real numbers as black boxes - university of auckland

NEW ZEALAND JOURNAL OF MATHEMATICS Volume 31 (2002), 189-202

REAL NUMBERS AS BLACK BOXES

P e t e r M . S c h u s t e r

(Received January 2002)

Abstract. We base a choice-free constructive model o f the continuum upon ‘black box ’ as primitive notion, putting the idea into action that any real number returns some rational number whenever it is asked for an approximate description of itself.

. . . wahrend der Stunde, ‘da der grofie Pan schlaft’, wie die Alten sagten.H e im it o v o n D o d e r e r 1

1. Introduction

This article is designed to contribute to a time-honoured problem that Feferman recently renewed as follows:

But as long as science takes the real number system for granted, its philosophers must eventually engage the basic foundational question of mathematics: ‘What are the real numbers, really?’ [5]

In a mathematical thought experiment, we try to capture the essence of the continuum with as little burden as possible. Disregarding any particular instantiation such as Cauchy sequences or Dedekind cuts, we take literally that a real number is something determined to arbitrary accuracy by rational numbers. Surprisingly, those ‘somethings’ prove to basically need only a single specification: different rational approximations of the same real number have to be sufficiently close to each other.

All this corresponds perfectly with the way in which one deals with real numbers when carrying out everyday calculations. Although the decimal expansion of 7r can be extended as far as we please, one has to be content with— and can get by on— finite decimal fractions like 3.1415927. Given this particular approximation, no one would accept 2.99999 as another good one: they differ in a manner disproportionate to the corresponding precision indicated by the respective number of digits.

The observation which gave the impetus to reflect upon this topic is that an arbitrary sequence (rn) of rational numbers is a Cauchy sequence if and only if for each rational number e > 0 there is some q 6 Q such that \rn — q\ < e whenever n is sufficiently large. As a subset of the generalised Cauchy reals going back to Schmieden and Laugwitz (see [18], also for further references), the true Cauchy reals are therefore distinguished by being approximable arbitrarily closely by rationals. A completely analogous observation can be made in the case of Dedekind reals, as

2000 AMS Mathematics Subject Classification: Primary 03F60; Secondary 00A30, 03A05, 26E40. Key words and phrases: Continuum, Constructive Analysis, Countable Choice, Set Comprehension, Dialogue Semantics, Choice Sequences.1 “ . . . during the hour ‘ in which the great Pan sleeps’ , as the ancients used to say.” Quoted from Die Strudlhofstiege oder Melzer und die Tiefe der Jahre.

190 PE TE R M. SCHUSTER

they are embedded into the generalised Dedekind reals investigated by Richman[is].

Why do we want to abstract from either of these time-honoured continuum models? As far as any sequential approach is concerned, the main reason is that we agree with Richman’s arguing for renouncing countable choice begun in [16]; see [17] and [19] for Richman’s thoughts and our adaptation, respectively. Therefore we have to refrain from ‘constructing’ sequences by an infinite number of possibly nonunique choices, a method employed rather frequently already within elementary analysis— at least as long as one sticks to Cauchy reals.

Working with Dedekind cuts, on the other hand, requires admittedly much less countable choice and turns often out to be rather straightforward, in particular when it comes to ordering the reals. Constructing Dedekind reals as pairs of infinite sets of rationals involves, however, a considerable amount of set comprehension that not only might cause foundational reservations but also appears to be quite far from any computational practice.

Guided by the firm belief that one cannot distinguish points on a geometrical line, or time instants, unless they are sufficiently distant from each other, we work within informal constructive mathematics in the spirit of Errett Bishop [1, 2, 4, 8]. For people still put off by constructive mathematics let us sound the all-clear, at least concerning Bishop’s variety: in its modern interpretation, as advocated by Bridges and Richman, it is simply mathematics dealing with basically the same objects as usual but by exclusive means of intuitionistic logic (see, for example, [14]). In particular, we may characterise the entities which we suppose to be given in a rather liberal way, as long as we specify all the properties we make use of later on; this applies, in particular, to higher-order entities like sets and sequences.

Consequently, we let ourselves be guided by the set of axioms for real numbers in constructive mathematics that was proposed by Bridges [3] as a foundation of any kind of computable analysis.2 We hope, however, to get beyond any purely axiomatic treatment, namely as close as possible to what Bridges loc.cit. called the ‘heuristical justification’ o f the axioms.

By the peculiar choice of intuitionistic logic, our approach is also open for future use as a mathematical theory, perhaps even as a point of reference, for how real numbers can be implemented on real-world computers. In fact, any good implementation of a particular real number x ought to be a program which converts any given rational e > 0 into some q € Q that is intended to be within e of x.

