aa r X i v : . [ m a t h . L O ] F e b AN INFINITE NATURAL SUM
PAOLO LIPPARINI
Abstract.
As far as algebraic properties are concerned, the usualaddition on the class of ordinal numbers is not really well behaved;for example, it is not commutative, nor left cancellative etc. Ina few cases, the natural Hessenberg sum is a better alternative,since it shares most of the usual properties of the addition on thenaturals.A countably infinite iteration of the natural sum has been usedin a recent paper by V¨a¨an¨anen and Wang, with applications toinfinitary logics. We present a detailed study of this infinitaryoperation, showing that there are many similarities with the ordi-nary infinitary sum, and providing connections with certain kindsof infinite mixed sums. Introduction
There are different ways to extend the addition operation from theset ω of natural numbers to the class of ordinals. The standard wayis to take α ` β as the ordinal which represents the order type of α with a copy of β added at the top. This operation can be introduced bythe customary inductive definition and satisfies only few of the familiarproperties shared by the addition on the naturals.On the other hand, again on the class of the ordinals, one can definethe (Hessenberg) natural sum α β of α and β by expressing α and β in Cantor normal form and “summing linearly”. See below for furtherdetails. The resulting operation is commutative, associative andcancellative. It can be given an inductive definition as follows.0 “ α β “ sup α ă αβ ă β t S p α β q , S p α β qu (1)where S denotes successor . Mathematics Subject Classification.
Primary 03E10; Secondary 06A05.
Key words and phrases.
Ordinal number, sum, infinite natural sum, left-finite,piecewise convex, infinite mixed sum.Work performed under the auspices of G.N.S.A.G.A.
It is relevant that the natural sum, too, admits an order theoreticaldefinition. If α , β and γ are ordinals, γ is said to be a mixed sum of α and β if there are disjoint subsets A and B of γ such that γ “ A Y B and A , B have order type, respectively, α and β , under the order inducedby γ . P. W. Carruth [C] showed that α β is the largest mixed sumof α and β . He also found many applications.In V¨a¨an¨anen and Wang [VW] the authors define a countably infiniteextension of by taking supremum at the limit stage. They provideapplications to infinitary logics. Subsequently, we have found appli-cations to compactness of topological spaces in the spirit of [Li], inparticular, with respect to Frol´ık sums.Carruth Theorem, as it stands, cannot be generalized to such aninfinite natural sum. Indeed, every countably infinite ordinal is an“infinite mixed sum” of countably many 1’s, hence in the infinite casethe maximum is not necessarily attained. See Definition 4.1 and thecomment after Theorem 4.2.However, we show that Carruth Theorem can indeed be generalized,provided we restrict ourselves to certain well behaved infinite mixedsums. In order to provide this generalization, we need a finer descrip-tion of the countably infinite natural sum. We show that any infinitenatural sum can be computed in two steps: in the first step one takesthe natural sum of some sufficiently large finite set of summands. Inthe second step one adds the infinite ordinary sum of the remainingsummands. In other words, the infinite natural sum and the moreusual infinite sum differ only for a finite “head” and they agree on theremaining “tail”. This is used in order to show that the infinite nat-ural sum of a sequence is the maximum of all possible infinite mixedsums made of elements from the sequence, provided one restricts onlyto mixed sums satisfying an appropriate finiteness condition.In the end, we show that the infinite natural sum can be actuallycomputed as some finite natural sum, and can be expressed in terms ofthe Cantor normal forms of the summands. We show that a sequencehas only a finite number of mixed sums satisfying an additional con-vexity property. This extends a classical theorem by Sierpinski [S1],asserting that one gets only a finite number of values for the sum ofsome fixed countable sequence of ordinals, by changing their order.2. Natural sums
We now give more details about the definitions hinted above andlist some simple facts about the natural sums. Here and below sums,products and exponentiations will be always intended in the ordinal
N INFINITE NATURAL SUM 3 sense. See, e. g., the books Bachmann [B] and Sierpinski [S2] for adetailed introduction to ordinal operations. Recall that every ordinal α ą Cantor normal form asfollows(2) α “ ω ξ k r k ` ω ξ k ´ r k ´ ` ¨ ¨ ¨ ` ω ξ r ` ω ξ r for integers k ě r k , . . . , r ą ξ k ą ξ k ´ ą ¨ ¨ ¨ ą ξ ą ξ . Definition 2.1.
