Logic, Foundations of Mathematics, and Computability Theory || Axioms for Intuitionistic Mathematics Incompatible with Classical Logic

A. S. TROELSTRA

AXIOMS FOR INTUITIONISTIC MATHEMATICS

INCOMPATIBLE WITH CLASSICAL LOGIC

1. INTRODUCTION

1.1. Standard formalizations of constructive mathematics ('constructive' here in the narrow sense of Bishop (1967): choice sequences are regarded as inacceptable, and Church's thesis is not assumed) can be carried out in formal systems based on intuitionistic logic which become classical formal systems on addition of the principle of the excluded third. The fact that in such systems for constructive mathematics the logical operations permit an interpretation different from the classical truth-functional one is then solely expressed by the fact that less axioms are assumed. As is well-known, this results in formal properties 1 such as

f-A v B ~ r A or r B,

f-3xAx ~ r- At for some term t (3xAx closed).

But given an interpretation of the logical operations different from the classical one, there is also the possibility of axioms which are valid on this interpretation, but not on the classical one. It goes without saying that, granted the fact that our formal system was intended to code (part of) mathematical practice based on this interpretation, special interest attaches to such axioms; and as we shall see from the discussion below, this is not the only reason for being interested in them.

As to the justification of such axioms, we can apply various standards of informal rigoul". The strictest requirement is a precise description of the mathematical objects of a theory ('conceptual analysis') followed by an informal, but rigorous derivation of the axioms proposed for these objects. We are rarely in a position to present such an analysis; an example taken from intuitionism is the theory of lawless sequences (see e.g. Kreisel, 1968; Troelstra, 1969 § 9, 1976 § 2).

In most cases, we can at best produce plausibility arguments, not unlike the way in which many new axioms for set theory (e.g. those on

Bu"s and Hintikka (eds.), Logic, Foundations of Mathematics and Computability Theory, 59-84. Copyri~ht © 1977 by Reidel Publishing Company, Dordrecht-Holland. All Rights Reseroed.

60 A. S. TROELSTRA

large cardinals) are proposed; the plausibility arguments lead us to expect consistency of the new axioms with the already accepted ones. Investigation of formal properties of systems including the proposed axioms, and exploration of the mathematical consequences may then help us to obtain a better insight into what is involved in the justification of such axioms.

For the examples of possible axioms conflicting with classical logic presented in this paper, we shall be content with plausibility arguments and formal analogies, without hiding (to ourselves) the weak epistemological basis for our axioms. It is worth mentioning however, that in one respect the status of our elamples is different from largecardinal axioms in set theory: for at least three of our five examples, the axioms conflicting with classical logic do not increase prooftheoretic strength, i.e. relative consistency proofs are available; for the other two examples, relative consistency is at least a plausible conjecture.

Our aim in this paper is not the presentation of new results, but to draw attention to what is at present a neglected area.

1.2. Contents of the Paper

In Sections 3-8 examples of axioms for intuitionistic mathematics incompatible with classical logic are presented. For some of these, such as Church's thesis and the continuity axioms, mathematical consequences and metamathematical properties have been studied extensively; others have scarcely been investigated. As we go along, we mention various open problems connected with these examples, and give reasons for being interested in axioms of this type.

Section 9 discusses admissible (or derived) rules corresponding to the axioms discussed in our examples; Section 10 connects the problem of the completeness of intuitionistic predicate logic with the addition of new axioms.

2. PRELIMINARIES

2.1. Variables

We shall use x, y, z, u, v, w to indicate numerical variables. We distinguish two disjoint classes of variables for number theoretic func-

AXIOMS FOR INTUITIONISTIC MATHEMATICS 61

tions: 'constructive-function' variables (a, b, c, d) and 'choicesequence' variables (0:, [3, 'Y, 5). X, Y, Z are used as meta-variables for variables for sets of natural numbers; similarly X', Y', Z' for sets of (choice) sequences.

2.2. Intuitionistic first-order arithmetic HA, intuitionistic second order arithmetic HAS, and elementary analysis EL are described in Troelstra (1973). EL is essentially HA with function variables and quantifiers, ranging over sequences closed under 'recursive in'.

EL * is actually the same as EL but with choice-sequence variables instead of constructive function variables. HASI is a combination of HAS and EL *, with variables for sets of sequences instead of sets of natural numbers.

