In set theory, a prewellordering on a set is a preorder (a transitive and reflexive) that is strongly connected (meaning that any two points are comparable) and well-founded is a well-founded relation.

Prewellordering on a set

A prewellordering on a set is a homogeneous binary relation that satisfies the following conditions: Reflexivity Transitivity Total/Strongly connected

• being a well-founded relation.

A homogeneous binary relation is a prewellordering if and only if there exists a surjection into a well-ordered set

Examples

denotes the set's cardinality) is a prewellordering.

Properties

is an equivalence relation 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 Conversely, every prewellordering is induced by a unique regular norm).

Prewellordering property

is a pointclass of Polish spaces closed under Cartesian product-prewellordering

is said to have the prewellordering property-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.

Examples

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

Consequences

Reduction

is an adequate pointclass with the prewellordering property, then it also has the reduction property

Separation

is an adequate pointclass whose dual pointclass has the separation property disjoint and its complement

are disjoint analytic then there is a Borel

See also

• – a graded poset is analogous to a prewellordering with a norm, replacing a map to the ordinals with a map to the natural numbers

References