18 Completion by Enlargement

There are many mathematical structures that are "incomplete" because they lack certain elements, such as the limit of a Cauchy sequence, the sum of infinite series, the least upper bound of a set of elements, a point "at infinity"n, and so on. A variety of standard techniques exist for completing such structures by adding the missing elements. Now, the enlargement of a structure in a nonstandard framework is a kind of completion, and we are going to explore ways in which enlargements give an alternative approach to standard completions. From this perspective there is some redundancy in the enlargement process because in a sense it "saturates" a structure with all the elements one could ever imagine adjoining to it. Some of these new elements are irrelevant to completion, while others may be distinct but indistinguishable in terms of their role in completing the original structure. Thus we need to factor out such redundancy, and as we shall see, standard completions can typically be obtained as quotients of certain kinds of enlargement. 18.1 Completing the Rationals The set *Q of hyperrationals contains infinitely close approximations of all real numbers. For if r E JR, then by transfer (Vx E *JR) [r < x 􀀆 (3q E *Q) (r < q < x)]. So, putting x = r + c with c a positive infinitesimal implies that there is some q E *Ql with r < q < r + c and hence q 􀀐 r. Thus q E *Q n hal(r) and  232 18. Completion by Enlargement r is the shadow of q. There are two points to immediately note here: • By this argument, for each x E hal(r) there is a member of *Q between x and r, and these members are all equally good as infinitely close hyperrational approximations to r. • There are many members of *Q, namely all the unlimited ones, that are "infinitely far away" and hence irrelevant to the issue of approximating reals. This suggests that we confine attention to the set *Qlim = {x E *Q : X is limited}n= *Q n JL of limited hyperrationals, and that we identify those that are in the same halo. The way to make this work is to use the shadow map sh : *Qlim --t lR introduced in Section 5.6. We have just seen in effect that this map is a surjection from *Qlim to IR: for each r E lR there is an element q E *Qlim with sh(q) = r. But *Qlim is closed under addition and multiplication, so forms a subring of *IR, and the shadow map preserves addition and multiplication (Theorem 5.6.2) so is a ring homomorphism from *Qlim onto JR. Thus by the fundamental homomorphism theorem for rings, lR is isomorphic to the quotient ring of *Qlim factored by the kernel {x E *Qlim : sh(x) = 0} of the shadow map. But this kernel is just the set *Qinf = {X E *Q : X 􀀒 0} = *Q n II of infinitesimal hyperrationals. So we have an isomorphism *Qlim /*Qinf 􀈵 JR (cf. Exercise 5.7(4)). The members of *Qlim/*Qinf are the cosets *Qinf + X = {q + X : q E *Qinf} of elements x E *Qlim . These are the same as the equivalence classes of *Q1im under the relation 􀀐 of infinite closeness, because the following conditions are all equivalent: X y, sh(x) sh(y), sh(x-y) 0, *Qinf (x -y) E 1 *Qinf +X *Qinf + y.  18.2 Metric Space Completion 233 Hence and *QlimI*Qinf can also be described as the quotient set *QlimI 􀄞. Its isomorphism with lR is given by the map *Qinf + x 1--7 sh(x). This construction can be viewed as providing an alternative way of building the reals out of the rationals. As it stands, we have assumed the existence of lR in the above analysis and used its Dedekind completeness in obtaining the shadow map on which the discussion was based. But we could try to prove directly that *QlimI*Qinf is a complete ordered field: since all complete ordered fields are isomorphic, this would show that the construction was independent of the choice of nonstandard framework in which *Q *Qlim, and *Qinf reside. The question of completeness of quotient structures , like *Qlim I*Qinf will be addressed in the next two sections. We will see that there are many structures X for which *XlimI􀀵 is a "completion" of X. 18.2 Metric Space Completion A metric on a set X is a function d : X x X-JR􀁴 = JR+ U {0} satisfying the axioms • d(x, y) = 0 iff x = y; • d(x, y) = d(y, x); • d(x, y) :::; d(x, z) + d(z, y) (triangle inequality). The pair (X, d) is a metric space, in which the number d( x, y) is to be thought of as the distance from x to y. The Euclidean metric on lR is given by d(x, y) = lx -Yl· When X carries a commutative ring structure, a metric sometimes comes from a norm, which is a function x 1--7 llxll E JR􀃃 satisfying • llxll = 0 iff x = 0; • llx · Yll = llxll · IIYII; • llx + Yll :::; llxll + IIYII· Then putting d(x, y) = llx-Yil induces a metric on X. The absolute value function lxl is a norm on lR that induces the Euclidean metric. A sequence (xn : n E N) in a metric space (X, d) is Cauchy if  234 18. Completion by Enlargement This is just like the definition of a Cauchy sequence in 􀃪nbut with d in ' place of the Euclidean metric on R In fact, many of the ideas and results about convergence etc. of sequences can be lifted to an abstract metric space in this way. For instance, the sequence (xn : n E N) converges to x in (X, d) if (Vc E 􀃪+) (3kc: EN) (Vn EN) [n 􀀛 kc:n---+ d(xn, x) < c ]. Then the metric space can be defined to be complete if