Prewellordering

In set theory, a prewellordering on a set ${\displaystyle X}$ is a preorder ${\displaystyle \leq }$ on ${\displaystyle X}$ (a transitive and reflexive relation on ${\displaystyle X}$) that is strongly connected (meaning that any two points are comparable) and well-founded in the sense that the induced relation ${\displaystyle x<y}$ defined by ${\displaystyle x\leq y{\text{ and }}y\nleq x}$ is a well-founded relation.

Prewellordering on a set

A prewellordering on a set ${\displaystyle X}$ is a homogeneous binary relation ${\displaystyle \,\leq \,}$ on ${\displaystyle X}$ that satisfies the following conditions:^[1]

1. Reflexivity: ${\displaystyle x\leq x}$ for all ${\displaystyle x\in X.}$
2. Transitivity: if ${\displaystyle x<y}$ and ${\displaystyle y<z}$ then ${\displaystyle x<z}$ for all ${\displaystyle x,y,z\in X.}$
3. Total/Strongly connected: ${\displaystyle x\leq y}$ or ${\displaystyle y\leq x}$ for all ${\displaystyle x,y\in X.}$
4. for every non-empty subset ${\displaystyle S\subseteq X,}$ there exists some ${\displaystyle m\in S}$ such that ${\displaystyle m\leq s}$ for all ${\displaystyle s\in S.}$
   • This condition is equivalent to the induced strict preorder ${\displaystyle x<y}$ defined by ${\displaystyle x\leq y}$ and ${\displaystyle y\nleq x}$ being a well-founded relation.

A homogeneous binary relation ${\displaystyle \,\leq \,}$ on ${\displaystyle X}$ is a prewellordering if and only if there exists a surjection ${\displaystyle \pi :X\to Y}$ into a well-ordered set ${\displaystyle (Y,\lesssim )}$ such that for all ${\displaystyle x,y\in X,}$ ${\textstyle x\leq y}$ if and only if ${\displaystyle \pi (x)\lesssim \pi (y).}$^[1]

Examples

Given a set ${\displaystyle A,}$ the binary relation on the set ${\displaystyle X:=\operatorname {Finite} (A)}$ of all finite subsets of ${\displaystyle A}$ defined by ${\displaystyle S\leq T}$ if and only if ${\displaystyle |S|\leq |T|}$ (where ${\displaystyle |\cdot |}$ denotes the set's cardinality) is a prewellordering.^[1]

Properties

If ${\displaystyle \leq }$ is a prewellordering on ${\displaystyle X,}$ then the relation ${\displaystyle \sim }$ defined by

is an equivalence relation on ${\displaystyle X,}$ and ${\displaystyle \leq }$ induces a wellordering on the quotient ${\displaystyle X/{\sim }.}$ The order-type of this induced wellordering is an ordinal, referred to as the length of the prewellordering.

A norm on a set ${\displaystyle X}$ is a map from ${\displaystyle X}$ into the ordinals. Every norm induces a prewellordering; if ${\displaystyle \phi :X\to Ord}$ is a norm, the associated prewellordering is given by

Conversely, every prewellordering is induced by a unique regular norm (a norm ${\displaystyle \phi :X\to Ord}$ is regular if, for any ${\displaystyle x\in X}$ and any ${\displaystyle \alpha <\phi (x),}$ there is ${\displaystyle y\in X}$ such that ${\displaystyle \phi (y)=\alpha }$).

