Axiom of constructibility
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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 up to a certain time. This process is transfinite and is considered to be continued up to any ordinal in time. Hence the class of all constructible sets is a proper class.
The axiom of constructibility implies the generalized continuum hypothesis and therefore 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.
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 (http://arxiv.org/PS_cache/math/pdf/0212/0212047.pdf)
- The length of infinite time turing machine computations (http://www2.maths.bris.ac.uk/~mapdw/pdw4.ps)