nLab axiom

Idea

An axiom is a proposition in logic that a given theory requires to be true: every model of the theory is required to make the axiom hold true. The sense however is that an axiom is a basic true proposition, used to prove other true propositions (the theorems) in the theory.

Definition

Given a language $L$ (perhaps specified by a signature: a collection of types, function symbols and relation symbols), a theory is the collection of assertions which are derivable (using the rules of deduction of the ambient logic or deductive system) from a given set of assertions, called axioms of the theory. In other words, a theory is generated from a set of axioms, by starting with those axioms and applying rules of deduction, much as terms in an algebraic system may be generated from a set of basic terms by applying operations. Axioms should therefore be considered as presenting a theory; different axiom sets may well give the same theory.

In terms of a deductive system, axioms can be regarded as “rules with zero hypotheses”. The form of such axioms depends on the details of the deductive system used: it could be natural deduction, sequent calculus, a Hilbert system, etc. If we take sequent calculus, for instance, then any collection of sequents written in the given language $L$

$$\vec{\phi} \vdash_{\vec x} \vec{\psi}$$

(asserting that “If every proposition $\phi_i$ is true in context $\vec{x}$ then also some $\psi_i$ is/has to be true”) can be taken as a collection of axioms for some theory. Models of the theory will then be those structures of the language in which the axioms are interpreted as true statements. For example, a model of group theory is a structure in the language of groups for which the group theory axioms hold, which is (of course) a group.

Assuming the deductive system is sound?, every sequent which is the conclusion of a valid sequent deduction, starting from the axioms, will also be true in every model. And if the deductive system is also complete, then every sequent of the language which is true in every model will in fact be provable from the axioms.

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

(…)

• pushout-product axiom

(…)

• Hilbert's sixth problem asks for an axiomatization of physics.

Related concepts

• axiom schema

mathematical statements

• theory

• judgement, assertion

• hypothesis, consequence

• axiom

• assignment

• definition (inductive, coinductive)

• lemma

• proposition/type (propositions as types)

• theorem

• proof/program (proofs as programs)

• example, counterexample

• conjecture, folklore

• exercise

• analogy

• tautology

• paradox

References

For instance def. D1.1.6 in

• Peter Johnstone, Sketches of an Elephant