axiom of choice See LÖWENHEIM -SKOLEM THEO -. REM , SET THEOR. axiom of comprehension, also called axiom of abstraction, the axiom that for every property, there is a corresponding set of things having that property; i.e., (f) (DA) (x) (x 1 A È f x), where f is a property and A is a set. The axiom was used in Frege’s formulation of set theory and is the axiom that yields Russell’s paradox, discovered in 1901. If fx is instantiated as x 2 x, then the result that A 1 A È A 2 A is easily obtained, which yields, in classical logic, the explicit contradiction A 1 A & A 2 A. The paradox can be avoided by modifying the comprehension axiom and using instead the separation axiom, (f) (DA) (x) (x 1 A È (fx & x 1 B)). This yields only the result that A 1 A È (A 2 A & A 1 B), which is not a contradiction. The paradox can also be avoided by retaining the comprehension axiom but restricting the symbolic language, so that ‘x 1 x’ is not a meaningful formula. Russell’s type theory, presented in Principia Mathematica, uses this approach. See also FREGE, RUSSELL , SET THEORY , TYPE THEOR. V.K. axiom of consistency, an axiom stating that a given set of sentences is consistent. Let L be a formal language, D a deductive system for L, S any set of sentences of L, and C the statement ‘S is consistent’ (i.e., ‘No contradiction is derivable from S via D’). For certain sets S (e.g., the theorems of D) it is interesting to ask: Can C be expressed in L? If so, can C be proved in D? If C can be expressed in L but not proved in D, can C be added (consistently) to D as a new axiom? Example (from Gödel): Let L and D be adequate for elementary number theory, and S be the axioms of D; then C can be expressed in L but not proved in D, but can be added as a new axiom to form a stronger system D’. Sometimes we can express in L an axiom of consistency in the semantic sense (i.e., ‘There is a universe in which all the sentences in S are true’). Trivial example: suppose the only non-logical axiom in D is ‘For any two sets B and B’, there exists the union of B and B’ ‘. Then C might be ‘There is a set U such that, for any sets B and B’ in U, there exists in U the union of B and B’ ‘. See also CONSISTENCY, PROOF THEORY. D.H.