every Cauchy sequence in the space converges to a point in the space. A completion of a metric space (X, d) is another space (X', d') such that • X􀃚 X' and dis the restriction of d' to X, i.e., (X, d) is a subspace of (X', d'); • (X', d') is complete; • X is dense in X'. The last condition means that any point x' in X' can be approximated arbitrarily closely by points of X, i.e., there is a sequence (xn : n E N) of points Xn E X that converges to x'.For this it suffices that for each c E 􀃪+ there is some Xc: E X with d' ( x', Xc:) < c. Thus 􀃪 is a completion of Q under the Euclidean metric. It can be shown that any two completions of a metric space are isometric, meaning that there is a bijection between them that preserves their metrics and leaves the original space fixed. In this sense a completion of a metric space is unique. In particular, 􀃪is the completion of Q. 18.3 Nonstandard Hulls Consider a nonstandard framework for a set X that carries a metric d. We will take this framework to be a sequentially comprehensive enlargement. The extended function *dnon *X is not a metric, because it takes values in *􀃪2: rather than JR2:. But it does satisfy the axioms of a metric, by transfer, and this is enough to ensure that we can define equivalence relationsn􀄞 and "' of infinitesimal and limited proximity in *X. Put x 􀄞 y iff *d(x,y) 􀄞 0, x "' y iff *d( x, y) is limited, for all x, y E *X. The equivalence classes under 􀄞 are the halos hal( X) ={y E *X : X 􀀌y}, while the equivalence classes under "' are the galaxies gal(x) = {y E *Xn: x"'y}.  18.3 Nonstandard Hulls 235 A member x of *X is limited if it is of limited distance from some member of X, i.e., if x"' y for some y EX. Let *Xlim = {x E *X: xis limited}. *Xlim proves to be a galaxy including X, and is sometimes called the principal galaxy. At first sight it might be thought that the metric d could be extended to *Xlim by taking the distance between limited points x, y to be the real number sh(*d(x,y)). But this number will be 0 whenever x 􀀌 y, so the first axiom for a metric is not satisfied. What we must do therefore is identify points that are infinitely close, by passing to the quotient set i X= (*Xl􀀁mj􀀌) = {hal(x) :xis limited} (note that if x E *Xlim, then hal(x) 􀃮 *Xlim, so *􀄟lim is partitioned by the halos of its points). A metric is then defined on X by d(hal(x), hal(y)) = sh(*d(x, y)) . .This is well-defined, since if hal(x) = hal(x') and hal(y) = hal(y'), then *d(x, y) 􀀌 *d(x', y'). The pair (X, d) is called the nonstandard hull of (X, d)n. Observe that if x, ynare distinct members of X, then hal(x) and hal(y) are distinct (indee􀄚, they are disjoint), so the mapping x ----+ hal(x) is an injection of X into X, allowing us to identify X with a subset of its nonstandard hull. Moreover, when x,y EX, d(x, y) = *d(x, y) = sh(*d(x, y)) = d(hal(x), hal(y)), so under this identification (X,d) becomes a subspace of (X, d), and t􀄙is is what justifies us continuing to use the symbol "d" for the metric on X. Theorem 18.3.1 The nonstandard hull (X, d) is complete. Proof. Let (hal(xn) : n E N) be a Cauchy sequence in X. The sequence (xn : n E N) of points in *Xlim extends to an internal hypersequence (xn : n E *N) in *X, by sequential comprehensiveness. We will show that (hal(xn) : n EN) converges to hal(xK) for some K E *N00• Now, for each n EN, by the Cauchy property there exists kn EN such that for all standard m 2: kn, and hence (i)  236 18. Completion by Enlargement But the set {mE *N: *d(xm,Xkn) < 1/(2n)} is internal, so by overflow we conclude that there is some unlimited Kn E *N such that (i) holds for all m E *N with kn :::; m :::; Kn . Invoking sequential comprehensiveness again, there is some unlimited K E *N that is smaller than every Kn (cf. Theorem 15.4.3). Then XK is limited (e.g., *d(xK,Xk1) duce the concept of a point x E *X being approachable from X, meaning that for each c E 􀂞+ there is some (standard) xc in X such that *d(x, xc) op(z) + op(w), min{ op ( z), Op(w)}.  18.4 p-adic Integers 239 Now put lzlp = 0 if z =I 0, if z = 0. The function I IP is a norm on Z (sometimes called the p-adic absolute value), and satisfies < lzlplwlp, max{lzlp, lwlp}· It gives rise to the p-adic metric the sense of this p-adic size of its terms. Thus an expression like n = p-, so the sequence p,p2,p3, ... converges to zero in with 0 􀂪 zi < p, can be seen as analogous to the decimal representation of certain real numbers in the form 1 ro + r1 (10 ) + r2 ( 2 + . with 0 􀂪 ri < 10. p-adic Integers A p-adic integer is a sequence a= (ano: n EN) such that for each n EN, (1) an E Zjpn, and (2) an+l = an(mod pn). This implies that (3) amo= an(mod pn) whenever m 􀀐 n (hence if am= 0, then ano= 0 for all n < m). The set Zp of all p-adic integers is a subset of the direct product ··· ·•· Zjp X Zjp2 X X Zjpn X and inherits operations of addition and multiplication from this direct product. Thus if a and b are p-adic integers, then a + b -(an EBn bn : n E N), ab (an ®n bn : n E N),  240 18. Completion by Enlargement where an ffin bn and an 0n bn are the sum and product modulo n. Zp proves to be an integral domain under these operations, and we now briefly review its basic structure. The divisibility relation I is defined in Zp just as it is in Z: for x, y E Zp, xly iff (::Jz E Zp) y = xz. Each integer z E Z (positive or negative) can be identified with the p-adic integer az = (z mod p, z mod p2 , .