The natural sum α β of two ordinals α and β is theonly operation satisfying α β “ ω ξ k p r k ` s k q ` ¨ ¨ ¨ ` ω ξ p r ` s q ` ω ξ p r ` s q whenever α “ ω ξ k r k ` ¨ ¨ ¨ ` ω ξ r ` ω ξ r β “ ω ξ k s k ` ¨ ¨ ¨ ` ω ξ s ` ω ξ s and k, r k , . . . , r , s k , . . . , s ă ω , ξ k ą ¨ ¨ ¨ ą ξ ą ξ .The definition is justified by the fact that we can represent everynonzero α and β in Cantor normal form and then insert some morenull coefficients for convenience just in order to make the indices match.The null coefficients do not affect the ordinals, hence the definition iswell-posed. See, e. g., [B, S2] for further details.An elegant way to introduce the natural sum is obtained by express-ing equation (2) in a conventional way as α “ ř ξ P F ω ξ , where F is thefinite multiset which contains each ξ ℓ exactly r ℓ times. This is justifiedby the fact that, say, ω ξ ` ω ξ “ ω ξ
2. In this way, α “ α “ ř ξ P F ω ξ and β “ ř ξ P T ω ξ , then α β is defined as ř ξ P F Y T ω ξ , where in theunion F Y T we take into account multiplicities. In this note, however,we shall follow the more conventional notations.It can be shown by induction on p max t α, β u , min t α, β uq , ordered lex-icographically, that Definition 2.1 is equivalent to the definition givenby means of equations (1). This shall not be needed in what follows.Notice that the assumption ξ k ą ¨ ¨ ¨ ą ξ ą ξ in Definition 2.1 isnecessary, since, for example, p ` ω q ` ω q is ω “ ω ` ` ω “ ω . However, theassumption that ξ k ą ¨ ¨ ¨ ą ξ ą ξ can be relaxed to ξ k ě ¨ ¨ ¨ ě ξ ě ξ . Proposition 2.2.
Let α , β and η be ordinals. PAOLO LIPPARINI (1) The operation is commutative, associative, both left and rightcancellative and strictly monotone in both arguments.(2) sup t α, β u ď α ` β ď α β .(3) If α, β ă ω η , then α β ă ω η .(4) If β ă ω η , then α β ă α ` ω η .(5) If β ă ω η , then p α β q ` ω η “ α ` ω η .Proof. Everything is almost immediate from Definition 2.1.For example, to prove (4), let α “ ω ξ k r k ` ¨ ¨ ¨ ` ω η r ` . . . with, asusual, the exponents of ω in decreasing order, and where we can allow r to be 0. Then α ` ω η “ ω ξ k r k ` ¨ ¨ ¨ ` ω η p r ` q , while, if β ă ω η , then α β “ ω ξ k r k `¨ ¨ ¨` ω η r ` . . . , since β does not contribute to summandswhere the exponent of ω is ě η . Thus surely α ` ω η ą α β , with noneed to compute explicitly those summands which are ă ω η . (cid:3) Parentheses are usually necessary in expressions involving both ` and ; for example, p q ` ω “ ω “ ω ` “ ` ω q , or p ` q ω “ ω ` “ ω “ ` p ω q . Definition 2.3.
Suppose that p α i q i ă ω is a countable sequence of ordi-nals, and set S n “ α . . . α n ´ , for every n ă ω . The natural sum of p α i q i ă ω is i ă ω α i “ sup n ă ω S n The above natural sum is denoted by ř i ă ω α i in [VW].In the above notation α . . . α n ´ we conventionally allow n “ Proposition 2.4.
Let α i , β i be ordinals and n, m ă ω .(1) ř i ă ω α i ď i ă ω α i (2) If β i ď α i , for every i ă ω , then i ă ω β i ď i ă ω α i (3) If n ă m , then S n ď S m ; equality holds if and only if α n “ ¨ ¨ ¨ “ α m ´ “ .(4) S n ď i ă ω α i ; equality holds if and only if α i “ , for every i ě n .(5) If π is a permutation of ω , then i ă ω α i “ i ă ω α π p i q (6) More generally, suppose that p F h q h ă ω is a partition of ω into finitesubsets, say, F h “ t j , . . . , j r p h q u , for every h P ω . Then i ă ω α i “ h ă ω j P F h α j “ h ă ω p α j α j . . . α j r p h q q Proof. (1)-(4) are immediate from the definitions and Proposition 2.2(1)-(2).
