Appendix 2: Ultrapowers That are Enlargements

& ULTRAPOWERS THAT ARE ENLARGEMENTS

APPENDIX

The

goal of this Appendix is to construct a special kind of

nonstandard models, in

the so-called enlargements,

that are needed

Section (0.4) as a first step in the process of defining

superstructure extensions (0.4.2),

(0.4.3)

that, we define

for which

the

the

important principles

and (0.4.4) are shown to hold.

In order to do

the ultrapower of a superstructure with respect

an ultrafilter, and then we show that for suitable ultra-

to

filters, this construction produces enlargements.

(APP.2.1) DEFINITION: J

Let U

5

be a nonempty set. A family of subsets of

U,V E U

(2) if

,

is called an ultrafilter when

P(J)

(1) if

J

€ U

U

,

then and

U n V EU V

,

satisfies U

V

C_

J

,

then

u ,

V € (3) for

every

subset

V ' SJ ,

complement J \ V € U ,

either

V € U

or

its

and not both.

(APP.2.2) REMARK: Nonempty (1)

and

prove

(2)

families of nonempty subsets of above are called filters.

J

It is not

that

satisfy

difficult

to

that 'ultrafilter' is the same as 'maximal filter' for the

relation

of inclusion, C

.

A standard application of

Zorn's

Lemma shows that each filter is contained in some ultrafilter.

Appendix

466

2

(APP.2.3) DEFINITION: Let

X

be a superstructure (see

infinite set, and

€u

g

J

x ,

into

f,g E

xJ ,

{j € J : f(j) = g(j)} € U

iff

J

an

.

If

call

{j € J : f(j) € g(j)} E U

iff

f =U g

J

an ultrafilter of subsets of

U

f,g are functions from f

(O.l.l)),

.

(APP.2.4) EXERCISE: are some simples exercises for the reader to get

Here

to these definitions. Show that for any (1)

=u

(2)

f

(3) f (4)

f, g

iff

{j E: J : f(j) f g(j)} E U ;

gU

iff

{j E: J : f(j)

g

f cUg

k

a l l €u-members of

(i.e.,

{j € J : f(j) 5 g(j))

,

f(j) = g(j)

and

then

f =

U

g

€ U ;

X

(3.8.3)],

x'

(see

, with x'(j)

that

in

transformed sentences,

into

=

want

to replace

.

super-

Stroyan-Luxemburg,

or Barwise [1978, Ch.

=

x

X

into

A.3, XJ

Theorem

given by

,

for all j E J

analog of the Transfer Principle (0.2.3)(a)

*

j's,

.

we know that for the embedding of --->

many

will be a fixed

Xo consequence of gas' Theorem

a

x an

are €u-members of

.

the rest of this Appendix,

[1976, Theorem 3.1]),

f

except for finitely

structure over a fixed ground set As

g(j)} 6 U ;

if the ultrafilter is free, that is, if n [ U : U E U ] =

0

For

,

f,g E X J

is an equivalence relation;

g) iff (5)

used

holds, provided

is changed into

eu

and

=

When we deal with nonstandard extension, though, we

EU,

=u

by actual 'belongs to'

and

'equals

2

Appendix

467

to'. That is why we make the next construction.

(APP.2.5) DEFINITION: Let

be any ultrafilter over

U

collapsing function

M

Xo

and call

Yo

class; let

.

The

Mostowski

is defined as follows.

(i) Take an element in

J

M(f)

f

of

whose images are

XJ

the corresponding

be the set of those

all

=U-equivalence

equivalence

classes

interpreted as new individuals. (ii) Assume

M(f)

for each

the elements in

xP

not in

p 2 0

that for some f & XpJ ; X

P+l

\

X P

we

have

defined

we are going to define it :

' M(f) = IM(g) : g

if

f

is in

J Xp

, g EU

X

for

and is

P+l

f}

(APP.2.6) DEFINITION: With the same notations as above, call structure on the ground set

where j E J

,

and define

* :x----

the constant function x ' ( j ) = x The ultrapower X J / U is the image of X

XI

.

Yo

the super-

Y

is

> Y

for

all

under

*.

(APP.2.7) REMARK: The

map

*

is a superstructure extension.

This means (see

its definition in the comments before (0.4.6)): (i)

*

is injective; the reader is invited to

supply

a

proof of this. (ii)

*

satisfies part (a) of Leibniz' Principle; this

is

Appendix

468

not

difficult

to

of

prove, but it is outside the scope

2

this

Appendix, since a much more careful treatment of bounded formulas is needed.

For a full proof of this and the Internal

Principle, we

refer

to any of the

following

Definition

books:

Stroyan-

Luxemburg [1976, Chapter 3, Sections 4 & 81, Davis [ 1977, Chapter Sections 7 & 81,

1,

Barwise [1978, Chapters A.6 & A.31; and for

an elementary version, to Keisler [1976, Section 1D*l

.