Prewellordering property

If ${\displaystyle {\boldsymbol {\Gamma }}}$ is a pointclass of subsets of some collection ${\displaystyle {\mathcal {F}}}$ of Polish spaces, ${\displaystyle {\mathcal {F}}}$ closed under Cartesian product, and if ${\displaystyle \leq }$ is a prewellordering of some subset ${\displaystyle P}$ of some element ${\displaystyle X}$ of ${\displaystyle {\mathcal {F}},}$ then ${\displaystyle \leq }$ is said to be a ${\displaystyle {\boldsymbol {\Gamma }}}$-prewellordering of ${\displaystyle P}$ if the relations ${\displaystyle <^{*}}$ and ${\displaystyle \leq ^{*}}$ are elements of ${\displaystyle {\boldsymbol {\Gamma }},}$ where for ${\displaystyle x,y\in X,}$

1. ${\displaystyle x<^{*}y{\text{ if and only if }}x\in P\land (y\notin P\lor (x\leq y\land y\not \leq x))}$
2. ${\displaystyle x\leq ^{*}y{\text{ if and only if }}x\in P\land (y\notin P\lor x\leq y)}$

${\displaystyle {\boldsymbol {\Gamma }}}$ is said to have the prewellordering property if every set in ${\displaystyle {\boldsymbol {\Gamma }}}$ admits a ${\displaystyle {\boldsymbol {\Gamma }}}$-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

${\displaystyle {\boldsymbol {\Pi }}_{1}^{1}}$ and ${\displaystyle {\boldsymbol {\Sigma }}_{2}^{1}}$ both have the prewellordering property; this is provable in ZFC alone. Assuming sufficient large cardinals, for every ${\displaystyle n\in \omega ,}$ ${\displaystyle {\boldsymbol {\Pi }}_{2n+1}^{1}}$ and ${\displaystyle {\boldsymbol {\Sigma }}_{2n+2}^{1}}$ have the prewellordering property.

Consequences

Reduction

If ${\displaystyle {\boldsymbol {\Gamma }}}$ is an adequate pointclass with the prewellordering property, then it also has the reduction property: For any space ${\displaystyle X\in {\mathcal {F}}}$ and any sets ${\displaystyle A,B\subseteq X,}$ ${\displaystyle A}$ and ${\displaystyle B}$ both in ${\displaystyle {\boldsymbol {\Gamma }},}$ the union ${\displaystyle A\cup B}$ may be partitioned into sets ${\displaystyle A^{*},B^{*},}$ both in ${\displaystyle {\boldsymbol {\Gamma }},}$ such that ${\displaystyle A^{*}\subseteq A}$ and ${\displaystyle B^{*}\subseteq B.}$

Separation

If ${\displaystyle {\boldsymbol {\Gamma }}}$ is an adequate pointclass whose dual pointclass has the prewellordering property, then ${\displaystyle {\boldsymbol {\Gamma }}}$ has the separation property: For any space ${\displaystyle X\in {\mathcal {F}}}$ and any sets ${\displaystyle A,B\subseteq X,}$ ${\displaystyle A}$ and ${\displaystyle B}$ disjoint sets both in ${\displaystyle {\boldsymbol {\Gamma }},}$ there is a set ${\displaystyle C\subseteq X}$ such that both ${\displaystyle C}$ and its complement ${\displaystyle X\setminus C}$ are in ${\displaystyle {\boldsymbol {\Gamma }},}$ with ${\displaystyle A\subseteq C}$ and ${\displaystyle B\cap C=\varnothing .}$

For example, ${\displaystyle {\boldsymbol {\Pi }}_{1}^{1}}$ has the prewellordering property, so ${\displaystyle {\boldsymbol {\Sigma }}_{1}^{1}}$ has the separation property. This means that if ${\displaystyle A}$ and ${\displaystyle B}$ are disjoint analytic subsets of some Polish space ${\displaystyle X,}$ then there is a Borel subset ${\displaystyle C}$ of ${\displaystyle X}$ such that ${\displaystyle C}$ includes ${\displaystyle A}$ and is disjoint from ${\displaystyle B.}$

See also

• Descriptive set theory – Subfield of mathematical logic
• Graded poset – partially ordered set equipped with a rank function, sometimes called a ranked posetPages displaying wikidata descriptions as a fallback – a graded poset is analogous to a prewellordering with a norm, replacing a map to the ordinals with a map to the natural numbers
• Scale property – kind of object in descriptive set theoryPages displaying wikidata descriptions as a fallback

References