A Tour with Constructive Real Numbers

The aim of this work is to characterize constructive real numbers through a minimal axiomatization. We introduce, discuss and justify 16 constructive axioms. Then we address their expressivity considering the alternative axiomatizations. 1 Overview of the work This work tries to understand (again) constructive real numbers. Our main contribution is a new system of axioms, synthesized with the aim of being minimal, i.e. of assuming the least number of primitive notions and properties. Such a system is consistent with respect to reference models we have in mind — (equivalence classes of) Cauchy sequences [TvD88] and co-inductive streams of digits [CDG00] — and will be compared to other proposals of the literature [Bri99, GN01]. In particular we will prove that our axiomatization has a sufficient deductive power. We have formalized and used our axioms inside the Logical Framework Coq [BB01]. However, the axioms can be stated and worked with in a general constructive logical setting, because we do not need all the richness of the Calculus of Constructions [CH88], the logic beneath Coq. In particular we do not require the use of dependent inductive types and universes. On the contrary, we should have available a logical system that accommodates second-order quantification (in order to axiomatize the existence of limit) and the Axiom of Choice (for defining the “reciprocal” function on reals different from zero). We define constructive real numbers through sixteen axioms organized in four groups: arithmetic operations, ordering, Archimedes’ postulate and completeness. Our axiomatization uses only three basic concepts: addition (+), multiplication (×) and strict order (<). In most of the constructive approaches to analysis [Bis67, Bee85, Wei00], real numbers are defined as a quotient of a set of representations (e.g. equivalence classes of Cauchy sequences, digit expansions, etc.). Hence, also in an axiomatic approach, it is necessary to see the reals as a set provided with an equivalence relation. In our proposal, this equivalence relation (∼) is not a primitive notion, but it is derived together with its fundamental properties through the strict ? Research partially supported by EEC Working Group “TYPES” and Italian MIUR COFIN “TOSCA” order relation. We define the equivalence by (x ∼ y) , ¬ [(x < y) ∨ (y < x)]. Similarly, it is not necessary to assume as basic the apartness relation (#) — which is a semi-decidable version of the inequality (6=) — as it is definable in terms of the order relation. The paper has the following structure. Section 2 introduces the constructive axioms, which are then explained and motivated. In Section 3 we start deducing some elementary consequences of the axioms. The following Section is devoted to a digression concerning possible models for the constructive real numbers. We conclude by articulating a detailed comparison between our axiomatization and other similar works in the literature. Acknowledgments. The authors are grateful to Herman Geuvers, Furio Honsell, Marino Miculan, Ivan Scagnetto and the anonymous referees for their interesting remarks. 2 Constructive axioms We introduce constructive real numbers as the mathematical entities satisfying four groups of axioms. The basic notions are the following: – a representation set R, with two elements 0R (zero) and 1R (one); – a binary relation < (strict order) over R; – two binary operations + (addition) and × (multiplication) over R. We do not assume the negation (−) and reciprocal (−1) as primitive functions. The main reason for this choice is that the reciprocal function cannot be defined in Coq: in fact, in Coq, each function has to be totally defined; and, in a constructive setting, each function has to be continuous (w.r.t. the Euclidean topology). On the contrary, it is impossible to make continuous by extension the reciprocal function. In order to state the axioms, it is convenient to define two relations and two functions: – a binary relation ∼ (equivalence) over R tells that two names represent the same number. It expresses the redundancy of the representation; – two recursively defined functions inj, exp : N → R (inj(n) = n, exp(n) = 2) are used in the Archimedean and completeness axioms; – a ternary relation near ⊆ R×R×N (near(x, y, n)⇔ |x−y| ≤ 2−n) expresses the Euclidean metric. Our axiomatization is parametric with respect to a set N of the natural numbers, that we suppose to be given. In our formalization in Coq, N is taken as the set of the inductive natural numbers. In a different context, N could be defined as a set satisfying the Peano’s arithmetic axioms. Finally, we claim that constructive real numbers are captured by the following axioms. Definition 1. (Axioms for constructive real numbers) Consts : R, {0R, 1R} ∈ R < ⊆ R×R + : R×R→ R × : R×R→ R Defs : ∼ ⊆ R×R (x ∼ y) , ¬ ((x < y) ∨ (y < x)) inj : N → R inj(0) , 0R, inj(n+ 1) , inj(n) + 1R exp : N → R exp(0) , 1R, exp(n+ 1) , exp(n)× (1R + 1R) near ⊆ R×R× N near(x, y, n) , ∀ ∈ R. (1R < × exp(n))→ (x < y + ) ∧ (y < x+ ) Add : +-associativity ∀x, y, z ∈ R. (x+ (y + z)) ∼ ((x+ y) + z) +-unit ∀x ∈ R. (x+ 0R) ∼ x negation ∀x ∈ R. ∃y ∈ R. (x+ y) ∼ 0R +-commutativity ∀x, y ∈ R. (x+ y) ∼ (y + x) Mult : ×-associativity ∀x, y, z ∈ R. (x× (y × z)) ∼ ((x× y)× z) ×-unit ∀x ∈ R. (x× 1R) ∼ x reciprocal ∀x ∈ R. (0R < x)→ ∃y ∈ R. (x× y) ∼ 1R ×-commutativity ∀x, y ∈ R. (x× y) ∼ (y × x) distribuitivity ∀x, y, z ∈ R. (x× (y + z)) ∼ (x× y) + (x× z) Order : non triviality 0R < 1R < -asymmetry ∀x, y ∈ R. (x < y)→ ¬(y < x) < -co-transitivity ∀x, y, z ∈ R. (x < y)→ (x < z) ∨ (z < y) +-reflects< ∀x, y, z ∈ R. (x+ z < y + z)→ (x < y) ×-reflects< ∀x, y ∈ R. (x× z < y × z)→ (x < y) ∨ ((y < x) ∧ (z < 0)) Archimedean ∀x ∈ R. ∃n ∈ N. x < inj(n) completeness ∀f : N → R. ∃x ∈ R. (∀n ∈ N. near(f(n), f(n+ 1), n+ 1))→ (∀m ∈ N. near(f(m), x,m)) Arithmetic operations. As the reader can see, the properties required for the arithmetic operations are just the same characterizing a classical abelian field; in [Bri99] this same set of properties is named “Heyting field”. Note that it is sufficient to assume the existence of the reciprocal only for positive reals. As we have already remarked, we do not assume the “negation” and the “reciprocal” functions: instead we assume the existence, for each real x, of its negation and, if 0 < x, of its reciprocal elements. In this way we have to postulate the Axiom of Choice for extracting effectively from a number x its negation and its reciprocal. The necessity of the Axiom of Choice can be seen as a weakness of our axiomatization. However, there is no simple way to avoid it: in fact, without Choice, the reciprocal function cannot be defined inside Coq (whereas the negation function and the limit functional are definable). An alternative axiomatization that does not require the Axiom of Choice could be obtained as follows. One postulates the existence of the negation and limit functions and, instead of a single inversion function, the existence of a series of approximations of the inversion function, inv : (N × R) → R, satisfying the axiom: ∀n ∈ N. ∀x ∈ R. (1R < x× exp(n))→ (x× inv(n, x) ∼ 1R) that is, the function λx. inv(n, x) behaves as the reciprocal function for all the real numbers bigger than 2−n. Given a suitable representation for the reals, the function inv is Coq-definable, and allows the evaluation of the reciprocal of any real number x for which it is possible to find a natural number n such that 2 < x. We did not pursue this alternative axiomatization for simplicity reasons. Order relation. Concerning the ordering, we make the following remarks. First, the classical trichotomy of total order (x < y)∨(x = y)∨(y < x) is not a constructive property: its substitute in the constructive setting is the property (x < y)→ (x < z) ∨ (z < y), which we name “co-transitivity”. Secondly, we have thought that it is cleaner to define only the relation of order: in fact, in constructive mathematics [TvD88, Bri99], the order is universally considered the most fundamental relation for the real numbers. The alternative would have been to start from the apartness relation — the constructive non equality — then to assume axioms for it, and further to introduce the order itself with its proper axioms. But this increases the length of the presentation of the constructive reals, and moreover introduces some redundancy, thus not permitting to carry out our declared purpose of being minimal. The equivalence and the apartness relations are defined using the basic strict order, therefore their properties are derived from the axioms. We define the equivalence (∼) and the apartness (#) in the following way: (x ∼ y) , ¬ [(x < y) ∨ (y < x)] (x# y) , (x < y) ∨ (y < x) There is still more, because we have been careful in the design of the relationship between the order and the operations. We are able to deduce all the basic properties relating the equivalence and the operations from the two reflection axioms: (x+ z < y + z)→ (x < y) (x× z < y × z)→ (x < y) ∨ [(y < x) ∧ (z < 0)] The fact that the equivalence is preserved by the basic notions (addition, multiplication and order) is an immediate consequence of these two axioms and the <-co-transitivity one. Notice that, on the contrary, the preservation of the equivalence does not follow from the more usual preservation axioms [Bri99, GN01]: (x < y)→ (x+ z < y + z) (0 < x) ∧ (0 < y)→ (0 < x× y) This particular phenomenon relies on the fact that the reflection of the order is more powerful than its preservation, as will be argued in Section 3. Archimedean property. The Archimedean axiom links real numbers to natural numbers, stating that the reals are standard with respect to the naturals. This axiom does not exclude the existence of non-standard reals, but in this case also the naturals must be non-standard. That is, it is possible to conceive non-standard models for our axioms: these models would contain infinitary and infinitesimal real numbers as well as infinitary naturals. Completeness. Finally, the completeness property for the field of the real numbers is postulated asking the existence of the limit of any Cauchy sequence 〈sn〉n∈N with an exponential convergency rate: ∀n ∈ N. |sn − sn+1| ≤ 2−(n+1) Many others choices for capturing the completeness are possible, and our axiom could appear weak at a first glance. Anyway, in order to evaluate constructively the limit of a Cauchy sequence S, it is necessary to know its convergency rate: from this convergency rate it is possible to extract (constructively) a subsequence of S having an exponential convergency rate. It follows that starting from our axiom we are able to derive the completeness properties that are found in the literature [Bri99, GN01]. Our choice has been motivated by simplicity reasons. The minimality of our axiomatization could be useful both for theoretical reasons — the mathematical curiosity about an essential characterization of the constructive reals is addressed — and practical ones — a simple test for possible models is provided. However, rather than pursuing a minimal set of axioms at all costs, we have chosen to axiomatize the different notions (order, addition, multiplication, etc.) separately, for the sake of the clarity of the axiomatization.