•., z mod pn, .n.. ) For instance, -1 corresponds to the p-adic integer n (p -1, p2 -1, ... , p(-1, ... ) The map z 􀁑 az is an injection of Z into Zp that preserves addition and multiplication and allows us to identify .Z with a subring of Zw A p-adic integer a has a multiplicative inverse in 'llp iff a1 # 0. When a1 # 0, then in general an "¥= O(mod p) and an has an inverse bn in Zjpn , i.e., anbn 1(mod pn). Then (bn : n E N) is the inverse of a in Zp. Invertible - elements of Zp are called p-adic units, and can also be characterised as those elements that divide 1 in Zp. Let a be a unit. Then any factor of a is also a unit. But p itself is not a unit, since it corresponds to the sequence ap = (0, p, p, ... ) , so p is not a factor of a, and therefore the only way to express a in the form pnb is to put n = 0 and b = a. On the other hand, if a is a nonunit, and nonzero, then taking the least n 2:: 0 such that an+l # 0, we havenn 2:: 1 and an(= 0, so an+m(-O(mod pn) for all mEN. Then proves to be a unit of Zp with a= pnb. This shows that the p-adic units are precisely those members of Zp that are not divisible by p. The representation of any nonzero a in the form pnb with b not divisible by p is unique, and this allows us to put op(a) = n. Then defining lalp = p-op (a) and IOIP = 0 gives a norm on Zp extending the p-adic norm on Z and inducing the associated extended metric dp on Zp. Note that a is a p-adic unit iff lalp = 1. Any p-adic integer a has in Zp, i.e., the difference a-an is equal to pnb for some b E Zp. This implies n op(a(-an) 2:: n, hence Ia(-anlp 􀂪 p-. Thus the sequence a1, a2, ag, . .. converges to a in the p-adic metric, showing that Z is dense in the metric  18.4 p-adic Integers 241 space (Zp, dp)· But in fact, using (1) and (2) it can be shown that a has the form (zo, Zo +ZIP, Zo +ZIP+ Z2P2, ... ) with 0 :S Zn < p, and so we can also write For instance, when p = 5, -1 (-1 mod 5, -1 mod 25, ... , -1 mod 5n, ... ) (4, 24, 124, 624, ... ) · 4 + 4 · 5 + 4 · 52 + 4 · 53 + · · · + 4 · 5n + · · . The Nonstandard Analysis Zp is complete under the p-adic metric dp, and is a completion of (Z, dp)· We are going to demonstrate this fact, not by appealing to any of the convergence results just claimed for Zp, but by showing: • (Zp, dp) is isometric to the nonstandard hull of (Z, dp)· • In (*Z, *dp), every limited point is approachable from Z. This implies that the nonstandard hull of (Z, dp) is equal to the completion of (Z, dp) based on approachable points, as given by Theorem 18.3.2. The symbols I IP and Op will continue to be used for the extension of these functions from Z to the commutative ring *Z, as provided by the nonstandard framework. In general, Op ( x) is a nonnegative hyperinteger, i.e., op takes values in the set *Z?: = *N U {0}. The basic properties and relationships of I IP and op are preserved by transfer. In particular, for all nonzero hyperintegers x, so I IP takes hyperreal values in the set {p-n : n E *Z?:(}. These values consist of positive infinitesimals (when n in unlimited) and real numbers :S 1 (when n is standard). These functions can then be used to define the sets of limited and infinitesimal elements in the p-adic sense as {x E *Z: lx(-zip is limited for some z E Z}, {x E *Z: lxiP 􀀌 0 in *R}. Now, lx-Olp = lxlp :S 1 holds for all standard integers x E Z, and hence for all hyperintegers x E *Z by transfer. This means that every member of *Z is p-adically limited, so *zlimv = *Z and the nonstandard hull here  242 18. Completion by Enlargement is just *Z/ 􀀐 with the metric induced by the shadows of *dp. To explain what the p-adic infinitesimals are, observe first that pN will be infinitesimal whenever N is unlimited because then by transfer IPNIP = p-N 􀀐 0. The idea that a p-adically "small" number is one that is divisible by a "large" power of p finds its ultimate expression in the following characterisation. Theorem 18.4.1 For any nonzero hyperinteger x E *Z, the following are equivalent. (1) x E *zinfp(. (2) op(x) is unlimited. (3) x is divisible by pn in *'ll for all n EN. (4) x is divisible by pN for some unlimited N E *N. Proof lxlp is the reciprocal of p0P(x), so is infinitesimal iff p0P(x) is unlimited, which holds iff op(x) is unlimited, asop is standard. Thus (1) and (2) are equivalent. Since the divisibility relation I is defined in 7l by xiy iff (3z E .Z) y = xz, it follows by transfer that xly for hyperintegers in *Z iffny = xz for some z E *Z. Now, the statement (ii) holds for all x E *Z and n E *Z􀁴, again by transfer. Hence if (2) holds, then for every n EN we havenn::; op(x), as op(x) is unlimited, and so pnlx by (ii). Thus (2) implies (3). Next, observe that for each x E *Z the set is internal, by the internal set definition principle, so if (3) holds, this set contains all members ofN, and hence by overflow it contains some unlimited N, establishing ( 4). Finally, if pNix with N unlimited, then (ii) gives N::; op(x), so op(x) is also unlimited. Thus (4) implies (2). This result shows that *zinfp = {pNq: N is unlimited and q E *Z}. The main properties of congruence relations lift to *Z by transfer. In particular, if