N INFINITE NATURAL SUM 5
Clause (5) is a remark in the proof of [VW, Proposition 4.4]. Anyway,(5) is the particular case of (6) when all the F h ’s are singletons.To prove (6), define, for h ă ωT h “ ˆ j P F α j ˙ . . . ˆ j P F h ´ α j ˙ Thus the right-hand of the equation in (6) is sup h ă ω T h . For h ă ω ,let m “ max ď ℓ ă h F ℓ . The maximum exists since each F ℓ is finite,and we are considering only a finite number of F ℓ ’s at a time. Theneach summand in the expansion of T h appears in S m ` (taking intoaccount multiplicities), hence, by (4) and monotonicity of the naturalsum, i ă ω α i ě S m ` ě T h . Hence i ă ω α i ě sup h ă ω T h . The reverseinequality is similar and easier. (cid:3) The assumption that each F h is finite in condition (6) above is neces-sary. For example, take α i “
1, for every i ă ω , thus i ă ω α i “ ω . Sup-pose that there is some infinite F ¯ h . Then j P F ¯ h α j “ ω . If ω z F ¯ h “ H ,then h ă ω ` j P F h α j ˘ ě ω ą ω .Not everything from Proposition 2.2 generalizes to infinite sums.For example, the operation i ă ω α i , though monotone, as stated in(2) above, is not strictly monotone. E. g., i ă ω “ i ă ω “ ω .Actually, i ă ω α i “ ω , for every choice of the α i ’s such that α i ă ω ,for every i ă ω , and such that there are infinitely many nonzero α i ’s.Condition (5) above can be interpreted as a version of commutativity,and (6) as a version of the generalized commutative-associative law.However, not all forms of associativity hold. We have seen that wecannot associate infinitely many summands inside some natural sum.Similarly, we are not allowed to “associate inside out”. Indeed, ω ` “ i ă ω “ i ă ω “ ω . This is a general and well-known fact. Forinfinitary operations, some very weak form of generalized associativityimplies some form of absorption. Example . Suppose that ‘ is a binary operation on some set X , and a P X is such that a ‘ x “ x , for every x P X . There is no infinitaryoperation À on X such that x ‘ à i P ω x i ` “ à i P ω x i for every sequence p x i q i P ω of elements of X . Indeed, taking x i “ a , forevery i P ω , and letting x “ À i P ω x i , we get a ‘ x “ x , a contradiction. PAOLO LIPPARINI Computing the infinite natural sum
Theorem 3.1. If p α i q i ă ω is a sequence of ordinals, then there is m ă ω such that the following hold, for every n ě m . (3) n ď i ă ω α i “ ÿ n ď i ă ω α i i ă ω α i “ p α . . . α n ´ q ` n ď i ă ω α i “ p α . . . α n ´ q ` ÿ n ď i ă ω α i (4) Proof.
Let ξ be the smallest ordinal such that the set t i P ω | α i ě ω ξ u is finite. Let m be the smallest index such that α i ă ω ξ , for every i ě m . The definition of ξ assures the existence of such an m . If ξ “ α i ’s are 0 and the proposition is trivial.Suppose that ξ is a successor ordinal, say ξ “ ε `
1. By the mini-mality of ξ , the set t i P ω | α i ě ω ε u is infinite, hence unbounded in ω .Then n ď i ă ω α i ě ř n ď i ă ω α i ě ω ε ω “ ω ε ` “ ω ξ . Suppose that ξ islimit. By the definition of ξ , we have that, for every ε ă ξ , there areinfinitely many i ă ω such that α i ě ω ε . In particular, we can choosesuch an i with i ě n . Then ř n ď i ă ω α i ě α i ě ω ε . Since this holds forevery ε ă ξ , we get n ď i ă ω α i ě ř n ď i ă ω α i ě sup ε ă ξ ω ε “ ω ξ . Theinequality n ď i ă ω α i ě ř n ď i ă ω α i ě ω ξ is proved, no matter whether ξ is successor or limit.On the other hand, because of the definition of m , if i ě n ě m ,then α i ă ω ξ . By Proposition 2.2(3), α n . . . α ℓ ´ ă ω ξ , for every ℓ ě n . Hence ř n ď i ă ω α i ď n ď i ă ω α i “ sup ℓ ă ω p α n . . . α ℓ ´ q ď ω ξ .In conclusion,(5) n ď i ă ω α i “ ÿ n ď i ă ω α i “ ω ξ thus we have proved (3).Let us now prove (4). The inequality i ă ω α i ě p α . . . α n ´ q ` n ď i ă ω α i is trivial, since every “partial sum” on the right is boundedby the partial sum on the left having the same length, by Proposi-tion 2.2(2). For the other direction, and recalling that S ℓ denotes α . . . α ℓ ´ , observe that, by associativity, for every ℓ ě n , wehave S ℓ “ S n α n . . . α ℓ ´ ă S n ` ω ξ “ S n ` n ď i ă ω α i , wherethe strict inequality follows from repeated applications of Proposition2.2(4), since α n , . . . , α ℓ ´ ă ω ξ . The last identity is from equation(5). Since i ă ω α i “ sup ℓ ă ω S ℓ and since S ℓ is increasing, we get i ă ω α i ď S n ` n ď i ă ω α i . N INFINITE NATURAL SUM 7
