Differential Order - math.helsinki.fi · Differential Order Wai Yan Pong (with Matthias...

Differential OrderWai Yan Pong

(with Matthias Aschenbrenner)

California State University

Dominguez Hills

Logic Colloquium 2003, Helsinki – p. 1/32

Orders of Differential Polynomials

Examples

1. is 1st order.

2. is 2nd order.

The situation gets complicate pretty quickly if we allowmore variables, derivations and equations.For examples,

3.

4.

What should be the “order” for these systems?



Examples

1.

� � � � � � ��

��

is 1st order.

2. � � � � � is 2nd order.


3.

4.




Examples

1.

� � � � � � ��

��

is 1st order.

2. � � � � � is 2nd order.


3. � � � ��

4.

� ��

� �



Differential Algebra

Basic Setup: Let

�

be a ring (commutative with 1). Aderivation on

�

is a map

��

such that

� � ��

� � ��

A differential ring is a ring together with finitelymany commuting derivations ( ). Anideal of is a differential ideal if for all .



Basic Setup: Let

�

be a ring (commutative with 1). Aderivation on

�

is a map

��

such that

� � ��

� � ��

A differential ring is a ring��

together with finitelymany commuting derivations

� �� (

�� ). Anideal

of

�

is a differential ideal if

� � �

for all �

.



Examples

1. .

2. .

3. Meromorphic Functions on .

4. Differential Polynomial Ring over ,

In general, let and



Examples

1.

��

.

2. .



In general, let and



Examples

1.

��

.

2.

� � ��

��

��

.



In general, let and



Examples

1.

��

.

2.

� � ��

��

��

.

3. Meromorphic Functions on

��

�� .


In general, let and



Examples

1.

��

.

2.

� � ��

��

��

.

3. Meromorphic Functions on

��

�� .

4. Differential Polynomial Ring over

�,

� � � � � � � � ��

In general, let

� � � � � ��

and

� � � ��



An element of

�

is called a differential operator. The orderof

� �� is defined to be � � .

Let be a differential field, a tuple in somedifferential extension . Let be the ideal

. It is a prime differential idealand is called the ideal of over .

Denote by the differential subring of generated byand and the field of fractions of .



An element of

�



Let

�

be a differential field, ��

a tuple in somedifferential extension

� � �

. Let

� � � � �

be the ideal��

. It is a prime differential idealand is called the ideal of � over

�.

Denote by the differential subring of generated byand and the field of fractions of .



An element of

�



Let

�

be a differential field, ��

a tuple in somedifferential extension

� � �

. Let

� � � � �

be the ideal��

. It is a prime differential idealand is called the ideal of � over

�.

Denote by

� � � �

the differential subring of

�

generated by�

and � and

� � � �

the field of fractions of

� � � �

.


Transcendence Degree

Let us go back to our examples. The order of a differentialpolynomial can be viewed as some sort of “dimension”.

1. Let be a generic solution over of .

2.




1. Let � be a generic solution over

�

of� � � � � � �

�

��

.

2.





�

of� � � � � � �

�

��

.

� � � ��

2.





�

of� � � � � � �

�

��

.

� � � ��

2. � � � � �� Logic Colloquium 2003, Helsinki – p. 6/32




�

of� � � � � � �

�

��

.

� � � ��

2. � � � � ��

� � � ��



3. � � � ��

4.

These last two examples both have infinite dimension. Canwe distinguish them somehow?



3. � � � ��

� � ��

4.




3. � � � ��

� � ��

4.

� ��

� ��



Kolchin Polynomial

For � �

, let

� � � ��

.

Consider the function , then for ,

1. 1

2. 2

3. (s+1)+2 = s+3

4. 2(s+1)=2s+2

All of them are polynomials in ! And we distinguish thelast two cases.


Kolchin Polynomial

For � �

, let

� � � ��

.Consider the function ��

�

� �

, then for � �

,

1. 1

2. 2

3. (s+1)+2 = s+3

4. 2(s+1)=2s+2

All of them are polynomials in ! And we distinguish thelast two cases.


Kolchin Polynomial

For � �

, let

� � � ��


�

� �

, then for � �

,

1. 1

2. 2

3. (s+1)+2 = s+3

4. 2(s+1)=2s+2

All of them are polynomials in �!

And we distinguish thelast two cases.