(APP.2.8) REMARK:

Moreover, whenever the ultrafilter chain

contains a descending

U

* is not

of sets with empty intersection, the map

(prove

so it becomes a meaningful extension.

it),

Examples

of

such ultrafilters are not hard to find; maybe the simplest one is a

maximal

filter

of parts of

N

that contains

the

so-called

Frbchet filter {A 5 N : N \ A

(apply

Zornls

is finite]

Lemma to prove the existence of

such

a

maximal

filter). However, we are interested in superstructure extensions that enjoy the stronger property of being enlargements, of

(0.4.6).

through

The

the

consistent

existence of enlargements can

Compactness

if

Theorem

("a

set

each of its finite subsets is

of

in the

be

sense

established

sentences

is

consistent"),

see

Robinson [1966, Chapter 21 for that approach; instead, we prefer to

show

stronger not

only

that

for

'adequate' ultrafilters

than that of the descending chain), a

true extension, but also

original superstructure.

an

(with a

property

the ultrapower enlargement

of

is the

2

Appendix

469

DEFINITION:

(APP.2.9)

We

say

the

ultrapower

e v e r y nonempty f a m i l y intersection

i s adequate

XJ/U

of s u b s e t s of

R

property ( i . e . ,

X

of

s € XJ

so t h a t

R

V B E R

3 U E U ,

for

with th e f i n i t e

such that every f i n i t e

h a s nonempty i n t e r s e c t i o n ) ,

family

when

sub-

there is a

map

.

B 2 s ( U )

(APP.2.10) EXAMPLES:

Let

(1)

X

,

be t h e s e t of f i n i t e p a r t s of t h e power s e t

J

J = Pf(P(X))

,

and

an u l t r a f i l t e r t h a t c o n t a i n s

U

of the

sets { j € J : A € j i s We c l a i m t h e u l t r a p o w e r

J X /U

A E X .

i s adequate.

t h e f i n i t e i n t e r s e c t i o n p r o p e r t y , then f o r each A . = n[B € R : B

s:J--->X

B € R

, i t is obvious t h a t

C a l l a binary relation

s u b s e t o f i t s domain,

finite

b € X

element

.

the

relations finite

x

concurrent i f f o r

,

S C_ dom(r)

every

t h e r e is always an

,

(a,b) € r

.

be t h e C a r t e s i a n p r o d u c t o f t h e f a m i l y

J

{Pf(dom(r)) : r € i.e.,

r €

such t h a t V a € S

Let

, the set

.

j E J

B z s ( { j € J : B € J}) (2)

J

be a c h o i c e f u n c t i o n

s ( j ) € Aj

Then f o r e v e r y

j

has

jl

J

i s n o t empty. Let

REP(X)

If

x

s e t o f a l l maps of

subset

i s a concurrent r e l a t i o n } j

d e f i n e d on t h e s e t o f

X

and such t h a t t h e image

of

i t s domain.

j(r)

( I t i s obvious

of

that

concurrent

r

is

a

concurrent

2

Appendix

470

r e l a t i o n s do e x i s t ,

so

i s n o t empty.) Next,

J

t a k e as

U

an

u l t r a f i l t e r that contains the s e t s i s a concurrent r e l a t i o n , then j o ( r ) c j ( r ) I . Then t h e u l t r a p o w e r X J / U i s adequate: l e t R E P ( X ) be a

U ( j ) = { j € J : i f r€X 0

family

the f i n i t e intersection property,

with

and

define

the

following binary r e l a t i o n : ( B , x ) E ro

ro i s a member of set

n j(ro)

and is c o n c u r r e n t :

X

n o t empty.

is

x E B € 8 ;

iff

Let

f o r each

--- >

s : J

j

be

X

€ J

, the

a

choice

function

s(j)e nj(ro) , For

every

B

if

j

Then,

therefore,

in

,

R

take a

,

cU(jB)

B 2 s(U(jB))

,

s

.

J

jBE J

the set

by d e f i n i t i o n of

E

j

so that

j B ( r o )= {B]

.

j(ro)

;

i s a member

B

s(j) € B

.

of

We have shown t h a t

.

(APP.2.11) PROPOSITION:

Every adequate u l t r a p o w e r i s an enlargement. PROOF: Let

A

be any member of

a *finite entity

in

F

X

.

We need t o show t h a t t h e r e i s

such t h a t

XJ/U

'A

C_ F

.

The f a m i l y IF € Pf(A) has

the

,

a E A

f i n i t e i n t e r s e c t i o n property.

a d e q u a t e , t h e r e i s a map

s : J --->X

s(U) 5 f o r some

: a € FI

U

in

U

.

{F

Pf(A)

Since th e ultrapower s o t h a t f o r each

is

a € A

: a € F]

Then { j € J : a E s ( j ) }€

By t h e d e f i n i t i o n of

,

M

,

u

it follows that

.*

a € M(s) €

*

Pf(A)

.

Appendix

Thus,

M(s)

471

is *finite and contains all

More simple ultrafilters, (APP.2.8),

‘A

= (*a

: a € A}

like the one mentioned in

. Remark

enjoy weaker forms of the Saturation and Comprehension

Principles (0.4.2) and (0.4.3). Stroyan-Luxemburg

[19761

properties of this type.

for

We refer the interested reader to

a

more

detailed

discussion

of