The identity i ă ω α i “ p α . . . α n ´ q ` ř n ď i ă ω α i is now imme-diate from (3). It can be also proved in a way similar to above. (cid:3) Notice that the sum ` in equation (4) cannot be replaced by a nat-ural sum , that is, we do not have, in general, i ă ω α i “ S n n ď i ă ω α i , nor we have i ă ω α i “ S n ř n ď i ă ω α i . This is similar tothe argument in Example 2.5: just take α i “
1, for every i P I ; then i ă ω α i “ ω but S n n ď i ă ω α i “ S n ř n ď i ă ω α i “ n ω “ ω ` n .However, in Corollary 5.1 we shall show that the computation of acountable natural sum can be actually reduced to the computation ofsome finite natural sum. Remark . Notice that equation (3) in Theorem 3.1, together withProposition 2.4(5), imply that if p α i q i ă ω is a sequence of ordinals, m is given by Theorem 3.1, and n ě m , then ř n ď i ă ω α i “ n ď i ă ω α i “ ř n ď i ă ω α π p i q , for every permutation π of r n, ω q . Actually, equation (5)in the proof of Theorem 3.1 shows that it is enough to assume that π is a bijection from r n, ω q to r n , ω q , for some n ě m (equation (5) doesnot hold if ξ “
0, but this case is trivial).The result in the present remark can be obtained also as a conse-quence of a theorem by Sierpinski [S1], asserting that a countable sumof nondecreasing ordinals is invariant under permutations. Just noticethat every sequence of ordinals is nondecreasing from some point on.On the other hand, Sierpinski’s result is immediate from equation (5).Thus parts of the present note can be seen as an extension of resultsfrom [S1] to natural sums.4.
Some kinds of mixed sums
The definition of a mixed sum of two ordinals can be obviously ex-tended to deal with infinitely many ordinals.
Definition 4.1.
Let p α i q i P I be any sequence of ordinals (with no re-striction on the cardinality of I ). An ordinal γ is a mixed sum of p α i q i P I if there are pairwise disjoint subsets p A i q i P I of γ such that Ť i P I A i “ γ and, for every i P I , A i has order type α i , with respect to the orderinduced on A i by γ .In the above situation, we say that γ is a mixed sum of p α i q i P I realizedby p A i q i P I , or simply that p A i q i P I is a realization of γ . Notice that α i can be recovered by A i , as embedded in γ .Notice that we could have given the above definition just under theassumption that γ and the α i ’s are linearly ordered sets, not necessarilywell ordered. In this respect, notice that any finite mixed sum of wellordered sets is itself necessarily well ordered; however, in case I is PAOLO LIPPARINI infinite, the α i ’s could “mix themselves” to a non well ordered set.For example, starting with countably many 1’s, we could obtain every countably infinite linear order as a mixed sum. Throughout this note,however, and no matter how interesting the general case of linear ordersis, we shall always assume that γ is an ordinal, that is, well ordered. Theorem 4.2. (Carruth [C], Neumer [N])
For every n ă ω and ordinalnumbers α , . . . , α n , the largest mixed sum of p α i q i ď n exists and is α α . . . α n . As we hinted in the introduction, and contrary to the finite case, theset of all the mixed sums of an infinite sequence of ordinals need nothave a maximum. If we take α i “ i ă ω , then every infinitecountable ordinal is a mixed sum of p α i q i P ω , thus the supremum of allthe mixed sums of p α i q i P ω is ω , which is not a mixed sum of p α i q i P ω .Hence there is some interest in restricting ourselves to well-behavedmixed sums Definition 4.3.