Kolchin Polynomial

For � �

, let

� � � ��


�

� �

, then for � �

,

1. 1

2. 2

3. (s+1)+2 = s+3

4. 2(s+1)=2s+2

All of them are polynomials in �! And we distinguish thelast two cases.


Kolchin Polynomial

Theorem (Kolchin). The function ��

� �

is

“polynomial-like”, i.e. there exists ��

such that

� � � � � � � � � � ��

� ��

.

Idea.

Char Set monomial ideal

Hilbert Polynomial of .


Kolchin Polynomial


� �

is


such that

� � � � � � � � � � ��

� ��

.

Idea.

� � � � � � � � � �

� � � � �

Char Set� � � � � �

� � � � ��

�

monomial ideal�

Hilbert Polynomial of .


Kolchin Polynomial


� �

is


such that

� � � � � � � � � � ��

� ��

.

Idea.

� � � � � � � � � �

� � � � �

Char Set� � � � � �

� � � � ��

�

monomial ideal�

�� Hilbert Polynomial of

� � � � �

.


Differential Order

For , let

monomial ideal in

is totally ordered by dominance:.


Differential Order

For � �

, let

� � � ��

monomial ideal in

� ��

��

� � � � � ��

is totally ordered by dominance:.


Differential Order

For � �

, let

� � � ��

monomial ideal in

� ��

��

� � � � � ��

is totally ordered by dominance:� � � ��

.


Sit Theorem

A rather surprising fact is that:

Theorem (Sit 75, A-P 02). is well ordered by dominance.

Define the differential order, , of over to bethe ordinal corresponding to .

In the single derivation case, Sit Theorem has an easyproof:


Sit Theorem

A rather surprising fact is that:

Theorem (Sit 75, A-P 02). is well ordered by dominance.

Define the differential order,

�� , of � over

�

to bethe ordinal corresponding to �� .

In the single derivation case, Sit Theorem has an easyproof:


Proof of Sit Theorem (easy case)

Proof. It is easy to see that ��

is of the form

� � � �where� � � � �

are integers (

�

is the differential transcendence degree of

� � �

). So � � � �

is actually in

� � � �

rather than just in� � � �

.

Thus

is an order preserving map from the Kolchin polynomials to theordinals.

We are going to explain our proof of this theorem. Let usstart with a result which seems to have very little to do withwhat we have discussed so far.


Proof of Sit Theorem (easy case)

Proof. It is easy to see that ��

is of the form

� � � �where� � � � �

are integers (

�

is the differential transcendence degree of

� � �

). So � � � �

is actually in

� � � �

rather than just in� � � �

. Thus

� � � ��

is an order preserving map from the Kolchin polynomials to theordinals.

We are going to explain our proof of this theorem. Let usstart with a result which seems to have very little to do withwhat we have discussed so far.


Dickson’s Lemma.

The following elementary fact is well-known (L. E. Dickson,1913):

Lemma. In any infinite sequence of monomialsin there are monomials ( )

with .

This has been called “the most frequently rediscoveredmathematical theorem”. It can be used to deduce theHilbert Basis Theorem for .


Dickson’s Lemma.


Lemma. In any infinite sequence � � � � � �� of monomialsin

� � � � � ��

there are monomials � � � � � (

� �

)

with � � � � � .

This has been called “the most frequently rediscoveredmathematical theorem”. It can be used to deduce theHilbert Basis Theorem for .


Dickson’s Lemma.


Lemma. In any infinite sequence � � � � � �� of monomialsin

� � � � � ��

there are monomials � � � � � (

� �

)

with � � � � � .

This has been called “the most frequently rediscoveredmathematical theorem”. It can be used to deduce theHilbert Basis Theorem for

� � �

.


Maclagan’s Finiteness Property

Diane Maclagan (2000) found a generalization of Dickson’sLemma to the set of monomial ideals in

� � �:

Proposition. In any infinite sequence of monomialideals in there are ideals , ( ) with .

This implies various finiteness statements in computationalcommutative algebra:


Maclagan’s Finiteness Property

Diane Maclagan (2000) found a generalization of Dickson’sLemma to the set of monomial ideals in

� � �:

Proposition. In any infinite sequence

� � � �� of monomialideals in

� � �

there are ideals

� , � (

� � ) with

� � � .

