Drag (physics)

In set theory, a prewellordering is a binary relation $\le$ that is transitive, total, and wellfounded (more precisely, the relation $x\le y\wedge y\nleqq x$ is wellfounded). In other words, if $\le$ is a prewellordering on a set $X$, and if we define $\sim$ by

$x\sim y⟺x\le y\wedge y\le x$

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

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

$x\le y⟺\varphi (x)\le \varphi (y)$

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

Prewellordering property

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

1. $x{<}^{*}y⟺x\in P\wedge [y\notin P\vee \{x\le y\wedge y\le ̸x\}]$
2. $x{\le }^{*}y⟺x\in P\wedge [y\notin P\vee x\le y]$

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

${\mathrm{\Pi }}_1^1$ and ${\mathrm{\Sigma }}_2^1$ both have the prewellordering property; this is provable in ZFC alone. Assuming sufficient large cardinals, for every $n\in \omega$, ${\mathrm{\Pi }}_{2n+1}^1$ and ${\mathrm{\Sigma }}_{2n+2}^1$ have the prewellordering property.

Consequences

Reduction

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

Separation

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

For example, ${\mathrm{\Pi }}_1^1$ has the prewellordering property, so ${\mathrm{\Sigma }}_1^1$ has the separation property. This means that if $A$ and $B$ are disjoint analytic subsets of some Polish space $X$, then there is a Borel subset $C$ of $X$ such that $C$ includes $A$ and is disjoint from $B$.

See also

• Descriptive set theory
• Scale property
• Graded poset – a graded poset is analogous to a prewellordering with a norm, replacing a map to the ordinals with a map to the integers

References

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