The ordered field R e as a complete metric space

150 P .J. Hammond Mathematical Social Sciences 38 1999 139 –156 `

6. The ordered field R e as a complete metric space

6.1. The ordered field Re In Hammond 1994, 1999c, the search for a minimal suitable range of probabilities suggested the particular ordered field Re, originally described by Robinson 1973. This is the smallest algebraic field that contains the real line R as well as a basic positive infinitesimal element e. One can regard e as a particular element of the ordered field of hyperreals used in non-standard analysis. Or perhaps more intuitively, as any vanishing ` sequence ke l of positive real numbers. n n 51 The typical non-zero element of Re is a ratio of two ‘‘polynomial expressions’’ – i.e., a ‘‘rational expression’’ whose normalized form is n k i a e 1 O a e k i 5k 11 i ]]]]]] f e 5 6 m j 1 1 O b e j 51 j for some unique integer k and non-zero leading coefficient a . Define the ordering . on k Re so that fe . 0 iff a . 0. This makes . a lexicographic linear ordering on Re, k in effect. It also makes Re a non-Archimedean ordered field, with fe infinitesimal iff k . 0. 6.2. A metric Section 7 will be concerned with finding weaker sufficient conditions for the algebraic continuity axiom AC of Section 4.2 to hold when probabilities are allowed to range over Re. So Re needs a topology such as that induced by Lightstone and Robinson’s 1975 metric. Given the normalized form 6 of f e, define k as the infinitesimal order v[ fe] of f e, with v[0]: 5 ` for the zero element of Re. If fe ± 0, note that: 1. if k . 0, then f e is an infinitesimal of order k; 2. if k , 0, then f e is really an infinite number of order 2 k, which can be regarded as an infinitesimal of negative order; 3. if k 5 0, then f e is infinitesimally different from a non-zero real number, in which case it is said to be of infinitesimal order 0. Note too that, for all non-zero pairs f e, ge [ Re, one has v [ f e ? ge] 5 v[ fe] 1 v[ge] and v [ f e 1 ge] min hv[ fe], v[ge]j It follows that the infinitesimal order is an instance of what Robinson 1973 describes as a ‘‘non-Archimedean valuation’’. Now define the function d: Re 3 Re → R so that 2v [ f e 2ge ] d f e, ge: 5 2 P .J. Hammond Mathematical Social Sciences 38 1999 139 –156 151 2` for every f e, ge [ Re, with the obvious convention that 2 : 5 0. Obviously, d f e, ge 5 d ge, fe and d fe, ge 5 0 if and only if v[ fe 2 ge] 5 `, which is true iff f e 5 ge. Finally, it is easy to verify that d is a metric because the triangle inequality is satisfied. 6.3. Convergence n n Consider any infinite sequence f e n 5 1, 2, . . . in Re. Evidently f e → f e as n n n n → ` iff d f e, fe → 0, so iff v[ f e 2 fe] → `. But after defining g e: 5 n n n f e 2 fe for n 5 1, 2, . . . , this holds iff g e → 0 because v[ g e] → `. Ignoring n any zero terms of the sequence g e n 5 1, 2, . . . , this requires that the infinitesimal n order k of n , n n k n i a e 1 O a e n n k i 5k 11 i n ]]]]]]] g e: 5 n m n j 1 1 O b e j 51 j should tend to 1 `. But then, for each number M 5 1, 2, . . . , there must exist nM such n n n that n . nM implies k . M because a 5 0 for all i M. That is, all coefficients a of i i the numerator must become 0 for n sufficiently large. This is necessary and sufficient for convergence. Note one important implication: An infinite sequence of real numbers converges iff it is eventually equal to a real constant. Thus, the very fine topology we have defined on Re induces the discrete topology on the subspace R, meaning that every subset of R is open in the subspace topology. 6.4. Completing the metric space n Any sequence f e n 5 1, 2, . . . of elements in Re is a Cauchy sequence iff for m n every real d . 0 there exists Md such that d f e, f e , d whenever m, n . Md . Just as ordinary continuity requires completing the space of rationals by going to the real line R in which Cauchy sequences converge, here I will consider a similar completion ` R e of Re. In Re many Cauchy sequences do not converge. Indeed, consider any sequence n n k ` f e 5 o a e n 5 1, 2, . . . with a non-recurring infinite sequence ka l of real k 50 k k k 51 2 n k coefficients – for example, o e n 5 1, 2, . . . . An analogy is the non-recurring k 50 ] Œ decimal expansion of any irrational real number such as 2, which has no limit among n the set of rationals. The obvious limit of the sequence f e should be the power series ] ` k Œ o a e , just as the limit of the decimal expansion 1.41421356 . . . is 2. Yet a k 50 k ` k non-recurring power series o a e does not correspond to a rational expression in k 50 k Re, because multiplying by any polynomial expression always leaves an infinite power series, never a polynomial. The obvious way to complete Re is to allow such ‘‘irrational’’ infinite power series. To retain the algebraic structure of an ordered field, the ratios of such power series must also be accommodated. However, the reciprocal of any power series is itself a power 152 P .J. Hammond Mathematical Social Sciences 38 1999 139 –156 ` k series, but of the form o a e where the leading power k could be negative. k 5k k ` 4 Accordingly, define R e as the set of all such power series. Of course, any rational ` ` expression in Re can be expanded as a power series in R e, so R e does extend Re. ` ` k Each member of R e can be conveniently written as o a e , where there must k 52` k exist k such that a 5 0 for all k , k . Then the metric d on Re can obviously be k ` n extended to R e. In fact, the infinite sequence f e n 5 1, 2, . . . of power series ` n k ` ` k ` o a e in R e will converge to the limit fe 5 o a e in R e iff for every k 52` k k 52` k n k there exists n such that a 5 a for all n n . k k k k ` n k ` Suppose o a e n 5 1, 2, . . . is any Cauchy sequence in R e. Then: i there k 52` k n exist k , n [ N such that a 5 0 whenever k , k and n . n ; ii for every k k there k n exist both n [ N and a [ R such that a 5 a for all n . n . Hence, the Cauchy k k k k k ` k ` sequence converges to f e: 5 o a e , so R e is a complete metric space. k 5k k ` n k ` ` n k ` Two Cauchy sequences ko a e l and ko b e l are said to be ‘‘limit k 52` k n 51 k 52` k n 51 ` ` n n k ` equivalent’’ in R e whenever o a 2 b e converges to 0 in R e as n → `. k 52` k k Because the quotient space of limit equivalence classes of Cauchy sequences in Re is ` ` easily seen to be an ordered field which is isomorphic to R e, it follows that R e is effectively the smallest complete metric space containing Re, just as R is the smallest complete metric space containing the ordered field of rationals.

7. Lexicographic expected utility