This implies various finiteness statements in computationalcommutative algebra:


Some Consequences of MFP

� Every ideal

in

� � �

has only finitely many initialideals with respect to various term orderings. (Hence

has a universal Gröbner basis.)

Let be any function. There exists a finite setof monomial ideals of such that for every

monomial ideal of , we have:

for for some

(Here Hilbert polynomial of .)


Some Consequences of MFP

� Every ideal

in

� � �

has only finitely many initialideals with respect to various term orderings. (Hence

has a universal Gröbner basis.)

� Let

� � � � �

be any function. There exists a finite set

� of monomial ideals of

� � �such that for every

monomial ideal

of

� � �

, we have:

��

� � � � � � � �

for � � ��

for some

� ��

(Here

�� Hilbert polynomial of

� � � �

.)


Noetherian Order

Given (partially) ordered sets

��

and

� �� ,

we canform their product and co-product:

(cartesian product) (disjoint union).

A final segment of is a subset such that

and

If is any subset, then

is the final segment generated by .


Noetherian Order


��

and

� �� , we can

form their product and co-product:

� � �

(cartesian product)��

(disjoint union).

A final segment of is a subset such that

and




Noetherian Order


��

and

� �� , we can


� � �


(disjoint union).

A final segment of

� � � � �

is a subset� � �

such that

��

and

��




Noetherian Order


��

and

� �� , we can


� � �


(disjoint union).

A final segment of

� � � � �

is a subset� � �

such that

��

and

��

If

� � �

is any subset, then

� � � � � � � � � ��

is the final segment generated by

�

.


Noetherian Orderings

The set

� � � �

of final segments of

�

can be viewed as anordered set itself by reverse inclusion:

� � � � ��

An anti-chain of is a subset such that for everyin , i.e.: neither nor .

We say that is well-founded if there is no infinite strictlydecreasing sequence in .

An ordered set is called Noetherian (or w.q.o) if it iswell-founded, and every anti-chain of is finite. (e.g. .)



The set

� � � �


�


� � � � �� An anti-chain of

�

is a subset

�

such that�� for every

� �� in

�

, i.e.: neither� � � nor � � � .

We say that is well-founded if there is no infinite strictlydecreasing sequence in .




The set

� � � �


�



�

is a subset

�


� �� in

�

, i.e.: neither� � � nor � � � .

We say that

�

is well-founded if there is no infinite strictlydecreasing sequence� � � � � �� in

�

.




The set

� � � �


�



�

is a subset

�


� �� in

�

, i.e.: neither� � � nor � � � .

We say that

�

is well-founded if there is no infinite strictlydecreasing sequence� � � � � �� in

�

.

An ordered set

��

is called Noetherian (or w.q.o) if it iswell-founded, and every anti-chain of

�

is finite. (e.g.

� �

.)


Characterizations of Noetherianity

Proposition. For an ordered set

� � � � �

TFAE:

1. is Noetherian.

2. Every infinite sequence in contains an increasing subsequence.

3. Every infinite sequence in contains ,with .

4. Any final segment of is finitely generated.

5. is well-founded.

6. Every total ordering on which extends is a well-ordering.


Characterizations of Noetherianity

Proposition. For an ordered set

� � � � �

TFAE:

1.

�

is Noetherian.

2. Every infinite sequence in

�

contains an increasing subsequence.

3. Every infinite sequence� � � � � �� in

�

contains� � ,� � � � � �

with� � � � � .4. Any final segment of

�

is finitely generated.

5.

� � � � � � � �

is well-founded.

6. Every total ordering on

�which extends

�

is a well-ordering.


Proof of (1) (2)

Proof. Consider the following partition of :

where

and .

Ramsey Theorem an infinite homogeneous set for this partition.Since is Noetherian, the only possibility is that .


Proof of (1) (2)

Proof. Consider the following partition of

� � � �

:

� � � � � � � � � � � � � � � � � � � � ��

where

��

and

��

.

Ramsey Theorem an infinite homogeneous set for this partition.Since is Noetherian, the only possibility is that .


Proof of (1) (2)


� � � �

:

� � � � � � � � � � � � � � � � � � � � ��

where

��

and

��

.

Ramsey Theorem

� �

an infinite homogeneous set

�

for this partition.

Since is Noetherian, the only possibility is that .


Proof of (1) (2)


� � � �

:

� � � � � � � � � � � � � � � � � � � � ��

where

��

and

��

.

