqueness of identity proofs
    • axiom K
    • boundary separation
    • equality reflection
    • axiom of circle type localization
  • homotopy type theoretic axioms:
    • univalence axiom
    • Whitehead's principle

• axioms of choice:

  • axiom of choice
  • axiom of countable choice
  • axiom of dependent choice
  • axiom of excluded middle
  • axiom of existence
  • axiom of multiple choice
  • Markov's axiom
  • presentation axiom
  • small cardinality selection axiom
  • axiom of small violations of choice
  • axiom of weakly initial sets of covers

• large cardinal axioms:

  • axiom of infinity
  • axiom of universes
  • regular extension axiom
  • inaccessible cardinal
  • measurable cardinal
  • elementary embedding
  • supercompact cardinal
  • Vopěnka's principle

• strong axioms

  • axiom of separation
  • axiom of replacement

• further

  • reflection principle

• axiom of inequality spaces