We say that γ is a left-finite mixed sum of p α i q i P I if γ can be realized as a mixed sum by p A i q i P I in such a way that, for every δ ă γ , the set t i P I | A i X δ “ Hu is finite; in words, for every δ ă γ ,the predecessors of δ are all taken from finitely many A i ’s.Given a realization p A i q i P I of γ and i P I , we say that A i is convex(in γ ) if r a, a s γ “ t δ P γ | a ď δ ď a u Ď A i , whenever a ă a P A i .We say that γ is a piecewise convex (resp., an almost piecewise con-vex ) mixed sum of p α i q i P I if γ can be realized in such a way that allthe A i ’s (resp., all but a finite number of the A i ’s) are convex in γ . Forbrevity, we shall write pw-convex in place of piecewise convex.If γ is a pw-convex mixed sum of p α i q i P I , as realized by p A i q i P I , then,for every i “ j P I and δ, ε P A i , δ , ε P A j , we have that δ ă δ ifand only if ε ă ε . In this way, if each A i is nonempty, the order on γ induces an order (in fact, a well order) on I . Hence we can reindex p A i q i P I as p A π p ι q q ι ă θ for some ordinal θ and some bijection π : θ Ñ I in such a way that δ ă δ , whenever δ P A π p ι q , δ P A π p ι q and ι ă ι .Then an easy induction shows that γ “ ř ι ă θ α π p ι q . If in addition γ isleft finite, then necessarily θ ď ω .Conversely, if γ “ ř ι ă θ α π p ι q , for some reindexing of the α i ’s, thentrivially γ is a pw-convex mixed sum of p α i q i P I , and if θ ď ω , then γ isalso left finite. We have proved the next proposition. Proposition 4.4.
Suppose that p α i q i P I is a sequence of ordinals, and α i ą , for every i P I . Then γ is a pw-convex (pw-convex and left-finite) mixed sum of p α i q i P I if and only if there are some ordinal θ (with θ ď ω ) and a bijection π : θ Ñ I such that γ “ ř ι ă θ α π p ι q . N INFINITE NATURAL SUM 9
Remark . There might be infinitely many left-finite mixed sums ofthe same sequence. Indeed, take α i “ ω , for every i ă ω . Since ω is theunion of countably many disjoint countably infinite sets, we see that ω is a (necessarily left-finite) mixed sum of p α i q i ă ω . By moving just onecopy of ω “to the bottom” we get that also ω ` ω is a left-finite mixedsum of p α i q i ă ω . Iterating, for every n ă ω we get ωn as a left-finitemixed sum of p α i q i ă ω . Also ω is a left-finite mixed sum of p α i q i ă ω ; byProposition 4.4 it is the only one which is left-finite and pw-convex;actually, it is the only one which is left-finite and almost pw-convex. Theorem 4.6. If p α i q i ă ω is a sequence of ordinals, then i ă ω α i is amixed sum of p α i q i ă ω . In fact, i ă ω α i is the largest left-finite mixedsum of p α i q i ă ω , and also the largest left-finite and almost pw-convexmixed sum of p α i q i ă ω .Proof. By equation (4) in Theorem 3.1, we have i ă ω α i “ p α . . . α n ´ q ` ř n ď i ă ω α i , for some n ă ω . By Theorem 4.2, γ “ α . . . α n ´ is a mixed sum of α , . . . , α n ´ . By the easy part of Proposition4.4, γ “ ř n ď i ă ω α i is a left-finite pw-convex mixed sum of p α i q n ď i ă ω .Putting the members of the realization of γ at the bottom, and themembers of the realization of γ at the top, we realize i ă ω α i “ γ ` γ as a left-finite and almost pw-convex mixed sum of p α i q i ă ω .To finish the proof of the theorem it is enough to show that if γ isany left-finite mixed sum of p α i q i ă ω , then γ ď i ă ω α i . Let γ be aleft-finite mixed sum of p α i q i ă ω as realized by p A i q i ă ω . If all but a finitenumber of the α i ’s are 0, then the result is immediate from Theorem4.2. Otherwise, left finiteness implies that γ is a limit ordinal. If γ ă γ ,then p γ X A i q i ă ω witnesses that γ is a mixed sum of p β i q i P I , where, forevery i ă ω , β i is the order type of γ X A i ; thus β i ď α i . Left finitenessimplies that only a finite number of the β i ’s are nonzero, thus, again byTheorem 4.2, γ ď β i . . . β i ℓ , for certain distinct indices i , . . . , i ℓ .Taking n “ sup t i , . . . , i ℓ u , we get γ ď β i . . . β i ℓ ď α i . . . α i ℓ ď α . . . α n ă i ă ω α i . Since γ is limit and γ ď i ă ω α i , for every γ ă γ , we get γ ď i ă ω α i . (cid:3) Expressing sums in terms of the normal form
The proof of Theorem 3.1 gives slightly more. Let α and ξ be or-dinals, and express α in Cantor normal form as ω η k r k ` ¨ ¨ ¨ ` ω η r .The ordinal α æ ξ , in words, α truncated at the ξ th exponent of ω , is ω η k r k ` ¨ ¨ ¨ ` ω η ℓ r ℓ , where ℓ is the smallest index such that ℓ ě ξ . Theabove definition should be intended in the sense that α æ ξ “ α ă ω ξ . Corollary 5.1.