m E N, then each x E *Z is congruent modulo m to a unique element r E Z/m, i.e., x-r is divisible by m(in *Z. We continue to denote this unique element r by x mod m. The map x ----+ x mod m, which is a ring  18.4 p-adic Integers 243 homomorphism from Z onto Z/m, thereby lifts to a ring homomorphism from *Z onto *(Z/m) = Zfm. This allows us to define a homomorphism by putting Bp(x) = (xnmodp, xmodp2, ••., xmodpn, ... )( . It is left as an instructive exercise to check that Bp(x) E Zp, i.e., x modpn+ln-x modpn(mod pn), and that Bp preserves addition and multiplication. Notice also that since we identify the standard integer z with the p-adic integer (z mod p, z mod p2, ••, z mod pn, ... ) , . it follows that Bp leaves all members of Z fixed. The kernel {x E *Z: Bp(x) = 0} of Bp consists of those x E *Z such that for all n E N we have x mod pn = 0, which means that pn divides x. By Theorem 18.4.1 this holds precisely when lxlv 􀄪 0. Thus the kernel is exactly the set *zinfp of p-adic infinitesimals, which is therefore an ideal of the ring *Z. Then the coset *zinfp + x = {pNq + x: N is unlimited and q E *Z} is the set of all hyperintegers that are infinitely close to x in the p-adic metric, because IY -xiP is infinitesimal if and only if y-x is of the form pNq with N unlimited. If we can show that Bp maps onto Zp, then by the homomorphism theorem we will have a ring isomorphism To prove that (}P is onto Zp requires us to invoke the concurrence version of enlargement (Theorem 14.2.1). If a= (an : n EN) E Zp, define a relation Ra 􀁕 N x Z by putting Ra has domain N and is concurrent: given integers n1, ... , nk E N, take any mEN with m 2:: n1, ... ,nk. Then since am(-an(mod pn) whenever nn m 2:: n (condition ( 3) of the definition of p-adic integer), it follows that n1Raam, ... , nkRaam. Hence as Ra is concurrent, there must be an x E *Z such that n(*Ra)x for all n E N. Transferring the definition of Ra then  244 18. Completion by Enlargement shows that for all such n, x = an(mod pn), and hence x mod pn = an. Thus ()P( x) = a, and the proof that ()P is onto is complete. Note that in proving the concurrence of Ra here we can choose m such that am > 0, so the proof shows that Ra is concurrent as a relation from N to N, and hence will produce a positive x (i.e., a member of *N) with Bp(x) = a. This fact will be used at the end of this chapter to derive a description of p-adic integers as certain hyperfinite formal sums. Another explanation of why such a positive x can be found is that the hyperintegers whose Bv-image is equal to a form a coset *zinfp + y, and any coset must contain positive elements. Indeed, for a given y, pN + y (which belongs to *zinfp + y) will be positive for large enough unlimited N. Preserving the Metric We have observed that the coset *zinfp + x is just the 􀄞-equivalence class halp(x) = {y E *Z: lx-YIP􀄞 0} of x in *Z. Hence (*Z/ 􀄞) = *Zj*zinfp ""nZp. This bijection between the nonstandard hull *Z/ 􀄞 and Zp is given by halp(x) Bp(x). If we can show that it preserves the metrics, we will H- have our desired demonstration that the nonstandard hull is isometric to (Zp, dp)· But the metric on *Z/ 􀄞 is induced by the norm function so we want shlxlp = !Bv(x)!p, or equivalently, !x!P 􀄞 IBp(x)nlp· There are two cases: (2) Bp(x) -=/= 0. Then by definition of Bp(x) there must be some standard n 2: 0 such that x mod pn+l -=/= 0, and op(Bp(x)) is the least such n by definition of the p-adic norm on Zp. Now, n Pnlx iff n 􀀢 op(x) for all n 2: 0 (cf. (ii) in the proof of Theorem 18.4.1), and op(x) is a standard integer because JxiP 'f. 0, so op(x) is the least standard n for which pn+l f x, i.e., the least standard n for which x mod pn+l -=/= 0. J Thus in this case op•(x) = op(Bp(x)) and lxiP = Op(x)lp· Having now shown that the nonstandard hull of (Z, dp) is isometric to (Zp, dp), it remains to show that this hull is a completion of (Z, dp), by  · 18.5 p-adic Numbers 245 showing that any point x E *Z is approachable from Z. But for each n EN, pn divides x-(x mod pn), so Then for any c E 􀂞+, by choosing a standard n large enough that p-n < c we get the standard integer x mod pn that is within c of x in the p-adic metric. This shows .that x is approachable from Z. Thus N earstandardness To round out this discussion, consider the points in *Z that are near to Z. If ixn-zip 􀀐 0, then pN divides x-z for some unlimited N E *N. So the nearstandard points of *Z are precisely those of the form x = pNy + z with N unlimited and z standard. Since (Z, dp) is incomplete, there must be points in *Z that are approachable from Z but not near to Z (Corollary 18.3.3). Indeed, if ixn-zip 􀀌 0 with z standard, then Op(xn-z) = 0, and so Op(x) = Op(z) = z. This shows that the nearstandard points in *Z are just the Op-preimages of members of Z. Any x E *Z with Op(x) E (Zp-Z) fails to be near to Z. 18.5 p-adic Numbers The ring Zp has a field of fractions Qlp = {􀁭 : a, b E Zp and b i-0}. Members of Qlp are called p-adic numbers, and equality between them is given by -a c iff adx= be in Zp. b d The field operations are given by the familiar formulae from rational arithmetic: a c ad+bc -+ b - d bd a c ac ' b d bdx - (􀁭)-1 ·  246 18. Completion by Enlargement Thus Qp stands in the same relation to Zp that Q stands to