2.3. Some Notations

Let j be a primitive recursive pairing from N2 onto N, with primitive recursive inverses it, h-

j(o:, [3)=defAx.j(o:X, [3x),

X(x, y) =defXj(X, y), (X)x =defAy.X(X, y),

X'(o:, [3) =defXj(a, f3), (X')a =def A[3.X(o:, [3),

For any formal system H let L(H) denote its language. For other notations, not explained in this paper, see Troelstra (1973).

3. EXAMPLE I: CHURCH'S THESIS

3.1. In L(EL) we can express Church's thesis formally as

CT Va3zVx3y[Tzxy & ax= Uy],

where T, U are Kleene's T-predicate and the result-extracting function respectively. In the presence of

ACoo Vx3yA(x, y) -+ 3aVxA(x, ax)

62 A. S. TROELSTRA

CT is equivalent to

CTo 'v'x3yA(x, y) --+ 3z'v'x3y[Tzxy I\A(x, Uy)].

Note that in EL, obtained by restricting ACoo of ELI to quantifier free A (call the restricted schema OF - ACoo), CT is unproblematic on the interpretation of the range of the function variables as ranging over total recursive functions and hence EL + CT is conservative over HA. CTo can be used to express Church's thesis within L(HA).

The intuitive justification for CTo (or CT+ACoo) is as follows: we think of the function variables as ranging over lawlike sequences (sequences given by a law for computing the value to each argument). The standard 'constructive' reading of ''v'x3yA(x, y)' is: we have a method for which we can show that it yields a y to each x such that A(x, y). If we leave non-lawlike, 'indefinite' or 'incomplete' objects such as choice sequences out of consideration, this means obviously the same as 3b'v'xA(x, bx); and this justifies ACoo. CT then expresses that those laws determining sequences must be identified with mechanical (recursive) laws. From an intuitionistic point of view, this amounts to the identification 'humanly computable' = 'mechanically computable' (cf. Kreisel, 1972).

3.2. CTo obviously conflicts with classical logic. By classical logic,

'v'x3z[(z = 0 & 13yTxxy)v(z~ 0 & Txx(z ..:...1)],

but there is no recursive b giving z for each x, since then bx = 0 ~ 13yTxxy, which would make {x: 3yTxxy} recursive. Thus CTo refutes an instance of 'v'x[Ax v-,Ax], namely for Ax ~3yTxxy.

Via the existence of r.e., recursively inseparable sets, which follows from CTo, we can refute e.g. an instance of (A --+ B)v(B --+ A). Many counterexamples to classical theorems on assumption of CT ° are in fact already consequences of the existence of an Enumerable but Not Decidable set

END 3a-,'v'x(3y(ay = x)v13y(ay = x»


or the more sophisticated 'existence of enumerable but not effectively separable sets':

INSEP 3a3b('Vxb3y(ay = x)/\3z(bz = x»/\

-8c'Vx[(3y(ay = x) ~ cx = 0)/\ (3y(by = x) ~ cx'" 0)].

The mathematical consequences of e.g. END + INSEP, END + -,INSEP have not been systematically investigated so far.

3.3. It is certainly not quite obvious that CTo can be consistently added to HA. There is no difficulty in interpreting 'Vx3y by recursive dependence of y from x, but can this consistently be extended to formulae of arbitrary complexity such that all schemata of intuitionistic predicate logic are validated?

The positive answer is provided by Kleene's realizability, which associates to each formula A an interpretation 3x(xrA) ('A is realizable') such that we have

THEOREM. (i) HA+ECTorA~3x(xrA)

(ii) HA+ECTorA~HAr3x(xrA),

where ECT 0 is a schema which generalizes CT 0:

ECTo 'Vx[Ax ~ 3yB(x, y)] ~ 3u'Vx[Ax ~ 3z(Tuxz /\ B(x, Uz))],

where A does not contain v, and 3 only in front of prime formulae (i.e. 3x(t = s».

Beeson (1972; see Troelstra 1973, 3.4.14) ha" shown that ECTo is essentially stronger: HA + CT off ECT o.

The mapping cp: A~3x(xrA) can be said to make explicit the recursive dependence of y on x in 'Vx3y, hereditarily. The crucial problem is here that we also have to define 3x(xr(A ~ B» for given 3x(xrA), 3x(xrB). This is done by

xr(A ~ B)=def'Vy(yrA ~ !{x}(y)/\{x}(y)rB).