Suppose that p α i q i ă ω is a sequence of ordinals, and let ξ be the smallest ordinal such that t i ă ω | α i ě ω ξ u is finite. Enumeratethose α i ’s such that α i ě ω ξ as α i , . . . , α i h , with i ă ¨ ¨ ¨ ă i h (thesequence might be empty). If ξ ą , then i ă ω α i “ p α i . . . α i h q ` ω ξ “ α æ ξi . . . α æ ξi h ω ξ and (6) ÿ i ă ω α i “ α i ` ¨ ¨ ¨ ` α i h ` ω ξ “ α æ ξi ` ¨ ¨ ¨ ` α æ ξi h ` ω ξ ;(7) moreover, for every ε ă ξ , we have i ă ω α i “ i ă ω α æ εi and ÿ i ă ω α i “ ÿ i ă ω α æ εi Proof.
The ξ defined in the statement of the present corollary is thesame as the ξ defined in the proof of Theorem 3.1; and the α i h definedhere is the same as a m ´ in that proof (if the sequence of the α i ℓ ’s is notempty). Equation (5) in the proof of Theorem 3.1 gives m ď i ă ω α i “ ω ξ . By commutativity and associativity of , and using Proposition2.2(5), equation (4) in Theorem 3.1 becomes exactly the first identityin equation (6). The second identity is easy ordinal arithmetic, noticingthat α ` ω ξ “ α æ ξ ` ω ξ and p α β q æ ξ “ α æ ξ β æ ξ , for every α and β .The proof of (7) is similar, using the fact that ř i ă ω α i “ α ` ¨ ¨ ¨ ` α m ´ ` ř m ď i ă ω α i . Then one should use the identity β ` γ ` ω ξ “ γ ` ω ξ ,holding whenever β ă ω ξ . Indeed, if γ ă ω ξ , then all sides are equalto ω ξ ; otherwise, if γ ě ω ξ , then β is absorbed by γ , since it is alreadyabsorbed by the leading term in the Cantor normal expression of γ .See [S1].To prove the last two identities, notice that if ε ă ξ , then ξ is alsothe least ordinal such that t i ă ω | α æ εi ě ω ξ u is finite. Hence wecan apply (6) twice to get i ă ω α æ εi “ p α æ εi q æ ξ . . . α æ εi h q æ ξ ω ξ “ α æ ξi . . . α æ ξi h ω ξ “ i ă ω α i . The last identity is proved in the sameway, using equation (7). (cid:3) Notice that Corollary 5.1 furnishes a method to compute i ă ω α i and ř i ă ω α i in terms of the Cantor normal forms of the α i ’s, in fact,of just finitely many α i ’s, once ξ has been determined.One cannot expect that, for every sequence p α i q i P ω of ordinals, thereis some permutation of ω such that i ă ω α i “ ř i ă ω α π p i q . The coun-terexample has little to do with infinity: just take two ordinals α and α such that α α “ α ` α and α α “ α ` α , for exam-ple, α “ α “ ω `
1. Then, setting α i “
0, for i ą
1, we have α α “ i ă ω α i “ ř i ă ω α π p i q , for every permutation π . Of course, N INFINITE NATURAL SUM 11 we can arrange things in order to have some really infinite sum, e. g.,take α “ α “ ω ` ω and α i “
1, for i ě Lemma 5.2.