Z. Moreover, since Z 􀃚 Zp, it follows that Q 􀃚 Qp. The p-adic order function extends to Qp by putting Op(ajb) = Op(a)x-Op(b). This is well-defined, because if ajb = cjd, then op(a) -op(b) = op(c) op( d) by the "logarithmic" law op(ad) = op(a) + op(d) etc. Then we put lxlp = p-ov(x) and dp(x, y) = lxx-Yip as before, but now for x, y E Qp. In particular, this gives a p-adic order function and norm on Q. To analyse this further, recall that nonzero p-adic integers a, b can be written uniquely in the form a = pnc and b = pmd with n, m 􀁂 0, c, d units in Zp, and p not a factor of c or d. Then a pnc n-mc = p b pmd =􀀁"d: Here op(ajb) = n-mE Z, and cjd is a unit in Zp· In general, then, any nonzero p-adic number has a representation in the form pmb with m a standard integer and b a unit in Zp, and this representation is unique (the case m 􀁂 0 giving the p-adic integers). Moreover, in view of the representation in Section 18.4 of p-adic units as power series with nonzero initial term, each p-adic number x =J. 0 is uniquely expressible in the form Pm(ZO + Z1P + Z2P2 + · + ZnPn + ' · ·· ' ) X where m is the integer op(x), 0 ::; Zn < p, and z0 􀁂 1. Since m can be negative here, it follows that a p-adic number can be written in the general form Z-kP-k + · · · + Z-lP-l + Zo +ZIP + Z2P2 + · · · + ZnPn + · · ·, with 0::; Zi < p, showing that it is the sum of a standard rational number z-kP-k + + Z-IP-1 and a p-adic integer. Note the analogy with the fact · · · that any real number can be represented as an infinite decimal expression r_k + .. ·+r-1 C􀀯)-1 +ro+r1 (I1o) +r2 .. ·+rn + · ... Under the metric dp, Qp is a completion of Q. Limited p-adics In a nonstandard framework, the functions Op and liP extend from Q to *Q by the transfer map and continue to satisfy the usual properties, including p-op(x), lxlp IYIP  18.5 p-adic Numbers 247 for x, y E *Q. The sets of hyperrationals that are limited or infinitesimal in the p-adic sense are given by { x E *Q : lx -qlp is limited for some q E Q} {x E *Qn: lxlp is limited} and {x E *Qn: lxlp 􀁏 0 in *IR}. The p-adic order op(x) of a nonzero hyperrational xis itself a hyperinteger, so falls under one of three cases: (1) op(x) is limited, and hence is a standard integer. Then lxlp is a nonnegative real number, equal to 0 when x = 0, and otherwise of the form pm with m E Z. ) (2) op(x) is positive unlimited (i.e., in *N00). Then pop(xxis positive unlimited, and lxlp is a positive infinitesimal: lxlp 􀀌 0. (3) op(x) is negative unlimited. Then -op(x) E *N00, and so lxlp is positive unlimited. This shows that x can fail to be p-adically limited only when case (3) occurs, so {x E *Q: op(x) is not negative unlimited} {x E *Qn: op(x) E Z U *N00}. This characterisation gives rise to a more useful one: p-adic limitedness of a hyperrational depends on the size of the denominator, as the next result indicates. Theorem 18.5.1 Let y, z E *Z. If !zip is not infinitesimal, then yjz is p-adically limited. Proof. op(z) is a nonnegative hyperinteger, so if lzlp = p-op(z) '/:. 0, then op(z) must be limited, i.e., op(z) EN U {0}. But then since op(y) 􀀛 0, Op(yjz) = Op(Y)-Op(z) cannot be negative unlimited. Hence as above, y/z E *Qlimp. The converse of this can fail. If lzlp 􀁏 0, then jyfzip will still be limited if Yip 􀀢 nzp for some n E N (in which case YIP is also infinitesimal). For !xiixN IxN instance, this happens when y = 2pxand z = pxwith N unlimited. Now, we can express any hyperrational as a ratio of hyperintegers Nthat have no factors on common. It is the presence of unlimited powers pxof p as factors that makes a hyperinteger p-adically infinitesimal, and it turns out that the absence of common factors of this particular kind is enough to give the converse to Theorem 18.5.1.  248 18. Completion by Enlargement Theorem 18.5.2 Let y, z be hyperintegers that have no common factors i of the form PN with N E *.Noo . If IY/zlv is limited, then Izxp is not in finitesimal. Proof Suppose that IY/zip is limited, but !ziP 􀀂 0. Then op(z) is positive unlimited (Theorem 18.4.1), while op(yjz) = ov(Y)x-op(z) is not negative unlimited. But this can be so only if op(y) is also positive unlimited. Then if N is the smaller of op(Y) and op(z), we have N E *.N00 and pN a factor of both y and z. However, this contradicts the hypothesis. 0 The Completion by exhibiting an isomorphism (iii) We are going to show that Qp is a completion of Q under the p-adic metric and demonstrating that all elements of *Qlimp are approachable from Q. For the isomorphism we need a homomorphism from *Qlimp onto QP. We already have a homomorphism ()P : *Z-+ Zp, and the relationships between *Q and *Z and QP and Zp suggest that we extend ()P to hyperrationals by putting (iv) Of course for this to be defined we need Bp(y) -=/= 0, but that is exactly where limitedness comes in. We apply the definition (iv) only when xjy is in reduced form, i.e., x andny have no proper factors in common. In particular, they have no common factors pN with N unlimited, so by Theorem 18.5.2 if xjy E *Qlimp , then !Yip '/:. 0, and so Bp(y) -=/= 0. Thus (iv) is well-defined for all members of *Qlimp . The fact that Bp : *Z -+ Zp preserves addition and multiplication and maps *Z onto Zp can be used to show: • 0-:J is a ring homomorphism from *Qlimp onto Qp that extends Bp. Thus to obtain the isomorphism (iii) we have only to show that *Qinfp is ixii/i the kernel of B:J. But for xjy E *Qlimp in reduced form, lx/yP = xpxYipwith !Yip '/:. 