2. Black Boxes

In [16], Richman preferred to understand any V3 statement appearing in the hypotheses of countable choice as a ‘black box’— an interpretation which fits perfectly to what we said above about the essence of a real number. Accordingly, by a real number x we mean in the following a black box that converts each positive rational number e into a rational number q\ the main request from any such x is the real number rule which says that \q — q'\ < e + s’ for the respective outputs q and q' whenever x has been fed at different times with inputs e and e '.

2 A similar treatment has been begun recently by Herman Geuvers et al. [7], who kindly informed us of this.

REAL NUMBERS AS B LA C K BOXES 191

We suppose furthermore that x never fails to act as long as the input is admissible, i.e. positive and rational, and that every output of x consists in nothing but a single rational number. Each rational number q is tacitly identified with the black box real that constantly produces q as its output, regardless of the input in question.

For the sake of convenience, we sometimes denote lx is a real number’ by

x G M

though we hesitate to think of M as the collection of all real numbers. If different types of reals have to be distinguished, we speak of real numbers in the sense mentioned above as black box reals.

What we do not expect from black box reals is, for instance, that each output is uniquely determined by the corresponding input: feeding x repeatedly with the same e might provide us with distinct outputs which, of course, have to be within 2e of each other according to the real number rule. Consequently, we do not assume that any black box real performs its action according to a deterministic algorithm definitely dictating which q has to be assigned to each e.

Always keeping this temporal and nondeterministic character of black box reals in mind, we write

e[x\q

as a shorthand for lx has already transformed £ into q\ This notation might not be completely satisfying, but allows at least to recognise the input-output order as left-to right order without immediately suggesting any functional interpretation.

In order to get a grip on the concept of black box reals, one might interpret it in several ways, e.g. as not necessarily deterministic algorithms possibly operating at random, as many-valued functions, or as binary relations. In dialogue semantics (proofs-as dialogues), each black box real can be related with a proponent claiming to represent a real number who defends this claim by returning some q according to the real number rule whenever the opponent comes out with an e (see Taschner [23]). If a rather solipsistic interpretation is sought instead, one might view any black box real as a particular instance of Brouwer’s creating subject. We anyhow have to stress that all those are mere interpretations which we do not suppose, though we sympathise with the dialogue version.

In particular, we do not want to understand black boxes simply as certain binary relations, because this view would neglect the dynamic aspect of black boxes. Admittedly, the £[x\q notation— and how it is mostly used— suggests such an identification; one could say, however, that black boxes are to binary relations as the potential is to the actual infinite: the intended meanings differ from each other, although in practice no such distinction hits anyone right in the face.

We rather follow Taschner ([21, Section 2], [22, 1.8]), according to whom a point in the completion of an arbitrary metric space is a ‘mathematical object’ that assigns to every e > 0 some element of the original space. ‘Mathematical object’ is, just as ‘black box’ , an unqualified notion except for the analogue of the real number

192 PE TE R M. SCH USTER

rule; whence black boxes constitute a linear model of this approach, which, by the way, is choice-free in essence.3

It is noteworthy that any such completion technique embodies the most primitive but most seminal way in which completeness can be expressed, and which simply is the converse of the principle that each real can be approximated by rationals: namely, if something can be approximated coherently by rationals, then this ‘something’ has to be a real.4

In spite of their somewhat nominalistic character, black box reals are reals insofar as every property characteristic of constructive models of the continuum can be shown for black box reals, too. As we shall see after the next section, this task can be performed in a rather natural way, invoking neither countable choice nor set comprehension, principles which seem to be essential for any plain treatment of Cauchy or Dedekind reals, respectively.

3. C om p arin g R eals

In the presence of countable choice— of at least countable choice, as we shall explain in a moment— each black box real x gives rise to a Cauchy real: applying x to 2~n for any n G N yields, by countable choice, some sequence (rn) of rational numbers such that \rn — rm\ < 2~n + 2~m for all m, n. In particular, (rn) is a Cauchy sequence of rationals, which is even equipped with a Cauchy modulus of convergence [24, Chapter 5, Definition 2.2] and, in a sense to be defined in Section5, actually converges to x.