Ramsey Theorem

� �


�

for this partition.Since

�

is Noetherian, the only possibility is that

� � � � � �

.


Two Corollaries

Corollary. If and are Noetherian, then so areand .

Corollary. Suppose is an order preserving surjection. Ifis Noetherian, so is .

Recall that one of the equivalent conditions for to beNoetherian is that “ is well-founded”. However,this does not imply that is Noetherian. In fact, wehave:


Two Corollaries

Corollary. If

��

and

� ��

are Noetherian, then so are

� � �

and

� � �

.

Corollary. Suppose is an order preserving surjection. Ifis Noetherian, so is .



Two Corollaries

Corollary. If

��

and

� ��


� � �

and

� � �

.

Corollary. Suppose

� � � � �

is an order preserving surjection. If

�

is Noetherian, so is

�

.



Two Corollaries

Corollary. If

��

and

� ��


� � �

and

� � �

.

Corollary. Suppose

� � � � �

is an order preserving surjection. If

�

is Noetherian, so is

�

.

Recall that one of the equivalent conditions for

�

to beNoetherian is that “

� � � � � � � �is well-founded”. However,

this does not imply that� � � �

is Noetherian. In fact, wehave:


Rado’s Example

Example (Rado). Let

� � � � � � �

, ordered by the rule

� � � � ��

either

� ��

and � � �

or

� � �

Then

� ��

is Noetherian, but

� � � � � � � �

is not:

�� , the final

segment of

�

generated by all

� � � � � �with

� �

, is an infinite

anti-chain in

� � � �

:

For , one checks directly that

and

Rado’s example is archetypical in the following sense:


Rado’s Example

Example (Rado). Let

� � � � � � �

, ordered by the rule

� � � � ��

either

� ��

and � � �

or

� � �

Then

� ��

is Noetherian, but

� � � � � � � �

is not:

�� , the final

segment of

�

generated by all

� � � � � �with

� �

, is an infinite

anti-chain in

� � � �

: For

� �

, one checks directly that

� � � � �� and

� � � � ��

� ��

Rado’s example is archetypical in the following sense:


Strong Noetherianity

Theorem. For a Noetherian ordered set

��

, TFAE:

1. is Noetherian.

2. For every function there exists such that

3. There exists no embedding .

A Noetherian ordered set is called stronglyNoetherian if it satisfies one of the above conditions.




��

, TFAE:

1.

� � � � � � � �

is Noetherian.

2. For every function

� � � � � � � �

there exists� � � �

such that� � � � � ��

3. There exists no embedding

� ��

.

A Noetherian ordered set is called stronglyNoetherian if it satisfies one of the above conditions.




��

, TFAE:

1.

� � � � � � � �

is Noetherian.

2. For every function

� � � � � � � �

there exists� � � �

such that� � � � � ��

3. There exists no embedding

� ��

.

A Noetherian ordered set

��

is called stronglyNoetherian if it satisfies one of the above conditions.



�(2)

� � (3). There exists

��

such that � � � � �� for all

� � � �

. Consider the partitions of

� � � � � � � �and

� � � ��

where

��

Applying Ramsey Theorem twice, we get�

infinite homogenous set forboth partitions. Since

�

Noetherian,� � � � � �

and

� � � � � � �

. By our

choice of

�

and

��

, one checks that

� � � � ��

for all

� � � � � ��

. So

� � � �� (identifying

�

and

�

) isan embedding of

�into

�.



� (3)

� � (1). An embedding of

� ��

into

� � � ��

induces an

order preserving surjection of from

� � � � � � � �

to

� � � � � � � �

. Since

� � � �

is not Noetherian, so does

� ��

.

(1) (2). is not Noetherian, so there exists asequence such that for all . Pick

. Since the ’s are initial segments, one checks

easily that is a function from to such thatfor all .



� (3)

� � (1). An embedding of

� ��

into

� � � ��

induces an

order preserving surjection of from

� � � � � � � �

to

� � � � � � � �

. Since

� � � �

is not Noetherian, so does

� ��

.

� (1)

� � (2).

��

is not Noetherian, so there exists asequence

� �� such that

�� for all

� �

. Pick� � � � � �� . Since the�� ’s are initial segments, one checks

easily that

�

is a function from� � � �

to

�

such that

� � � � � � ��

for all

� � � �

.