0, and hence !Yip is a standard real number. Therefore f !xxyiP 􀀌 0 iff !x!p 􀀂 0 iff Bp(x) = 0 iff o:(xjy) = 0, so indeed xjy belongs to *Qinfp iff it is in the kernel of 0-:J. Preserving the Metric In order to show that the isomorphism (iii) preserves the metric of the space *Qlimp /*Qinfv , and hence show that Qp is isomorphic to the nonstandard  18.6 Power Series 249 hull of (Q, dp), we need to show that shivip = iOt(v)ip, or equivalently, iviv 􀁐 iOt(v)iv, for all v E *Qiimp . As with the integer case in Section 18.4, there are two parts to this: (1) e:(v) = 0. Then v E *Qinfp and ivip 􀁏 0 = iOip = iOt(v)ip· (2) o:(v) =I= 0. Then in fact, iOt(v)ip = ivip, because if v = xjy in reduced form with x, y E *Z, then Op(x), Op(Y) =/= 0, so ixip = iOv(x)iv and iYiv = iOp(Y)ip by the integer case, and therefore ix/yiP = ixip/iYiv = i0p(x)iv/i0p(Y)ip = iOp(x)/Op(Y)ip = iOt(xjy)iv· It remains now to prove that each v E *Qlimp is approachable from Q. Again there are two parts: (1) If Ot(v) = 0, then iv-Oip 􀀐 0, so vis actually near to Q. (2) If o:(v) =I= 0, then as just shown, ivip = iOt(v)iv-Since o:(v) is a p-adic number, it is equal to pmb for some m E Z and some bE Zp with p f b, and so ibiv = 1. Now choose an x E *Z with Op(x) = b. Then o: leaves pm fixed because it is a standard integer, and o:(vx-pmx) = o:(v)x-o:(pmx) = e:(v)x-(pmb) = 0, so ivx-pmxip 􀁐 0. But given any E E JR.+ , since x E *Z is approachable from Z, there must be a standard z E Z such that cpm ix-zip < T· Then pm z is a standard rational number that is p-adically within c of v, since iv -pmzip < iv -pmxiv + iPmX -Pmzip ivx-pmxip + iPmipixx-zip < ivx-pmxiv + p-m(cpm)/2 < c because iv -pmxip is infinitesimal. 18.6 Power Series Polynomials Let (R, +, -, , 0, 1) be a commutative ring. A polynomial in x of degree n · over R is a "finite formal sum" · ao + a1•x + a2x2 + · · + anxn,  250 18. Completion by Enlargement where ao, .x.x. , an are elements of R, called the coefficients of the polynomial. Coefficient ai is of degree i. The leading coefficient is an, which is required to be nonzero if n =J. 0. The set of all polynomials in x over R of all possible degrees n E z􀁴 is denoted by R[x]. When n = 0, a single element a0 of R is regarded as a polynomial, and has degree 0 (unless ao = 0: the zero polynomial 0 will not be assigned a degree). Thus we havenRn􀃚 R[x]. Members of Rnare constant polynomials. Two polynomials are equal if they have the same degree and corresponding coefficents (i.e., those of the same degree) are identical. Thus a polynomial is uniquely determined by its list of coefficents, and this suggests that a more explicit way to define a polynomial is to view it as a sequence a = (a0, •.., an, . . . ) of elements of R, or equivalently, a function a : z􀃃 --+ R, that is ultimately zero in the sense that (3n E Z􀁴) (Vm EN) (m > n--+ am = 0). The least such n is the degree of a. The inclusion of R in R[x] arises by identifying each r E R with the sequence (r, 0, 0, ... , 0, .. . ) . The set R[x] of polynomials over R forms a commutative ring under the operations a + b -( ao + bo, ... , an + bn, ... ) , -a (-a0, , -an, ... ) , •.. ab -(aobo, aob1 + a1bo, ... , aobn + a1bn-1 + · · · + anbo, ... ) . Power Series A power series over R is an "infinite formal sum" with coefficients from R. Thus we may simply say that a power series is any sequence a = (ao, ... , an, ... ) of elements of R, or equivalently, any function a : z􀁴 --+ R. The set of all power series over R will be denoted by R[x]n. It forms a ring under the operations defined as for R[x] and has R[x] as a subring. Altogether now we have R 􀁽 R[x] 􀁽 R[x]. If a power series a is nonzero, then it must have a nonzero coefficient. The least n such that an =J. 0 is called the order of a, denoted by o(a). Put { 2-o(a) !xif a =J. 0, !ax= 0 if a= 0. Then d(a, b)x= la-b! defines a metric on R[x]. Note that Ia! 􀂪 1 in general.  18.6 Power Series 251 A power series a as above determines the sequence of partial sums, which are polynomials. The order of ax-(ao + a1x + · · · + anxn) is at least n + 1, and so It follows that the sequence of partial sums converges to a in the metric just defined. This implies that R[x] is dense in R[x]. In fact, R[x] is a complete metric space, hence a completion of R[x], as we will now show by invoking the nonstandard hull construction again. Enlargement Let *R[x] abbreviate the enlargement *(R[x]) of R[x] in a nonstandard framework for R. Since members of R[x] are functions from z2: to R, the members of *R[x] are internal functions from *Z2': to *R (Exercise 13.13(3)), or alternatively internal hypersequences a= (an : n E *Z2':). Since polynomials are ultimately zero, so too are members of * R[x]. This is because (:In E Z2) ('i/m E(N) (m > n -tax= 0) mx is true for all a E R[x], so (:In E *Z2':) (Vm E *N) (m > n -tax= 0) m is true for all a E *R[x]. But now the largest n for which an =I= 0 may be unlimited, so in general a member of *R[x] may be thought of as a hyperfinite formal sum with its degree N E *N possibly being unlimited. The coefficients an can be nonstandard here, even