nLab
foundations - contents

foundations

The basis of it all

• mathematical logic

• first-order logic

• type theory, homotopy type theory

• set theory

  • material set theory

    • ZFC
    • ZFA

  • structural set theory

    • ETCS
    • SEAR

• universe

Foundational axioms

foundational axiom

• 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
  • Mostowski's axiom
  • axiom of pairing
  • axiom of transitive closure
  • axiom of union

• 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

• Whitehead's principle

• 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

Removing axioms

• constructive mathematics
• predicative mathematics

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