Product of SNOS

Proposition. If

�

and

�

are strongly Noetherian, then so is� � �

.

Proof. For any . Let and .

Consider the partition where,

Ramsey Theoreman infinite homogeneous set for this partition.is strongly Noetherian .is strongly Noetherian there are in with

. Hence .


Product of SNOS

Proposition. If

�

and

�


.

Proof. For any

� � � � � � � � � �

. Let

� � � �� and� � � �� .

Consider the partition where,


. Hence .


Product of SNOS

Proposition. If

�

and

�


.

Proof. For any

� � � � � � � � � �

. Let

� � � �� and� � � �� .

Consider the partition

� � � � � � � �

where,

��


. Hence .


Product of SNOS

Proposition. If

�

and

�


.

Proof. For any

� � � � � � � � � �

. Let

� � � �� and� � � �� .


� � � � � � � �

where,

��

Ramsey Theorem

�

�


� � �for this partition.

is strongly Noetherian .is strongly Noetherian there are in with

. Hence .


Product of SNOS

Proposition. If

�

and

�


.

Proof. For any

� � � � � � � � � �

. Let

� � � �� and� � � �� .


� � � � � � � �

where,

��

Ramsey Theorem

�

�


� � �for this partition.�

is strongly Noetherian

� � � � � � �

.

is strongly Noetherian there are in with. Hence .


Product of SNOS

Proposition. If

�

and

�


.

Proof. For any

� � � � � � � � � �

. Let

� � � �� and� � � �� .


� � � � � � � �

where,

��

Ramsey Theorem

�

�


� � �for this partition.�


� � � � � � �

.�


�there are

� � � �

in

�

with� � � � � � ��

. Hence� � � � � � � � � � �

.


Connection with Monomial Ideals

monomials in

monomial ideals in

(Dickson’s Lemma) The ordered set is Noetherian.

(Maclagan) The ordered set is strongly Noetherian.



�

monomials in

� � ��

� ��

� ��

monomial ideals in





�

monomials in

� � ��

� ��

� ��

�

monomial ideals in

� ��





�

monomials in

� � ��

� ��

� ��

�

monomial ideals in

� ��

(Dickson’s Lemma) The ordered set

� �

is Noetherian.




�

monomials in

� � ��

� ��

� ��

�

monomial ideals in

� ��

(Dickson’s Lemma) The ordered set

� �

is Noetherian.

(Maclagan) The ordered set

� �

is strongly Noetherian.


Proof of Sit Theorem

Proof. Fix , one checks easily that

is an order preserving surjection from to .

Maclagan Theorem is Noetherian isNoetherian dominance ( ) is a well order.

Now suppose is a decreasing chain in . Thenis a decreasing chain for natural numbers.

Hence there exists such that for all . So we aredone.



Proof. Fix �, one checks easily that

��

is an order preserving surjection from

� � � � � � � � �to

� � � � �

.

Maclagan Theorem is Noetherian isNoetherian dominance ( ) is a well order.






��


� � � � � � � � �to

� � � � �

.

Maclagan Theorem

� � � � ��

is Noetherian

isNoetherian dominance ( ) is a well order.






��


� � � � � � � � �to

� � � � �

.

Maclagan Theorem

� � � � ��

is Noetherian

� � � � � �

isNoetherian

dominance ( ) is a well order.






��


� � � � � � � � �to

� � � � �

.

Maclagan Theorem

� � � � ��

is Noetherian

� � � � � �

isNoetherian

�

dominance (

�

) is a well order.






��


� � � � � � � � �to

� � � � �

.

Maclagan Theorem

� � � � ��

is Noetherian

� � � � � �

isNoetherian

�

dominance (

�

) is a well order.

Now suppose � � � � �

�� is a decreasing chain in .

Thenis a decreasing chain for natural numbers.





��


� � � � � � � � �to

� � � � �

.

Maclagan Theorem

� � � � ��

is Noetherian

� � � � � �

isNoetherian

�

dominance (

�

) is a well order.

Now suppose � � � � �

�� is a decreasing chain in . Then

��

�� is a decreasing chain for natural numbers.Hence there exists � such that � � � for all

� �

. So we aredone.


How to Compute RD?

Usually it’s hard!

There are algorithms to compute the characteristic set of adifferential ideal. So suppose the Kolchin polynomial of

is given.