when n is standard. Thus a member of * R[x] is an internal hyperpolynomial with coefficients from * R (note that * R is a commutative ring, by transfer of the fact that R is). *R[x] is not the same thing as (*R)[x]. The latter is the ring of (finite) polynomials a0+a1x+a2x2+· · ·+anxn with coefficients from *R. Of course we can view a polynomial as a special case of a hyperpolynomial, and so identify each member of (*R)[x] with a member of *R[x]. To be precise this requires a use of transfer: for a fixed n E z;:::, the statement ('ifao, ... , an E R) (3b E R[x] ) [ bo = ao I\··· I\ bn =an I\ ('i/m EnN) (m > n -t bx= 0)x] m  252 18. Completion by Enlargement asserts (correctly) that for any list a0, •.., an of elements of R there is a polynomial b in R[x] having this list as its coefficients. By transfer then, for any list a0, ... , an of elements of *R (possibly including nonstandard elements) there is a hyperpolynomial bin *R[x] with a0, ... , an as its coefficients. In particular, * R[x] includes all members ofn*R as constant hyperpolynomials. Also, if a ER[x], then a is regarded as being in *R(x] by identifying it with its extension to *Z􀃭 having an = 0for all unlimited n. The functions o(a), lal, a+ b, ab all extend from R[x] to *R[x], preserving many of their properties by transfer. Theorem 18.6.1 a E*R[x] is approachable from R[x] if and only if the coefficient an belongs to R for all standard n. Proof. Fix a standard n EZ< Then if two polynomials a, b ER[x] are closer than 2-n to each other (i.e., Ia-bl < 2-n), the order of a-b must be at least n + 1, so (a-b)n = 0and hence anx= bn. Thus the statement holds for all a, bER[x], and so by transfer holds for all a, bE*R[x]. Now suppose that a is in *R[x]ap, the set of all members of * R[x] approachable from R[x]. Then for each standard n there must be some polynomial bER[x] with Ia-bl < 2-n. From the previous paragraph it then follows that an = bnER. Thus the coefficient an is in R for each standard n. Conversely, suppose a E*R[x] has an ER for all standard n. For each such n, the polynomial belongs to R[x]. But a r n is within 2-n of a, because the statement !a-a r nl< 2-n holds for all a E R[x] (see above) so holds for all a E*R[x] by transfer. This shows that a is approachable from R[x]. D At the end of Section 18.3 we promised to provide an example of a metric space having limited entities that are not approachable. The theorem just proved furnishes many examples. All members of *R[x] are limited, and indeed satisfy IaI :S 1 by transfer. But if R is infinite then * R[x] will have members that have some coefficents of standard degree that are nonstandard, i.e., belong to * R -R. Such hyperpolynomials are not approachable from R[x], as Theorem 18.6.1 shows.  18.6 Power Series 253 Infinitesimals The infinitesimal members of * R[x] can be characterised as those internal hyperpolynomials whose coefficients of standard degree all vanish: Theorem 18.6.2 For any nonzero a E * R[x], the following are equivalent. (1) lal 􀀌 0. (2) o(a) is unlimited. (3) There is an unlimited N E *N such that ano= 0 for all n < N. (4) an = 0 for all standard n. Proof In general, lal = 2-o(a) and o(a) is a nonnegative hyperinteger, so IaI will be appreciable iff o(a) is limited, or equivalently, IaI will be infinitesimal iff o(a) is unlimited. Thus (1) and (2) are equivalent. Now, by transfer we have that for any nonzero a E *R[x], (Vm E *Z􀀤) [m < o(a) +--+ (Vn E *Z􀀤) (no:::; m-an = 0)]x. From this, (2) implies (3) by putting N = o(a). It is immediate that (3) implies (4). Finally, if (4) holds, then the above transferred sentence ensures that each standard m is smaller than o(a), so (2) follows. 0 Corollary 18.6.3 In *R[x], two hyperpolynomials are infinitely close precisely when their coefficients of standard degree are identical: a 􀀌 b if and only if an = bn for all standard n. 0 The Completion Letn(}: *R[x]apx-R[x] be the restriction map a= (an : n E *Z􀄕) ------+ (an : n E Z􀀤), i.e., O(a) is the standard power series defined by putting O(a)n =an for all standard n. By Theorem 18.6.1, O(a) is indeed a member of R[x] whenever a E *R[x]aP. The map (} is a ring homomorphism. To see that it preserves addition, notice that (a + b )n = an + bn holds for all n E *Z􀀤 and all a, bE *R[x], by transfer, and this is more than enough to guarantee ()(a+ b) =()(a)o+ ()(b). For multiplication, observe that for any fixed standard n,  254 18. Completion by Enlargement for all a, b E R[x], and hence for all a, bE *R[x]. But this equation asserts that (O(ab))n = (O(a)O(b))n· As this holds for all standard n, we conclude that O(ab) = O(a)O(b). Next we want to establish that 0 maps onto R[x]. Given a power series a E R[x], then a is a function from z􀀤 to R, and so it transforms to a function *a : *Z􀀤 --+ * R that has *an = an E R for all standard n. In spite of Theorem 18.6.1 we cannot conclude from this that O(*a) = a, because we do not know whether *a is in the domain * R[x]ap ofn(} at all. Indeed, *a will not even be in *R[x] unless it is ultimately zero, and if all the coefficients of a are nonzero, then we will have *am # 0 for all m E *Z􀀤 by transfer. To overcome this, consider the statement • for any function c E Rz􀀤 and any n E z􀀤 there is a polynomial bE R[x] that agrees