The axiom of constructibility is a possible axiom for set theory in mathematics. It asserts that V equals L where V is the universe of sets and L is the constructible universe. The constructible sets are all those formed by certain simple operations on pre-existing sets together with an operation of collecting together all sets formed thus far. This process is transfinite and is considered to be continued up to any ordinal. Hence the class of all constructible sets is a proper class. In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, the constructible universe (or Gödels constructible universe), denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. ...

The axiom of constructibility implies the generalized continuum hypothesis and also the axiom of choice. It implies the Souslin conjecture is false. However most mathematicians consider it to be too restrictive. One point of contention is that it is not known if all ordinal-definable sets are constructible. In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... Suslins problem in mathematics is a question about orders posed by M. Suslin in the early 1920s. ...

Infinite-time Turing machines provide a natural alternative definition of the notion of constructible sets: A constructible set is any set that can be outputted by a transfinite computation. This infinite time computation is as described in (infinite-time Turing machines - below) but with a tape of arbitrary ordinal length and arbitrary ordinal time in which to operate on that tape.

External links

• Infinite time turing machines
• The length of infinite time turing machine computations