Note that in order to apply countable choice as it is formulated set-theoretically, one had to suppose in addition that for any given e > 0 all rational numbers q with e[x\q, or all q lying within e of x in a sense yet to be defined, do form a set— assumptions which are foreign to how we imagine black box reals. More generally, some set comprehension would be necessary for transforming x into a rational Dedekind cut, which in turn gives a rational Cauchy sequence by the usual form of countable choice: all the rationals strictly less (greater) than x form the only possible candidate for the corresponding lower (upper) cut, where order of black box reals is defined as in Section 4 below. Similar reservations had to be neglected for assigning x to a formal real in the sense of Negri and Soravia [11], made up of all the ordered pairs of rationals that encircle x.

Conversely, there are at least as many black box reals as there are reals in whatever sense. Since this argument serves well as a case study for how black box reals are to be constructed, let us go into detail and consider any element x of some valued field containing Q as a dense subfield. For every rational e > 0, there is some q G Q with \x — q\ < e ; by assigning q to e only in such a situation we get a black box real, which we call x again. Indeed, if also \x — q'\ < then \q — q'\ < e + e' by the triangle inequality, which is to say that x obeys the real number rule.

What is the way in which we define black box reals like this x l We give, first, a precise description of the situation, depending on e, in which any q is admissible as an output of x , and derive, secondly, the real number rule from these admissibility

3Although Taschner starts from the Cauchy reals as the range of the metric, his method can also be employed beforehand for completing the rationals to the reals— distances o f rationals are rational anyway.4We owe this insight to Fred Richman.

REAL NUMBERS AS B LA CK BOXES 193

conditions, whatever rational is chosen as the respective output. Of course, for every input e we must be able to find some admissible q\ sometimes we have to say how to do this, and how to start from the initial data, pieces of information which in our example are encoded in the hypothesis that Q is a dense subfield.

Let us stress once and forever that we understand the admissibility conditions as only necessary ones: when x transforms e into q, then q must be an admissible output of x for e as input. On the other hand, if we know just that q and £ meet the admissibility conditions, then x might well produce another admissible rational p whenever applied to £\ the only fact guaranteed by the real number rule is that \p — q\ < 2e in any such case.

The suspicious reader might now argue that by defining black box reals in this way we make use of nothing but countable choice, that x simply is a choice function extracted from the fact that for all the count ably-infinitely many inputs we are able to choose a corresponding output. Of course, this is of particular relevance in view of that no generality would be lost if black box reals exclusively processed rationals indexed by natural numbers, such as 1/n, 2~n, or 10- n . 5

We are nevertheless not able to agree with that suspicion because x is not even intended to represent a function from the classical point of view; what we have in mind is something like ‘ lazy evaluation’ or ‘just-in-tim e calculation’ rather than ‘an infinite number of choices carried out once and forever’ . Let us remember that x does not have to operate according to a general rule, a behaviour that every strict constructivist would expect from any (choice) function.

After all this, one could still suspect that black box reals are Brouwer’s (free) choice sequences in sheep’s clothing. Freely quoting from [4], page 107, “ . . . for the intuitionist, a real number (rn) need not be given by a rule: its terms r i, 7*2,... are simply rational numbers, successively constructed, subject only to the restriction \rn — rm\ < m ~l + n -1 . . . ” . The difference between choice sequences and black box reals is admittedly smaller than between the latter and Cauchy reals— does one not get a choice sequence by successively applying any given black box real to 1 ,1 / 2 ,1 /3 , . . . and denoting the results by r i , r 2, . . . ?

We think, however, that any such act of identification again requires some countable choice, if also neither any deterministic nor ‘actual infinite’ one: as soon as the construction of rn is completed, it has to be fixed as the n-th term of this particular choice sequence. To be more explicit, choice sequences are said to be equal if and only if the respective initial sequences coincide up to arbitrary length, whereas black box reals may well produce a different output whenever applied once more to the same input. Black box reals are thus a notion even less restrictive than free choice sequences.

It should be mentioned that ‘real numbers as black boxes’ have already been proposed by Fletcher [6, Chapter 11], who wanted to get rid of time as an intrinsic aspect of choice sequences.6 Disregarding whether time is really constitutive for a true continuum (unlike Fletcher, we are inclined to think so), our version of black box reals differs from Fletcher’s from the outset: he supposes black boxes to be deterministic and thus history-independent— we do not need this hypothesis.

5This observation stems from George Kapoulas.6Fred Richman has kindly communicated this reference to us.

194 P E TE R M. SCH USTER

As long as one pretends to know nothing about what happens inside a black box, it is reasonable to request only very little: the real number rule is just enough, any determinism assumption might be too much. Moreover, we are not able to follow Fletcher when he tries to establish a one-to-one correspondence between black boxes and choice sequences; as far as we see, black boxes in Fletcher’s sense, deterministic as they are supposed to be, could at most be constructed from lawlike sequences, but not from any possibly lawless one.

