How do we know that *JR contains new entities in addition to the real numbers? In the ultrapower construction of * in Chapter 3, this was deduced from the fact that the nonprincipal ultrafilter :F contains no finite sets. That allowed us to show that *A -A is nonempty whenever A is an infinite subset ofn. If * is the transform of lR under a universe embedding, we may not be able to conclude that * #IR (for instance, the identity function on lU is a universe embedding lU --+ lU making * = IR). The condition that *JRn-IR be nonempty will have to be added as a new requirement (as was done in the previous chapter by assuming *N -N =I 0), or else derived from some other principle. In Section 11.11 it was shown that countable saturation guarantees the existence of hyperreal numbers that are characterised by countably many internal conditions. Saturation will be discussed further in Chapter 15. In this chapter we look at another principle, known as "enlargement"n, which is particularly convenient in the way it allows us to obtain nonstandard entities. We will then see how enlargements can be constructed by forming ultrapowers of superstructures. 14.1 Enlargements Assume from now on that lU is a universe over some set that includes . lU' is called an enlargement of lU if there exists a universe embedding lU lU' such that: 184 14. The Existence of Nonstandard Entities if A E 1U is a collection of sets with the finite intersection property, then there exists an element b E U' that belongs to * B for every BE A, i.e., b E n{* B : B E A} = nimA. Enlargements have an abundance of nonstandard internal entities: • Let A be the set of intervals (0, r) JR. for all positive real r. Then A E l!h(:IR.) and A has the finite intersection property. If bE n{*(O, r) : r E JR.+}, then b is a positive infinitesimal member of *R Indeed, in this case n im A is precisely the set of positive infinitesimals, and the enlargement principle ensures that it is nonempty. If we take A instead to consist of the intervals ( -r, r) -{0}, then n imA is the set of all nonzero infinitesimals. • For r E JR.+, let (r, oo) = { x E JR.: r < x }. Then by transfer *(r, oo) = { x E *JR. : r < x}. The collection of intervals ( r, oo) has the finite intersection property, and any member of is a positive unlimited hyperreal. • Let A E U be an infinite set. Then the collection {A-{a}o:aEA} has the finite intersection property. Since *(A -{a}) = *A -{*a}, it follows that in any enlargement of U there must be an entity b that belongs to *A but is distinct from *a for all a E A. Thus bE *Ao-{*ao: a E A}o= *A-imA. Such a b will be nonstandard, because if b =*a for some a E U, then *a E *A, implying a E A (recall from Theorem 13.9.2 that imA is the set of standard members of *A). So we see that in an enlargement, any infinite standard set has nonstandard members. In particular, if A is any infinite subset of JR. (e.g., A = N, Z, Q, etc.), then as A = imA, we deduce that A is a proper subset of *A. • If a E A, let Aa = {Z E PF(A) : a E Z} be the collection of finite subsets of A that contain a. Then { Aa : a E A} has the finite 14.2 Concurrence and Hyperfinite Approximation 185 intersection property, for if a1, ... , an EA, put Z = { a1, ... , an} to get Z EAa1 n n Aan . But if · · · BEn{*(Aa)o: a EA}, then B E *Pp(A), since in general *(Aa) *Pp(A), and *a E B for each a EA by transfer of the sentence (VZ E Aa)(a EZ). Thus Bis a hyperfinite subset of *A that contains {*ao: a EA}, i.e., imA Bo*A. The last example shows that enlargement is a stronger property than we had considered hitherto, since we were previously only able to establish the existence of such hyperfinite approximating sets in relation to countable subsets ofn:IR (cf. Section 12.3). We are left then with the question of whether enlargements themselves exist. In fact, they can be obtained by applying the ultrapower construction to the superstructure 1U(X), as will be explained below. The outcome is this: Enlargement Theorem. For any set X there exists an enlargement of IU(X) that is of the form 1U(*X). 