Transnite Ordinals and Their Notations

For The Uninitiated

Version

Hilbert Levitz

Department of Computer Science

Florida State University

levitzcsfsuedu

Into duction

This is supp osed to b e a primer to give you a feel for the sub ject fast

Youre not exp ected to understand immediately why everything that is as

serted is true but theres enough there so you could ll in the gaps if you

wanted to

The theory of transnite ordinals is a part of set theory While the

concept is tied up with the completed innite and high cardinalities well

emphasize more constructive asp ects of the theory There have b een appli

cations of constructive treatments of ordinals to recursive function theory

and pro of theory

Pretty much everything we need to know ab out ordinals follows from the

following three prop erties of the class of all ordinals O Just how the exis

tence of such a class can b e shown well regard as a matter of mathematical

foundations

Basic Prop erties of the System of Ordinal Numb ers

O has a well ordering which will b e denoted by

Any well ordered set whatso ever is order isomorphic to a unique

initial segment of O The ordinal determining the segment is called the

ordinal of the set This makes the system of ordinals in some sense the

mother of all well ordered sets

Any set of memb ers of O has a strict upp er b ound and therefore by

well ordering a least upp er b ound in O

Warning The collection of all ordinals itself is not a set Youll get

hit with a famous paradox if you treat it like one Subcollections of initial

segments of the ordinals are sets and these are all we use

As a consequence of the well ordering we see easily that

i One can do arguments by induction over the ordinals or over any

initial segment of them if we use the form of the induction principle

Recall that for natural numb ers we also have induction in the n to n

form but that fails as so on as one has an ob ject greater than every natural

numb er Induction in the form frequently go es under the fancy name

transnite induction

ii We let denote the smallest ordinal Every ordinal has an immedi

ate successor so by taking rep eated successors of zero we generate all the

natural numb ers as an initial segment of the system of ordinals Not all non

zero ordinals have an immediate predecessor Those without an immediate

predecessor are called limit ordinals The smallest numb er to follow all the

natural numb ers is denoted by and it is a limit ordinal Each limit ordinal

is the least upp er b ound supremum of the set of smaller ordinals

iii On account of the well ordering we can dene by recursion op erations

of addition multiplication and exp onentiation The only added feature over

recursion on natural numb ers is that we need to sp ecify what to do at limit

ordinals

x x

0 0

x y x y

x y sup x y when y is a limit ordinal

y y

x

0

xy xy x

xy sup xy when y is limit ordinal

y y

0

x

0

y y

x x x

y y

x sup x when y is a limit ordinal

y y

It turns out that

a Addition and Multiplication are asso ciative Neither is commutative

Multiplication distributes from the left over addition that is

xy z xy xz

Right distributivity fails but we do have the inequality

x y z xz y z

b Addition multiplication and exp onentiation are with trivial excep

tions

i Continuous strictly increasing function of the right argument

ii Weakly monotone increasing functions of the left argument

Cantor Normal Form

The theorem of ordinary numb er theory that justies writing any non

zero numb er to a numb er base like base or base applies to transnite

ordinals as well

Theorem Any nonzero ordinal can b e written uniquely as a

p olynomial to any base greater than with descending exp onents

and co ecients less than the base The co ecients are written

to the right of the base Such a representation is called a Cantor

normal form

Its common to use as the base the numb er in which case the co e

cients are natural numb ers thus a typical normal form lo oks like

1 2

k

n n n

1 2

k

where

1 2

k

The rule for comparing two normal forms to see which represents the

bigger ordinal is as follows One rst lo oks to see which has the highest

leading exp onent if they are the same then you lo ok to see which has the

highest leading co ecient etc Such a metho d could go into an innite

regress on account of the fact shown in the next section that some ordinals

can equal their own leading exp onent

x

Numb ers of the form determine initial segments closed under addi

x

dertermine initial segments of the ordinals tion and numb ers of the form