4. Order

The strict partial order < , as an existential statement of positive character, is fundamental for every constructive treatment of the continuum, in particular if one wants to be cautious with employing negation within any definiens or antecedens.

In the case of black box reals, let us set

x < y (and y > x) whenever s[x\q and S[y]p with q + e x < z

Cotransitivity7 x < y = > ( x < z \ / z < y )

Asymmetry —>{x < y A y < x)Irreflexivity —<(x < x)

for all x ,y , z € R.Indeed, provided x < y and y < z, we must have e[x]q, S[y]p, S'[y\p', and p[z]r,

with q + e < p — S and p' + S' < r — p. By the real number rule, p — 5 < p' + hence q + e < r — p and therefore x < z.

For the proof of cotransitivity, suppose that x < y, which is to say that e[x]q and 5[y]p with q + e 0 yields some rational r with p[z]r. Since q + e + p < p — 5 — p, we get q + e + p < r or r < p — 5 — p; in other words, x < z or z < y.

Finally, if x < y and y < x then, by transitivity, x < x, which is impossible: q + e < q' — e' with e[x\q and e'\x\q' contradicts the real number rule.

It is noteworthy that asymmetry implies irreflexivity and, by transitivity, vice versa. In absence of transitivity, however, the latter is strictly weaker than the former: von Plato presented in [12, 13] a constructive theory and some models of order where transitivity fails and asymmetry is replaced by irreflexivity.

The weak partial order < is best defined by setting, for any x, y G M, x < y (and y > x) whenever

z < x =>■ 2 < y and z > y => z > x

for all z € M. According to its very definition, < is a reflexive and transitive relation.Note that we already avoid negative propositions: we use neither asymmetry for

deriving transitivity of < from cotransitivity, nor irreflexivity for reducing (again by cotransitivity) the definition of x < y to either half, let alone to -*(x > y). This is part and parcel of Bishop’s philosophy [2, p. 23] according to which we make no

7This constructive substitute for the law of dichotomy is also called ‘comparability’ or ‘splitting’

REAL NUMBERS AS BLACK BOXES 195

constitutive use of negative propositions; compare with the definition of equality in Section 6.

We define the maximum m ax{x, y} of black box reals x and y by assigning e to m ax{g,p} whenever e[x]q and e[y]p. Of course, this construction bears the usual constructive properties, which are

M A X - V x < m ax{x, y} A y < m ax{x, y}M A X - 3 z < m ax{x, y } = ^ z < x \ / z < y

for all x ,y , z G M; confer also Negri [9] and von Plato [12],As we shall see later on, maxima constitute a special case of suprema, or least

upper bounds, as constructed for black box reals in Section 5. Let us nevertheless deal with maxima separately, if only for the reason that maxima are determined by their construction whereas suprema are only unique up to equality (Corollary 6.2).

First, we have to show that m ax{x, y} as defined above obeys the real number rule. Given also that £'{x]q' and £'[y\p' , we may assume that q < p. In case q' < p' we are done: by the real number rule \q — q'\ < e + s' and \p — p'\ < £ + e'. If, however, p' < q', then

q < p < p ' - \ - £ + e' and p' < q' < q + e + e'\

hence \q — p'\ < £ + s' as required.For the proof of M AX-V, suppose that z < x — that is r + p < q — £ where p[z\r

andefx]*/— and set 8 = (q — £ — r — p)/3. Applying m axjx, y} to 8 yields m a x {g ',;/} with 8[x]q' and 8[y\p'. We have q - £ — 8 < q' according to the real number rule; hence r + p < max{q' :p'} — 5 and therefore 2 < m a x{x ,y }. An analogous argument serves to prove that y < m ax{x, y}.

For the proof of M A X -3, suppose that z < m ax{x, y}; that is, r+p < max{</,p} — £ where p[z]r, e[x\q, and e[y]p. If p < q, this amounts to r + p < q — £ and hence z < x\ if q < p, then r + p < p — £ and therefore z < y.

Minima can, of course, be treated similarly to maxima, and both constructions can be generalised in an obvious way to an arbitrary finite number of arguments.

The following observations enable us to handle the ordering of black box reals as we are used to.

Lemma 4.1. For any i e 8 and q G Q we have.q — £ < x < q + £ whenever e\x]q.