14.2 Concurrence and Hyperfinite Approximation Abinary relation R is called concurrent, or finitely satisfiable, if for any finite subset { x1, ... , Xn} of the domain of R there exists an element y with XiRY for all i between 1 and n. If a concurrent relation R belongs to 1U, then an enlargement will contain entities b with *x(* R)b for all x Edam R. This provides another language for describing the above examples: • Let R be the "greater than" relation on n +: R = {(r,oy) EJR+ x JR+o: r > y}. R is concurrent, and by transfer * R is the "greater than" relation on *JR+. If r(* R)b for all r EJR+, then b is a positive infinitesimal. • Let R be the "less than" relation on JR+: R = { (r, y) E JR+ X JR+ : r < y }. If r(* R)b for all r EJR+, then b is a positive unlimited hyperreal. 186 14. The Existence of Nonstandard Entities • Let R be the nonidentity relation on a set A E lU: R = {(a,y) E Ax A: a# y}. Then R is concurrent precisely when A is infinite, and *R is the nonidentity relation on *A. If *a(* R)b for all a E A, then bE *Ao-{*ao: a E A}, and so b is a nonstandard member of *A. • Let R be the membership relation between A and PF(A): R = {(a,Z)o: a E Z E PF(A)}. For any A E lU, R is a concurrent relation in lU. If *a(* R)B for all a E A, then B is a hyperfinite subset of *A that includes imA. These notions yield alternative characterisations of the concept of enlargement: Theorem 14.2.1 If lU liJ' is a universe embedding, then the following are equivalent. (1) lU' is an enlargement of 1U relative to ( 2) For any concurrent relation R E lU there exists an entity bE liJ' such that *x(*R)b for all x E domnR. (3) For each set A E 1U there exists a hyperfinite subset B of *A that contains all the standard entities ofo*A: imA = {*a : a E A} B E *PF(A). Proof First assume (1). If R is a binary relation in 1U, for each x E domnR let R[x] = {y E ran R : xRy}. Then if R is concurrent, the collection {R[x] : x E domnR} has the finite intersection property. Also, this collection is a subset of the · power set P(ranR), so belongs to lU. Hence by (1) there is abE 1U' that belongs to every *(R[x]). By transfer of (Vy E R[x]) (xRy) we then have *x(*R)b for all x E domR, establishing (2). The proof that (2) implies (3) was indicated in the discussion of the last example above. To show that (3) implies (1), let A E 1U be a collection of sets with the finite intersection property. Take a transitive T E 1U with A T. Then the sentence (VZ E 'PF(A)) (3y E T) (Vz E Z) (y E z) is true. But assuming (3), there exists a hyperfinite B *A containing imA. Then by transfer of this sentence, since B E *PF(A), there exists some b E *T with b E z for all z E B, and hence b E *C for all C E A. Therefore ( 1) holds. D 14.3 Enlargements as Ultrapowers 187 14.3 Enlargements as Ultrapowers In Chapter 3 we constructed * by starting with certain real-valued sequences, i.e., functions r, 8 : N -4 ' and identifying them when the set [r = 8] = {n E N : rn = 8n} on which they agree belongs to an ultrafilter :F on N. This method of construction can be applied much more widely by allow ing :F to be an ultrafilter on a set I other than N. Then any two functions with domain I can be identified if the subset of I on which they agree belongs to :F. The approach can be used to build enlargements of a super structure lU(X), and we will now sketch out the way in which it works. So, let I be an infinite set and F a nonprincipal ultrafilter on I. Then lU(X)1 is the set of all functions from I to U(X). For a E lU(X), let a1 E lU(.X:)f be the function with constant value a. For j, g E lU(X)f, put [! = g] {i E I: f(i) = g(i)}, [/ E g] {i E I: f(i) E g(i)}, [/ E a] -[f E a1] = {i E I: j(i) E a}, [a E!] [aI E /] = { i E I: a E f(i)}, Zn {f E lU(X)1 : [j E Un(X)] E F}, z U{Zn : n 2: 0}. Z is then the union of the increasing chain Z0 <;;;; Z1 <;;;; The members • of Z may be viewed as functions of "bounded rank" in the sense that if f E Zn, then f(i) E Un(X) for :F-almost all i in I. In fact, such an f can itself be assigned a rank, because [/ E IUn(X)] <;;;; {i E I: f(i) has rank 0} U U {i E I: f(i) has rank