closed under multiplication

Denition of the ordinal 

0

The limit of the sequence

was named by Cantor We can see that its a solution of the the

0

x

equation x by the following simple argument

Consider the recursively dened sequence

i a

o

a

n

ii a

n+1

then

a l im a

n n

l im a l im a l im

n+1 n

x

Another way to say that a numb er is a solution of x is to say that

x

its a xed p oint of the function

is the smallest ordinal bigger than that determines an initial

0

segment closed under ordinal addition multiplication and exp onentiation

From this it follows that you cant denote it just using symb ols plus

times and exp onentiation One can show using Cantor Normal Form that

anything smaller can b e so represented

Actually a similar argument can b e used to show additional xed

p oints and they run clear through the ordinals They can b e arranged in a

transnite sequence as

0 1 2

Computation With Ordinals On Real Computers

We describ e b elow a system of formal terms T as follows

i The symb ol a is an element of T

ii If x and y are elements of T then so is the formal term f x y

Our intention here is that a denote the ordinal numb er The meaning

of the function symb ol f here we shall leave mysterious

Recursive Denition of an ordering on T

i a f x y for all terms x and y

ii f x y f u v i one of the following holds

a x u and y v

b x u and y f u v

c u x and f x y v or f x y v

This ordering is easily shown to b e linear It turns out to b e a well

ordering whose ordinal is It is known that it cant b e proven to b e a

0

well ordering in rst order numb er theory yet any initial segment can b e

Gentzen

Each term can b e regarded as a notation for an ordinal less than To

0

demonstrate that we are dealing with ob jects that can b e manipulated by

computer we give b elow a PROLOG program for telling of two terms u and

v whether u v

l essa f U V

l essf X Y f U V X U l essY V

l essf X Y f U V l essX U l essY f U V

l essf X Y f U V l essU X f X Y V

l essf X Y f U V l essU X l essf X Y V

Now if u and v are are formal terms of T the query

lessuv

will cause he machine to return the answer yes if the ordinal denoted

by u is less than the ordinal denoted by v and no otherwise

Connections to the Multiset Orderings of Term Rewriting

Theory

i The set of natural numb ers ordered by the multiset ordering has

ordinal

In fact in such a case the multiset fm m m g with m m

1 2 1 2

k

m corresp onds to the ordinal

k

m m m

1 2

k

ii If we p ermit multisets of multisets of multisets etc of natural

numb ers the ordinal is

0

In fact in such a case the multiset fM M M g where M

1 2 1

k

M M are multisets corresp onds to the ordinal

2

k

or dM or dM or dM

1 2

k

Problem of Skolem

Nothing to do with logic really but just ab out every one whos made a

contribution works in logic

Consider the set of formal terms S dened inductively b elow

i The symb ols and X are in S

v

ii If u and v are in S then so are u v uv and u

Each term determines in a natural way a function of one variable on

p ositive natural numb ers

Consider the ordering of the functions

f g i f x g x for suciently large x

Questions

Is it a linear ordering Yes Richardson

Is it a well ordering Yes Ehrenfeucht using Kruskals Theorem

If yes to the ab ove whats the ordinal Unknown

Skolem showed is a lower b ound Levitz showed that the rst critical

0

epsilon numb er is an upp er b ound The rst critical epsilon numb er is

x

dened as follows Arrange the solutions of x in order and call them

etc Then the rst critical epsilon numb er is the smallest

0 1 2

memb er of the sequence equal to own subscript

Levitz Van den Dries and Dahn have partial results supp orting the

conjecture that the actual ordinal is

0

The Least Uncountable Ordinal

It is a theorem of general set theory that there are uncountable sets

and a further theorem that any set can b e well ordered Let denote

the smallest ordinal that is the ordinal of an uncountable set The initial

segment determined by has the following prop erties

I If then are only countably many ordinals less than

II Every countable set of ordinals less than has a strict upp er b ound

less than

It turns out that any well ordered set with these prop erties is order

isomorphic to the initial segment determined by More precisely if W is

a well ordered set each of whose memb ers has only countably many smaller

memb ers in W and each countable subset of W has a strict upp er b ound in

W then W is order isomorphix to the initial segment determined

Normal Functions and Ordinal Notations