Proof. For any z G M with z < q — £, we must have p[z\r with r + p < (q — e ) — 5 for some rational <5 > 0; note that S\p]p for all p G Q and 8 > 0. It follows that r + p < q — £ and thus z < x. The second inequality can be demonstrated similarly. □

Proposition 4.2.1. For any x, y G M with x < y there are u, v G Q such that x 0. Since x < q + £ and p — 8 < y by Lemma 4.1, u = q + £ + r) and v = p — 8 — rj are as required. Part 2 is an immediate consequence of Lemma 4.1. □

5. C om pleten ess

By the completeness of the reals one usually understands its sequential version, which is to say that every Cauchy sequence of real numbers converges. In the absence of countable choice, however, this seems to be strictly weaker than the well-known constructive version of order completeness a la Dedekind, which in the context of black box reals happens to be provable independently of the former.

Let us call any black box real a a least upper bound of a set8 S of black box realsif

whenever r)[a]c then s < c + 77 for all s G S and r > c — 77 for some r G S.

In particular, such a a is a least upper bound of S in the usual sense, which is to say that

LUB -V s < a for every sG S', andLUB - 3 for each i G R with x < a there is r G S' with r > x.

For the proof of LUB-V, suppose that z < s. There must be u, 77 G Q with z < u < u + 2 r ) < s , according to Proposition 4.2. Applying a to 77 yields c G Q such that c -|- 77 is an upper bound of S; in particular s < c + rj. We conclude that u < c — T) < a, by means of Lemma 4.1, and therefore z < a. For the proof of LUB-3, suppose that x < a. We must have e[x]q and t][(t]c with q + e < c — 77, and so there is r G S with c — 77 < r. Hence x < q + e < r, again by Lemma 4.1.

T h eorem 5.1 (Least-upper-bound principle). Let S be a nonempty set of black box reals that is bounded above. If for any pair of rationals a a for some r G S,

then there is a least upper bound of S.

P ro o f. First, we construct such a a. Given a rational 77 > 0, we claim that there are rational numbers a a for some r G S.

To this end, pick e G S and / G R with s < f for all s G S. By Proposition 4.2, there are d, g G Q such that d < e < f < g. Choose c i , . . . , cn G Q with n > 1,

d Co C\ "C . . . cn-\-\ g ,

and Cj — Cj-i < 77/2 for all j . By hypothesis, there is k G { 1 , . . . , n] such that

both s < Cfc+1 for all s G S and r > Cfc_ 1 for some r G S;

then a = c^-i and b = Ck+ 1 are as required. We get a black box real a by assigning 77 to c = (b— a)/2 only in such a situation. Indeed, given 77' > 0, let a' < b' be rational numbers with b' — a' < rj' such that

both s a' for some r' G S',

8In case one hesitates to quantify over an arbitrary subset S o f K, one might instead think of S as a family indexed by a sufficiently neat set; cf. [11 , Proposition 6.3].

and set c' — (b' — a')/2. By combining the inequalities

b > r , > a ! > b , — rf and b ' > r > a > b — rj,we get

b —r } < b ' < b + 7j' and a — rj < a' < a + 77; whence max{|a — a'|, |6 — 6'|} < max{?7, 77' } and therefore |c — c'| < m ax{7y, 77'} . By its very construction, cr is a least upper bound of S. □

Needless to say, analogous statements can be proven for greatest lower bounds, alias infima. As in [3], it is readily seen by means of induction and cotransitivity that the least-upper-bound principle can be applied to finitely enumerable sets, which is to say that maximum and minimum are particular cases of supremum and infimum. In view of cotransitivity, LUB-V and LUB-3 ensure that, among the hypotheses of Theorem 5.1, the alternative ‘for any pair of rationals a < b, either . . . or . . . ’ is really necessary for the existence of a supremum.

The derivation of sequential completeness from order completeness given by Bridges in [3] does not make use of any sophisticated set comprehension and can easily be rendered choice-free.9 If only to illustrate once more the features of the black box approach, let us nevertheless demonstrate the sequential completeness of black box reals directly without invoking the least-upper-bound principle; we can even do this in a rather elementary way.

Deliberately postponing the metric structure of R to Section 7, we base the following definitions almost exclusively on the ordering of R. Let us call a sequence (xn) of black box reals

[> a Cauchy sequence whenever for every rational e > 0 there are q G Q and N G N such that q — e < xn < q + e for all n > N, and

> convergent to x G R if for every rational e > 0 there is N G N such that if 5[x]p, then p — £ — S < xn N.

One can easily be convinced that these somewhat unusual definitions correspond perfectly to the usual ones. Of course, convergent sequences are Cauchy sequences.