Suppose that γ is a mixed sum of α and α with α ă α “ ω ξ , for some ξ ą , and the mixed sum is realized by A , A insuch a way that A is cofinal in γ . Then γ “ ω ξ .Proof. γ ě α “ ω ξ is trivial. For the other direction, let δ ă γ .Thus δ is a mixed sum of β , β , where β , β are the order types of,respectively, A X δ , A X δ . Since δ ă γ , A is cofinal in γ and A hasorder type ω ξ , then A X δ has order type ă ω ξ . Moreover A X δ hasorder type ď α ă ω ξ . Since δ is a mixed sum of β and β , then, byTheorem 4.2, δ ď β β . We have showed that β , β ă ω ξ , hence δ ď β β ă ω ξ , by Proposition 2.2(3). Since δ ă ω ξ , for every δ ă γ ,then γ ď ω ξ . (cid:3) Theorem 5.3. If p α i q i P I is a sequence of ordinals, then there are atmost a finite number of left-finite and almost pw-convex mixed sums of p α i q i P I .Proof. If there is some left-finite mixed sum of p α i q i P I , then necessarilyall but countably many α i ’s are 0. Hence it is no loss of generalityto assume that | I | ď ω . If all but a finite number of the α i ’s are 0,then the corollary follows from a theorem by L¨auchli [L¨a], assertingthat a finite set of ordinals has only a finite number of mixed sums. Inconclusion, we can assume that I “ ω and α i “
0, for every i ă ω .Suppose that γ is a left-finite and almost pw-convex mixed sum of p α i q i ă ω , as realized by p A i q i ă ω . Assume the notation in Corollary 5.1.We shall show that γ is a mixed sum of α i , . . . , α i h , ω ξ . This will givethe result by the mentioned theorem from L¨auchli [L¨a].Since γ is realized as an almost pw-convex mixed sum, there is n such that A i is convex, for every i ě n . Without loss of generality,choose n ą i h , or, which is the same, n ě m , where m is given byTheorem 3.1. Let A “ Ť i ě n A i , thus γ is realized as a finite mixed sumby A , . . . , A n ´ , A . Moreover, by the left finiteness of the realization p A i q i P I , and since we have assumed that the A i ’s are nonempty, we getthat A is cofinal in γ . The order type of A is ř n ď i ă ω α π p i q , for somepermutation of π of r n, ω q , by Proposition 4.4, since A “ Ť i ě n A i ,the A i ’s are convex (in γ , hence, a fortiori, in A ), for i ě n , and therealization p A i q i ă ω is left-finite. By Remark 3.2 and equation (5) inthe proof of Theorem 3.1, we get that A has order type ω ξ . Now let F “ t , . . . , n ´ uzt i , . . . , i h u , and set A “ A Y Ť j P F A j . By repeatedapplications of Lemma 5.2, we get that A has order type ω ξ . Since γ “ A Y A i Y ¨ ¨ ¨ Y A i h , then γ is a mixed sum of ω ξ , α i , . . . , α i h , whatwe wanted to prove. (cid:3) In view of Proposition 4.4, Theorem 5.3 extends a classical resultby Sierpinski [S1], asserting that ř i ă ω α π p i q assumes only finitely manyvalues, π varying among all permutations of ω .By Remark 4.5, the assumption of almost pw-convexity is necessaryin Theorem 5.3. Remark . A shifted sum is defined in the same way as a mixed sum,except that we do not require the A i ’s to be pairwise disjoint. See[Li] for applications of finite shifted sums. Since every mixed sum isa shifted sum and, on the other hand, given a shifted sum of p α i q i P I ,there is always some larger mixed sum of p α i q i P I , we get that Theorem4.2 holds for shifted sums in place of mixed sums. Then the proof ofTheorem 4.6, too, carries over to get the result for shifted sums in placeof mixed sums. On the other hand, the analogue of Theorem 5.3 doesnot hold for shifted sums. Indeed, let α i “ ω , for i ă ω . Then ω can berealized as a left-finite pw-convex shifted sum of p α i q i ă ω by A i “ r i, ω q .Then we can get infinitely many left-finite pw-convex shifted sums of p α i q i ă ω arguing as in Remark 4.5. Remark . It is clear that, when restricted to the class of ordinals, thesurreal number addition is the natural sum. See Alling [A], Ehrlich [E]and Gonshor [G] for information about the surreal numbers. Corollary5.1 suggests the possibility of extending the infinitary natural sum tothe class of those surreal numbers which have positive coefficients intheir Conway normal representation. Recall that every surreal number x can be uniquely expressed in Conway normal form as x “ ř s P S ω s r s ,where S is a reverse-well-ordered set of surreal numbers and the r s arenonzero real numbers. In case x is an ordinal the Conway and theCantor normal forms coincide.Let p x i q i ă ω be a countable sequence of surreal numbers with normalforms x i “ ř s P S i ω s r s,i and such that all the r s,i ’s are positive (weshall show later that this request can be somewhat weakened). Let S “ Ť i ă ω S i and define a subset S ˚ of S by declaring s P S ˚ if andonly if t t P S | t ě s u is reverse-well-ordered. It might well happenthat S ˚ “ H ; however, in any case, S ˚ is reverse-well-ordered. For s P S , let c s “ ř i r s,i , where the sum is taken among all i ă ω suchthat s P S i . This might be either a finite sum or a countably infinitesum of positive real numbers; in the latter case we consider it as aninfinite sum in the sense of classical analysis. We allow the possibility c s “ 8 , that is, c s “ ω in the surreal sense. N INFINITE NATURAL SUM 13
