In set theory, a prewellordering on a set is a preorder on (a transitive and strongly connected relation on ) that is wellfounded in the sense that the relation is wellfounded. If is a prewellordering on , then the relation defined by

is an equivalence relation on , and induces a wellordering on the quotient . The order-type of this induced wellordering is an ordinal, referred to as the length of the prewellordering.

A norm on a set is a map from into the ordinals. Every norm induces a prewellordering; if is a norm, the associated prewellordering is given by

Conversely, every prewellordering is induced by a unique regular norm (a norm is regular if, for any and any , there is such that ).

Prewellordering propertyEdit

If   is a pointclass of subsets of some collection   of Polish spaces,   closed under Cartesian product, and if   is a prewellordering of some subset   of some element   of  , then   is said to be a  -prewellordering of   if the relations   and   are elements of  , where for  ,


  is said to have the prewellordering property if every set in   admits a  -prewellordering.

The prewellordering property is related to the stronger scale property; in practice, many pointclasses having the prewellordering property also have the scale property, which allows drawing stronger conclusions.


  and   both have the prewellordering property; this is provable in ZFC alone. Assuming sufficient large cardinals, for every  ,   and   have the prewellordering property.



If   is an adequate pointclass with the prewellordering property, then it also has the reduction property: For any space   and any sets  ,   and   both in  , the union   may be partitioned into sets  , both in  , such that   and  .


If   is an adequate pointclass whose dual pointclass has the prewellordering property, then   has the separation property: For any space   and any sets  ,   and   disjoint sets both in  , there is a set   such that both   and its complement   are in  , with   and  .

For example,   has the prewellordering property, so   has the separation property. This means that if   and   are disjoint analytic subsets of some Polish space  , then there is a Borel subset   of   such that   includes   and is disjoint from  .

See alsoEdit


  • Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.