T h eorem 5.2 (Sequential completeness). Each Cauchy sequence of black box reals converges.

P ro o f. Let (x n) be a Cauchy sequence. Given a rational e > 0, we can find ] V g N and q G Q such that q — £ < x n < q + £ for all n > N. By assigning q to £ only in such a situation we get a black box real x. In fact, if also q' — £r < xn < q' + £f for all n > N' and some N' G N, we can assume that N' > N and get \q — q'\ < £ + e' from the particular case n = N' of the foregoing inequalities.

Finally, let us show that (xn) converges to x. Given e > 0, and using the Cauchy property, we compute K G N and r G Q such that r — e /2 < x n < r + £/2 for every n > K. Provided 8[x]p, by the construction of x we must have M > K such that

9To this end, it suffices to observe that any Cauchy sequence (x n) converges to a given real number y provided that

(*) for all e and N there is n > N such that x n is within e o f y.Different from [3, Lemma 3] there is no need to suppose, instead o f (*), the existence o f a subsequence o f (xn) converging to y. O f course, such a subsequence can be extracted from (*) by means of countable choice, and a fortiori every subsequence o f (xn) converges to y.

p — 8 < xn M. From these inequalities with n substituted by M , we can conclude as above that \p — r\ < 8 + e j 2 , and thus obtain

p - 8 - e < r - e/2 < xn < r + e/2 K. □

6. Equality and Inequality

For any x, y G R, set

x = y if both x < y and y < x, and x ^ y if either x < y or else y < x;

one could say that = is the ‘antisymmetrisation’ of < , and ^ the ‘symmetrisation’ of <.

Of course, = is an equivalence relation, and < , < , ^ are extensional— that is, they respect = . Furthermore, ^ is a tight apartness relation in the sense of [8 , pp. 7-8]; in other words, we have

Consistency —>(x ^ x)

Symmetry x ^ y & y ^ x

Cotransitivity x ^ y ^ {z ^ x\l z ^ y)

Tightness ~ {x j - y) x = y

for all x ,y , z G R.Once more, these relations turn out to behave just in the way one is used to.

Proposition 6.1.1. For any x ,y G M, we have x ^ y if and only if e[x\q and S[y\p such that

\p - q\ > S + e.2. For any x, y G we have x = y if and only if \p — q\ < 8 -\- e whenever e[x]q

and 8[y]p.

Proof. Concerning Part 1, observe that x < y if and only if e[x\q and 8[y\p with q + e 8 + e. For the proof of Part 2, suppose first that x = y. If e[x\q and 8[y]p, then

q — e < x < y < p - \ - 8 and p — 8 < y < x < q + e,

by Lemma 4.1; hence \p — q\ < 8 + e. Conversely, given z < x, we have p[z]r and e[x]q with r + p < q — e. Setting 8 = (q — e — r — p)/3, we obtain r + p + 5 < q - e — 8. Apply y to 8 and pick the result p , which is to say that 8[y\p. Since q ~ p < 8 + e by hypothesis, we get r + p y goes similarly. □

Note that we again avoid negation insofar as we neither define x = y as its equivalent -i(x 7 y) nor make any further use of tightness (for example, for deriving Part 2 from Part 1 in the foregoing proof of Proposition 6.1).

Although LUB-V and LUB-3 ensure that least upper bounds are uniquely determined up to = , let us prove this fact in the way particular to black box reals.

C orolla ry 6.2.1. If a set S of black box reals has least upper bounds a and a', then a = a'.

2. If (xn) is a sequence of black box reals converging to x and x ', then x = x '.

P roo f. Given r}[a\c and rf[o']d, we can find r ,r f E S such that

c — Tj < r < c' + r)' and c' — rj' < r' < c + 77.

Hence |c — c'\ < 77 + 7/, and the proof of Part 1 is completed. For the proof of Part2, given e > 0, we compute N > 0 such that if 8[x\p and 8'\x']p', then

p — 5 — e < xn < p + 5 + e and p' — 5' — £ < xn < p' + S' + £.

Hence \p — p'\ < £ + 8' + 2e:, from which we get \p — p'\ < 8 + 8' because £ is arbitrary. □

7. A rith m etic and V alue

It is obvious that, for each x, y G R,D> the sum x + y, defined by assigning r] to q + p whenever £[x]q and 8[y]p with

£ + <5 < 77, and

[> the additive inverse —x, defined by assigning £ to — q whenever £[x\q,

are black box reals. The absolute value |ar| G 1 of some x € M is defined as max{x, —x} , or, equivalently, as assigning £ to |g| whenever £[x\q.