Suppose first that S “ S ˚ . In this case we define(8) i ă ω x i “ ÿ s P S ω s c s This is a well-defined surreal number, since S “ S ˚ , hence S is reverse-well-ordered. Strictly speaking, equation (8) does not necessarily givea normal form representation, due to the possibility that some c s is ω .Formally, i ă ω x i “ ř s P T ω s c s ` ř s P U ω s ` , where U “ t s P S | c s “ ω u and T “ S z U .Notice also that, in order to give the definition in equation (8) weonly need that S “ S ˚ and that the sums ř i r s,i are well-defined in thesense of classical analysis, no matter whether all the r s,i are positive.Now suppose that S “ S ˚ and let ¯ s be the surreal number t S z S ˚ |Hu . In this case we define(9) i ă ω x i “ ω ¯ s ` ÿ s P S ˚ ω s c s The definition in equation (9) makes sense just under the assumptionthat c s “ ř i r s,i is well-defined, for every s P S ˚ . However, it isnatural to ask that there is s P S z S ˚ such that ř i r t,i is well-defined and strictly positive , for every t P S z S ˚ with t ě s . In case thatthere is s P S z S ˚ such that ř i r t,i is well-defined and strictly negative,for every t P S z S ˚ with t ě s , a more natural definition would be i ă ω x i “ ´ ω ¯ s ` ř s P S ˚ ω s c s .In the special case when each x i is an ordinal, the definitions givenby (8) and (9) coincide with Definition 2.3, by Corollary 5.1.The definition given in the present remark is quite tentative, and isnot the only possible one. Notice that the definition of i ă ω x i givenhere for surreal numbers does not satisfy the analogue of Proposition2.4(2). Indeed, let x i “ ω i and y i “ ω ω ´ , for i ă ω . Then x i ă y i , forevery i ă ω ; however, i ă ω x i “ ω ω ą ω ω ´ “ ω ω ´ ω “ i ă ω y i . Disclaimer
Though the author has done his best efforts to compile the followinglist of references in the most accurate way, he acknowledges that thelist might turn out to be incomplete or partially inaccurate, possibly forreasons not depending on him. It is not intended that each work in thelist has given equally significant contributions to the discipline. Hence-forth the author disagrees with the use of the list (even in aggregateforms in combination with similar lists) in order to determine rank-ings or other indicators of, e. g., journals, individuals or institutions.In particular, the author considers that it is highly inappropriate, and strongly discourages, the use (even in partial, preliminary or auxiliaryforms) of indicators extracted from the list in decisions about individu-als (especially, job opportunities, career progressions etc.), attributionsof funds, and selections or evaluations of research projects.
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Foundations of analysis over surreal number fields , North-HollandMathematics Studies, . Notas de Matem´atica , North-Holland Pub-lishing Co., Amsterdam, 1987.[B] H. Bachmann,
Transfinite Zahlen, Zweite, neubearbeitete Auflage , Ergebnisseder Mathematik und ihrer Grenzgebiete, Band 1, Springer-Verlag, Berlin-New York, 1967.[C] P. W. Carruth,
Arithmetic of ordinals with applications to the theory of or-dered Abelian groups , Bull. Amer. Math. Soc. (1942), 262–271.[E] P. Ehrlich, The absolute arithmetic continuum and the unification of all num-bers great and small , Bull. Symbolic Logic (2012), 1–45.[G] H. Gonshor, An introduction to the theory of surreal numbers , London Mathe-matical Society Lecture Note Series, , Cambridge University Press, Cam-bridge, 1986.[L¨a] H. L¨auchli,
Mischsummen von Ordnungszahlen , Arch. Math. (1959), 356–359.[Li] P. Lipparini, Ordinal compactness , arXiv:1012.4737 (2011), 1–46.[N] W. Neumer, ¨Uber Mischsummen von Ordnungszahlen , Arch. Math. (1954),244–248.[S1] W. Sierpinski, Sur les s´eries infinies de nombres ordinaux , Fund. Math. (1949), 248–253.[S2] W. Sierpinski, Cardinal and ordinal numbers. Second revised edition , Mono-grafie Matematyczne, Vol. 34, Panstowe Wydawnictwo Naukowe, Warsaw1965.[VW] J. V¨a¨an¨anen, T. Wang,
An Ehrenfeucht-Fra¨ıss´e game for L ω ω , MLQ Math.Log. Q. (2013), 357–370. Dipartimento Diaccio di Matematica, Viale della Ricerca Scien-tifica, II Universit`a di Roma (Tor Vergata), I 00133 Rome, Italy
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