n}, ··· and the union on the right is made up of pairwise disjoint sets, so {i E I: f(i) has rank k} E :F for exactly one k ::; n, allowing us to define f to be of rank k. Then Zn -Zn-1 consists of the functions of rank n, and the rank of the function a1 in Z is the same as the rank of the entity a in U(X). For f E Zo, let (!] = {g E Zo : [/ = g] E :F}. Put lf = { [/] : f E Zo}. There is a map f [!]nfrom Z to the superstructure IU(li) over lf that composes with the map a a1 of IU(X) into Z to give a universe embedding. The definition of [!] is by induction on the rank of f. 188 14. The Existence of Nonstandard Entities For f E Zo, [f] has just been defined and is a member of Uo (Y). Thus for n 2 0 we can make the inductive hypothesis that for all f E Zn, [f] has been defined and is a member of Un (Y) . Then forngoE Zn+l -Zn put [g] = { (!] : f E Zn and [f E g] E F}. This specifies [g] as a subset of Un (Y), and hence a member of Un+l (Y), and completes the inductive construction. · Now, for f, g E Zit can be shown that (!] E (g] iff {i E I: f(i) E g(i)} E F and [f] = [g] iff {i E I: f(i) = g(i)} E F. For each a E UJ(X) define *a to be [ai] E UJ(Y). For a EX, a1 is of rank 0, i.e., a1 E Z0, and we let a be identified with *a E Y. For a of rank n + 1 we find that *a = { [f] : f E Zn and [f E a]E F}. Consequently, - *Un (X) { (!] : [f E Un(X)] E F} { [f]o: f has rank n}. In particular, *X = { (!] : [f E X] E F} = Y, showing that UJ(Y) is just UJ(*X) . Also, since [f E 0] = 0, *0 = {[f] : [f E 0] E F} = 0. To show that the transfer principle holds for this construction requires the demonstration of a version of Los's theorem. This takes the following form. For any .Cucx)-formula cp(x1, ... ,xp) and any /I, .o. .,fp E Z, the sentence *cp([h] , .n.o. , [fp]) is true if and only if {i E I: cp(fi(i), ... , fp(i)) is true} E F. If the ultrafilter F is non principal, then *JR will have nonstandard members, as shown in Sections 3.8 and 3.9. But to obtain UJ(*X) as an enlargement of 1IJ(X) a special ultrafilter Fhas to be used. To define this, let I = PF (UJ (X)), the set of all finite subsets of U(X). Each member of I belongs to UJ(X), but I itself does not. For each a E I, define Ia ={bE I: a b}. 14.4 Exercises on the Ultrapower Construction 189 Then if a1, ... , an E I, putting b = a1 U · · · U an E I gives b E Ia1 n · · · n Ian . This shows that the collection {Ia : a E I} has the finite intersection property, and so is included in some ultrafilter F on I (Theorem 2.6.1). To show that the superstructure lU(*X) associated with this particular F is an enlargement, let A E lU(X) be a collection of sets that has the finite intersection property. For each b E I, b n A is a finite subcollection of A, so if bn A is nonempty, it has nonempty intersection. In that case, let f(b) be any member of this intersection. Hence f(b) E nCb n A) E P(UA). If, however, b n A= 0, let f(b) = 0. The resulting function f has bounded rank, and so determines an element [!]nof lU(*X). It will suffice to show that [/] E n{* B : B E A}. Now if BE A, then {B} E I, and so I{B} ={bE I: BE b} E F. But now if BE b, then BE b n A, and so f(b) E B. This shows that I{B} {bE I: f(b) E B} = [/ E B], giving [/ E B] E F, and therefore [/J E * B, as desired. This completes the proof of the enlargement theorem. An alternative proof that lU(*X) is an enlargement can be given by directly proving the hyperfinite approximation property that for any set A E lU(X) there exists a B with imA BE *Pp(A). (i) For such an A put g(b) = b nAoto define a function g :I Pp(A). Then [g E PF(A)] =IE F, making [g] E *PF(A). But for a E A, we have a E b n A when a E b, so I{a} ={bE I: a E b n A}o= [a E g] , implying that * a E [g] . Thus putting B = [g] fulfills (i). 14.4 Exercises on the Ultrapower Construction (1) Verify the details of the ultrapower construction of enlargements. (2) Suppose FEZ is such that F(i) is a function for (almost all) i E I. Show that [F] is a function satisfying [F]([h]) = [k] iff {io: F(i)(h(i)) = k(i)} E F. (3) Is the ultrafilter used in the above construction nonprincipal?