Again, for each the following items constitute black box reals:> the product xy , defined by assigning 77 to qp whenever £[x]q and 8[y]p with

£, 8 < 1 and eL + 8K < 77,

where K ,L > 1 are fixed integers such that \x\ < K — 1 and \y\< L — 1;

[> the multiplicative inverse l/x in case x ^ 0, defined by assigning £ to 1/q whenever 8[x]q with <5 < min{p,Ep2}, where p > 0 is a fixed rational such that |x| > 2p.

Let us show that the real number rule is obeyed by these operations. Concerning xy , given also 7/ > 0 and £f[x\q\ 8'[y\p' with e',8' < 1 and £rL + 8'K < rj', we have |o| < K — 1 + £ < K by Lemma 4.1; similarly, \p'\ < L. Hence

\qp-q'p'\ < \q\ ■ \p~p'\ + \p'\ • \q~q'\< K(8 + 8 ' )+L (£ + £')

< v + v',as required. In the case of l/x, given also e’ and 8'[x\q' with 8' < min{p,£rp2}, we have | <71 > 2 p - 8 > p by Lemma 4.1; similarly, |gr| > p. From the inequalities

\q - q'\ < 8 + 8' < (e + £')p2

we conclude that1 1 \q — q'\ ,q q' \qq'\

By means of Proposition 6.1, it is routine to verify that, with respect to the equality = defined above, all the well-known algebraic laws also hold for black box reals.

200 P E TE R M. SCHUSTER

Besides commutativity and associativity of either operation, as well as distribu- tivity, we get, for each black box real x, that —x and 1/x are the additive and multiplicative inverse, respectively.

Moreover, the strict partial order < is preserved by addition and multiplication; that is

AD D < x < y = ^ x + z < y + z

MULT< x < y, z > 0 =>■ xz < yz

for all x ,y , z € R.For the proof of ADD<, observe first that if x < y, then x 0 with u + 4r = v. According to the construction of these sums, we must have r[x + z](q + r ) and r[y + z](p + s), where e[x\q, S[y]p, p[z]r, and o[z\s with e + p < r and 5 + a < r. Again by Lemma 4.1, q — e —a — p. All this yields

p + s — 5 — a — (q + r + e + p ) > v — u — 4r = 0;

whence q + r + e + p 0

and y > 0, then xy > 0. To this end, note first that if x, y > 0, then x, y > 2 Jr\ for some rational r\ > 0 with rj < 1. What we have to show for xy > 0 is that r > rj whenever rj[xy]r. According to the construction of xy , we must indeed have e[x\q and S[y\p with e,S < ij < fr\ and r = qp. By Lemma 4.1, we get

Q > 2 y / r j - e > 0 7 and p > 2y/rj — S >

whence qp > T).In particular, the equality = is a congruence relation with respect to all algebraic

operations on black box reals.Leaving it to the reader to fill in the few missing details of proof, let us summarise

the following properties characteristic of black box reals. (For this purpose only, we may denote by R the collection consisting of all black box reals, though we expressed before some reluctance to carry out any such act of comprehension.)

D> R is a constructively ordered field as considered in [10];

> R is archimedean ordered and order complete (Proposition 4.2, Theorem 5.1);

> R is a Heyting field equipped with an archimedean valuation in the sense of [8];

> R is a complete metric space containing Q as a dense subset (Lemma 4.1, Theorem 5.2).

It is noteworthy that, according to Lemma 4.1, we have \x — q\ < £ whenever e[x]q, which is to say that any black box real is approximated by its output rationals in a way that perfectly meets the intended meaning.


8. C om p lem en ta ry R em arks

We conclude this article of a somewhat tentative character by outlining a few directions worth future examination.

First, dropping the real number rule from black box reals would provide a constructive model of the continuum that contains infinitesimal and infinite numbers as well; compare the passing from Cauchy sequences to arbitrary sequences of rational in the approach due to Schmieden and Laugwitz (see [18]). Secondly, the behaviour of black box reals ought to be formalised with particular regard to their temporal character, perhaps following the example of Kripke semantics. Brouwe- rian counterexamples could also be re-justified by this temporal interpretation of the continuum. Thirdly, it is tempting to hope that the search for an appropriate notion of continuous function within the context of black box reals would shed some light on the (uniform) continuity principles particular to intuitionism, such as the fan theorem and the principle of continuous choice. Last but not least, black box reals are by their very definition most suitable for any approximate constructive analysis as described in [19].