with c up toon, i.e., Cm =_bm for all m 􀂪 n. Since this is manifestly true, so is its *-transform. But *a is a standard, hence internal, function from *Z􀀤 to *R, so belongs to *(RZ􀀤). Therefore if we take an unlimited N E *N, by this *-transform we deduce that there is some bE *R[x] that agrees with *a upntonN. Hence b agrees with *a on all standard n, so that bn = an E R for all such n, implying both that bE *R[x]ap (Theorem 18.6.1) and O(b) =a. Thusn(} maps onto R[x]. From the definition of (} we have that O(a) = 0 iff ano= 0 for all n E Z􀄖. Theorem 18.6.2 then gives O(a) = 0 iff lal 􀀵 0, so the kernel ofn(} is the set *R[xpnr of infinitesimal elements of *R[x]. As with previous cases, the cosets of the kernel are the equivalence classes under the infinite closeness relation 􀀵, and we conclude that R[x] is isomorphic to the corresponding quotient: (*R[x]apI􀁿)x"' R[x]. It remains only to show that this isomorphism preserves metrics, in the sense that shlal = IO(a)i, to conclude that the space R[x] of power series over R is isometric to the completion (*R[x]ap I􀁿, d) of (R[x], d). This is left as an exercise: Exercise 18.6.4 If a is a nonzero member of * R[x]ap, show: (1) If o(a) is limited, then lal = IO(a)l. (2) If o(a) is unlimited, then ial 􀁿 IB(a)l.  18.7 Hyperfinite Expansions in Base p 255 18.7 Hyperfinite Expansions in Base p Any p-adic integer can be represented as an infinite sum :x+ ZnPn + · · · · · 2:x=0 ZnPn = Zo + ZIP + Z2P2 + · with coefficients Zn from Zjp. In view of our discussion of power series in the last section, this suggests that we could view it instead as a hyperfinite sum. To see how this works, recall that each standard positive integer z EN has a unique expansion in base p of the form = · · · Z ZQ + Z1P + Z2P2 + + ZnPn and so is represented in this base by the sequence (zi : 0 :::; i :::; n) of numbers that are between 0 and p-1. The representation gives a bijection between N and the set Seq(p) of all finite sequences of elements of Zjp. This bijection is provided by the operator 2: : Seq(p) 􀀆 N taking (zi : 0 :::; i :::; n) to the number E7=o ZiPi. In a nonstandard frame­work 2: will lift to a bijection 2:: *Seq(p) 􀀆 *N. By appropriate transfer arguments we can see that *Seq(p) is the set of all internal hyperfinite sequences of elements of *(Z/p) = Zjp. A typical member of *Seq(p) is an internal function of the form (zi : i E *Z􀆮 and i :::; n), with 0 :::; Zi < p and n possibly unlimited. The operator 2: takes this hypersequence to an element of *N that we denote by E7=o ZiPi. Every member of *N is represented in this way as a hyperfinite sum determined by a unique member of *Seq(p), and so has an expansion in base p. Now, within z􀜐n' if n < m, then the difference (E::o ZiPi) -(E􀄢o ZiPi) is divisible by pn+l, so By transfer, (v) holds for all n, m E *Z􀃭 with n < m when these sums (v) are defined. This property can be used to analyse the relation of infinite closeness of hyperintegers in terms of the behaviour of the coefficients of their base p expansions. Consider two hyperintegers that have base p expansions  256 18. Completion by Enlargement with N, M unlimited. If z and w are infinitely close in the p-adic metric, i.e., jz-wlv 􀀵 0, then for any standard n :2: 0, pn+I divides z-w (Theorem 18.4.1), so But n < N, M, so applying result (v) gives i z·pxz·pi(mod pn+l) and likewise Consequently, But then because both sums belong to Z/pn+I, and so the uniqueness of the base p expansion of standard integers implies that Zi = Wi for i 􀂪 n, and in particular, Zn = Wn. This argument can be worked in reverse, to establish the following analogue of Corollary 18.6.3. Theorem 18. 7.1 Two positive hyperintegers are p-adically infinitely close precisely when their base p expansions have identical coefficients of standard degree: D Now, we saw in Section 18.4 that if a = zo + ZtP + Z2P2 + .x.. + ZnPn + ..x. is a p-adic integer, then there exists a positive hyperinteger x with 2 a= Ov(x) = (x mod p, x mod px, ... , x mod pn, .x.n. ) . Hence X mod pn+l = zo + ZIP + Z2P2 + .n.. + ZnPn for all n E z􀀤. But x has a base-p expansion X= Xo +XtP + + XNPN · · · for some N E *Z􀀤, and for each standard n 􀀛 0 we get by result (v) that x -= x􀉏 ·pi(mod pn+l)' so XiP X mo d pn+l = ZiP ,  18.8 Exercises 257 and therefore Xi = Zi for all i :::; n. Thus the p-adic integer a and the hyperinteger x have the same coefficients of standard degree in these basep expansions. Of course for any given a E Zp there will be more than one x E *N representing a in this way, i.e., having Op(x) =a, but all such x's will be infinitely close in the p-adic metric. Altogether, this discussion shows that we can view any p-adic integer as a hyperfinite base-p expansion N zo + Z1P + · · · + ZNP with 0 ::; Zi < p, provided that we identify any two such expansions that differ only at coefficients of unlimited degree. 18.8 Exercises (1) Write out in full the transfer arguments showing that members of *Seq(p) are internal hyperfinite sequences of members of '11../p (cf. the proof of Theorem 13.17.1 for guidance). (2) Complete the proof of Theorem 18.7.1 by showing that if two positive hyperintegers have identical coefficients of standard degree in their hyperfinite base p expansions, then they are p-adically infinitely close.