Prewellordering

x\sim y\iff x\leq y \land y\leq x

then

\sim

X

, and

\leq

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 Ordinal s. Every norm induces a prewellordering; if

\phi:X	o Ord

is a norm, the associated prewellordering is given by
:

x\leq y\iff\phi(x)\leq\phi(y)

Conversely, every prewellordering is induced by a unique regular norm (a norm

\phi:X	o Ord

is regular if, for any

x\in X

and any

\alpha<\phi(x)

, there is

y\in X

such that

\phi(y)=\alpha

).

PREWELLORDERING PROPERTY

\, and $\leq^---$ are elements of $\boldsymbol{\Gamma}$ , where for $x,y\in X$ ,

ot\leq x\}]

\boldsymbol{\Gamma}

is said to have the prewellordering property if every set in

\boldsymbol{\Gamma}

admits a

\boldsymbol{\Gamma}

-prewellordering.

Examples

\boldsymbol{\Pi}^1_1\,

and

\boldsymbol{\Sigma}^1_2

both have the prewellordering property; this is provable in ZFC alone. Assuming sufficient Large Cardinal s, for every

n\in\omega

\boldsymbol{\Pi}^1_{2n+1}

and

\boldsymbol{\Sigma}^1_{2n+2}

have the prewellordering property.

Consequences

Reduction

,B^---\,, both in $\boldsymbol{\Gamma}$ , such that $A^---\subseteq A$ and $B^---\subseteq B$ .

Separation

If

\boldsymbol{\Gamma}

is an

X\setminus C

are in

\boldsymbol{\Gamma}

, with

A\subseteq C

and

B\cap C=\emptyset

.

For example,

\boldsymbol{\Pi}^1_1

has the prewellordering property, so

\boldsymbol{\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

X

such that

C

includes

A

and is disjoint from

B

.

REFERENCES

	CATEGORIES ABOUT PREWELLORDERING

	descriptive set theory

	wellfoundedness

	order theory

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