A ck n ow ledgem en ts. The author owes various inspirations to many participants of the symposion Reuniting the Antipodes— Constructive and Nonstandard Views of the Continuum, Venice, Italy, May 17-22, 1999. This meeting, which he had the pleasure to co-organise, was only possible because of being generously supported by the Istituto Italiano per gli Studi Filosofici, Naples and Venice, and by the Volks wagen-Stiftung, Hannover. Last but not least, the author is indebted to Giovanni Curi, Hajime Ishihara, Georgios Kapoulas, and Helmut Schwichtenberg for fruitful discussions, and, in particular, to Douglas Bridges and Fred Richman for kindly commenting on drafts of this article.

R eferen ces

1. Errett Bishop, Foundations of Constructive Analysis, McGraw-Hill, New York, 1967.

2. Errett Bishop, and Douglas Bridges, Constructive Analysis, Grundlehren der math. Wiss., Bd. 279, Springer, Berlin and Heidelberg, 1985.

3. Douglas S. Bridges, Constructive mathematics: a foundation for computable analysis, Theor. Comp. Sci. 219 (1999), 95-109.

4. Douglas S. Bridges and Fred Richman, Varieties of Constructive Mathematics, Cambridge University Press, 1987.

5. Solomon Feferman, Weyl vindicated: ‘Das Kontinuum’ seventy years later, in Temi e prospettive della logica e della scienza contemporanea, (C. Cel- luci and G. Sambin, eds), Coop. Libr. Univ. Editr., Bologna, 1988, vol. 1, pp. 59-93. Also in In the light of logic (S. Feferman, ed.), Oxford University Press, 1998, p. 298.

6. Peter Fletcher, Truth, Proof and Infinity. A Theory of Constructions and Constructive Reasoning, Synthese Library, vol. 276. Kluwer, Dordrecht etc., 1998.

7. Herman Geuvers, Randy Pollack, Freek Wiedijk and Jan Zwanenburg, Skeleton for the proof development leading to the fundamental theorem of algebra, preprint, Katolieke Universiteit, Nijmegen, 2000.


8. Ray Mines, Fred Richman and Wim Ruitenburg, A Course in Constructive Algebra, Springer, New York, 1987.

9. Sara Negri, Sequent calculus proof theory of intuitionistic apartness and order relations, Archive Math. Logic, 38 (1999), 521-547.

10. Sara Negri, A sequent calculus for constructively ordered fields, [20], 143-155.11. Sara Negri and Daniele Soravia, The continuum as a formal space, Archive

Math. Logic, 38 (1999), 423-447.12. Jan von Plato, Order in open intervals of computable reals, Math. Struct, in

Comp. Science, 9 (1999), 103-108.13. Jan von Plato, Positive lattices, [20], 185-197.14. Fred Richman, Intuitionism as generalization. Philos. Math. 5 (1990), 124-128.15. Fred Richman, Generalized real numbers in constructive mathematics, Indag.

Math. N. S. 9 (1998), 595-606.16. Fred Richman, The fundamental theorem of algebra: a constructive treatment

without choice, Pacific J. Math. 196 (2000), 213-230.17. Fred Richman, Constructive mathematics without choice, [20], 199-205.18. Peter M. Schuster, A constructive look at generalised Cauchy reals, Math. Logic

Quart. 46 (2000), 125-134.19. Peter M. Schuster, Elementary choiceless constructive analysis, in Computer

Science Logic (P. Clote and H. Schwichtenberg, eds), Proc. 2000 Fischbachau Conf., Lect. Notes Comp. Sci., vol. 1862, Springer, Berlin and Heidelberg, 2000, pp. 512-526.

20. Peter Schuster, Ulrich Berger and Horst Osswald, Reuniting the Antipodes— Constructive and Nonstandard Views of the Continuum, Proc. 1999 Venice Symp., Synthese Library, vol. 306. Kluwer, Dordrecht etc., 2001.

21. Rudolf Taschner, Entwurf einer konstruktiven Topologie, Osterreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II, 199 (1990), 161-192.

22. Rudolf Taschner, Lehrgang der konstruktiven Mathematik. 3. Teil: Funktionen Manz and Holder-Pichler-Tempsky, Wien, 1993.

23. Rudolf Taschner, Real numbers and functions exhibited in dialogues, [20], 257- 269.

24. Anne S. Troelstra and Dirk van Dalen, Constructivism in Mathematics. An Introduction, Two volumes, North Holland, Amsterdam, 1988.

Peter M. SchusterMathematisches InstitutUniversitat MiinchenTheresienstrafie 3980333 MiinchenGERM AN [email protected]

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