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Transfinite Ordinals and Their Notations

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Transfinite Ordinals and Their Notations 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
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