This is not intended to give a rigorous denition of a system of ordi

nal notations but rather to help develop some insight and intuition for a

commonly used one

x

That cute little argument used to show that has xed p oints can b e

abstracted to show that if a sequence of successive iterations of a continuous

function approaches a limit then that limit is a xed p oint Whats more if

the function is strictly increasing and f the iterations are increasing

and therefore approach a limit This brings us to the notion of a normal

function

Denition A normal function f from into is a strictly

increasing continuous function which has the additional prop erty

that f

i Normal functions have xed p oints and the set of such is order iso

morphic to the set of ordinals less than

ii The function which enumerates the xed p oints of a normal function

is itself a normal function This suggests a hierarchy of normal functions

for given by

x

a x

0

b For the function enumerates in order the simultaneous

xed p oints of all for all

One might attempt to dene a function in this way however the

attempt to do so fails as it turns out that the set of simultaneous xed

p oints of all for all is empty

Bachmanns idea essentially was to get around this collapse through

the following considerations The function turns out to b e a normal

x

function so he let b e dened as the function which enumerates in order

the set of xed p oints of Now we can dene to b e the function

x +1

which enumerates in order the xed p oints of We can continue this way

but face a similar collapse in trying to dene To keep things going

+

let b e dened as the function which enumerates in order the xed

+

p oints of

+x

With no more than what was said ab ove you can without a general rule

and and governing this phenomenon work your way to things like

2

and even

Feferman indicated how one can give a general rule for describing the hi

erarchy His ideas have b een worked out and rened variously by Weyrauch

Aczel Bridge and Bucholz Not only is the least uncountable ordinal

used in subscripts for this hierarchy but ordinals from even higher cardi

nality classes are used For example ordinals like

2 3

app ear in subscripts

Despite the presence of uncountable ordinals in the description of the

hierarchy the ordinals we present b elow are countable and in fact recursive

To show this one has to show how all ordinals in initial segments they

determine can b e represented by means of smaller ordinals using functions

of the hierarchy and work out recursion relations that show how to compare

two representations assuming knowledge of how to compare their subterms

x

If one starts the hierarchy using the function as we did here then one

x

also needs the addition function on ordinals to get representations since

is additively prime

One could now if one so desires build up the representations from

scratch as purely formal terms and recursively dene the ordering on them

thus throwing away the whole scaolding of ordinal functions and un

countable ordinals used in the construction

An Ordinal Menagerie

Below we list some ordinals in order of size There is an imp ortant

b o dy of literature in pro of theory relating some of these to the strength

of various formal systems To keep things simple we wont get into it here

the ordinal numb er

0

1 0

the least critical epsilon numb er that is the smallest solution

2

of x

x

the closure ordinal for primitive recursive functions on the ordi

nals

the FefermanSchutte ordinal Maximal ordinal for recursive

0

path orderings Dershowitz

ordinal of the Ackermann notation system

2

the maximal ordinal for simplication orderings Dershowitz

Okada An ordering of the natural numb ers having this ordinal is presented

in the German edition of Schuttes b o ok Beweistheorie

Closure ordinal for nite structured lab eled trees under

+1

homeomorphic emb edding as dened by Kruskal with lab els coming from

a well quasiordered set This is the closure ordinal in the sense that if the

maximal ordinal of well ordered extensions of the set of lab els is smaller than

this ordinal then so will b e the maximal ordinal for well ordered extensions

of the set of structured lab eled trees DSchmidt

Veblens ordinal

Howards Ordinal

+1

the limiting ordinal of Takeutis ordinal notations of nite order

Levitz

References

A nice intro duction to transnite numb ers generally is P Halmos Naive

Set Theory Princeton NJ Van Nostrand

An excellent treatment of ordinal notations is in K Schutte Proof The

ory SpringerVerlag BerlinHeidelb ergNew York