A classical theorem of Macaulay characterizes exactlythose functions form to which arise as Hilbert functionsof homogeneous ideals of . Using this we get:


How to Compute RD?



� � �

is given.

A classical theorem of Macaulay characterizes exactlythose functions form to which arise as Hilbert functionsof homogeneous ideals of . Using this we get:


How to Compute RD?



� � �

is given.

A classical theorem of Macaulay characterizes exactlythose functions form

�

to�

which arise as Hilbert functionsof homogeneous ideals of

� � �

. Using this we get:


How to Compute RD?

Theorem. Every Kolchin polynomial is of the form

� � � ��

� � � �

� �

�

� � � � ��

� �

��

� � � � ��

� �

� �

for certain integers � � � � � � �

��

� �

and � � �

.

Theorem.

where is the number of occurrences of amongthe coefficients .


How to Compute RD?

Theorem. Every Kolchin polynomial is of the form

� � � ��

� � � �

� �

�

� � � � ��

� �

��

� � � � ��

� �

� �

for certain integers � � � � � � �

��

� �

and � � �

.

Theorem.

��

� ��

where � � is the number of occurrences of

� � � ��

amongthe coefficients � � � � � �� .


What does RD good for?

Don’t know yet!

But I would argue that it is something that we shouldknow about -DCF . And now we have a nice way ofunderstanding it.

Unifies proofs of many results using the sameprinciple: Noetherianity of certain ordered sets.

Proposition (P 96). For a complete type in DCF , we have

where are integers and is the differential transcendence degreeof .



� Don’t know yet!

But I would argue that it is something that we shouldknow about -DCF . And now we have a nice way ofunderstanding it.







� But I would argue that it is something that we shouldknow about �-DCF� . And now we have a nice way ofunderstanding it.








� Unifies proofs of many results using the sameprinciple: Noetherianity of certain ordered sets.







� Unifies proofs of many results using the sameprinciple: Noetherianity of certain ordered sets.

Proposition (P 96). For a complete type � in DCF� , we have

� � � � � � � � � � � � � � � ��

where

� � �

are integers and

�

is the differential transcendence degreeof �.


Ranks Inequalities

Proposition (McGrail 00). For a complete type in -DCF , we have

where and are the differential type and differential transcendencedegree of respectively.

Proposition (Pillay-P 01). RU=RM for the generic types of definablegroups in -DCF .

Proposition (A-P 01).


Ranks Inequalities

Proposition (McGrail 00). For a complete type � in �-DCF� , we have

� � � � � � � � � � � � � � � � � � � � � � � � � � � � �

where � and

�

are the differential type and differential transcendencedegree of � respectively.

Proposition (Pillay-P 01). RU=RM for the generic types of definablegroups in -DCF .



Ranks Inequalities


� � � � � � � � � � � � � � � � � � � � � � � � � � � � �

where � and

�


Proposition (Pillay-P 01). RU=RM for the generic types of definablegroups in �-DCF� .



Ranks Inequalities


� � � � � � � � � � � � � � � � � � � � � � � � � � � � �

where � and

�


Proposition (Pillay-P 01). RU=RM for the generic types of definablegroups in �-DCF� .


� � � � � � � � � � � � � � � � � � � � � � ��


Variation of RD in definable families

We end with two rather technical results:

Theorem. Let be a formula in the language of -DCF withparameters in a differential field . Let be an ordinal such that

for all tuples of suitable length. Then there

exists an ordinal such that .

Morley Rank is a bit trickier:Corollary. Given and . If for all ,

then there exists an ordinal such that

for all .




Theorem. Let

� � ��

be a formula in the language of �-DCF� withparameters in a differential field

�

. Let � be an ordinal such that�� for all tuples

�

of suitable length. Then there

exists an ordinal

� � � such that

��

.

Morley Rank is a bit trickier:Corollary. Given and . If for all ,


for all .




Theorem. Let

� � ��


�


�


exists an ordinal


��

.

Morley Rank is a bit trickier:

Corollary. Given and . If for all ,


for all .




Theorem. Let

� � ��


�


�


exists an ordinal


��

.

Morley Rank is a bit trickier:Corollary. Given

� � ��

and � � � �

. If for all

�

,

� � � � � ��

then there exists an ordinal

� � � � � � � � �

such that� � � � � ��

for all

�

.


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