64 A. S. TROELSTRA

It is an open problem whether there also exists a mapping cp such that

HA+CToI-A~cpA, HA +CToI-A<:>HAl-cpA.

3.4. Markov's Schema

M 'v'x[Ax v,Ax]1\ 'Il3xAx ~ 3xAx

is modulo CT 0 equivalent to

MpR 'Il3xAx ~ 3xAx (A primitive recursive) or equivalently

'v'xy('Il3zTxyz ~ 3zTxyz).

The mathematical consequences of HA + ECTo+ M have been extensively explored - this system may be regarded as a codification of mathematical practice of the Russian constructivist school. Some ex

amples

(i) Mappings from complete separable metric spaces into separable metric spaces are continuous. (Ceitin, 1959; Y. N. Moschovakis, 1964)

(ii) Real-valued functions on [0, 1] are not always uniformly continuous. (Beeson A)

(iii) There exists a continuous mapping cp of the square [2 == {(x, y): x E [0, 1], y E [0, I]} into [2 such that p«x, y), cp(x, y»;a. 8-1 for all (x, y) E [\ p an Euclidean metric; and there is a uniformly continuous mapping rfJ of 12 into 12 such that 'v' z E 12 (rfJz'l- z). (Refutation of Brouwer's fixed point theorem, due to Orevkov (1963, 1964).)

3.5. We note in passing that instead of ECTo one sometimes considers (e.g. Dragalin, 1973)

ECT' 'v'x[ ,Ax ~ 3yB(x, y)] ~ 3u[ ,Ax ~ 3z(Tuxz 1\ B(x, Uz»)].

However, this is equivalent to ECTo relative to HA + M, since a


formula A' with 3 only in front of prime formulae is via M equivalent to a negative formula A (i.e. a formula without v, 3) and for such A, A~...,..,A.

4. EXAMPLE II: CONTINUITY AXIOMS

4.1. In the formulation of the continuity schemata we shall assume all predicates to be extensional w.r.t. numerical and function variables, i.e.

x = x' /\a = a ' /\A(a, x) -+ (a', x').

This is automatically ensured if for example we think of the language L(EL *). The following continuity schemata are of special interest here.

CONTo! Va3!xA(a, x) -+ 3'Y E KoVaA(a, 'Y(a»,

CONTo Va3xA(a, x) -+ 3" E KoVaA(a, ,,(a»,

CONTI Va3(3A(a, (3) -+ 3" E KoVaA(a, 'Y I a).

Here Ko,,==Va3x(,,(ax);t: 0)/\ Vnm("n;t: O-+"n = ,,(n * m»

and ,,(a) = x ==3y( ,,(ay) = x + 1)

" I a = (3 == Vx3y( ,,«x) * ay) = (3x + 1).

Ko" expresses: " is a neighbourhood function; i.e. " codes a continuous functional r: NN -+ N or r: NN -+ NN. (If we permit functionals we could e.g. express CONTo as Va3xA(a, x)-+3reCont VaA(a, ra).)

4.2. There is a strong analogy between CONTo, CONT}, and CTo; CONT 0 and CONTI can be used in the same way as CT 0 in refuting schemata of classical predicate logic; so is e.g. Va(Aa v-,Aa) refuted by CONTo for Aa ==3x(ax = 0). On the other hand there is a conflict between CONT 0 and CT; in

CT Va3xVy3z(Txyz & ay = Vz)

66 A. S. TROELSTRA

it is required by CO NT 0 that the x depends continuously on a (Le. can be found from an initial segment) which is obviously impossible. CONTo! +CT on the other hand is consistent (see, e.g. Troelstra, 1973, 2.6.15, 3.2).

4.3. Consistency of EL+CONT1 presents similar problems as the consistency of HA +CTo did before, which can be resolved by similar methods: in this case Kleene's realizability of functions (for a short account see Troelstra, 1973, 3.3). However, in this case there is another method available which opens up different perspectives: the elimination of choice sequences. It is well known that the usual intuitive justifications and plausibility arguments for the continuity schemata depend on the interpretation of the function variables as ranging over choice sequences. But the so called elimination theorems provide a quite different solution: they permit us to regard quantification over choice sequences2 as a 'figure of speech'. A bit more precisely, we consider systems H (in the language of EL + constant K for the class of neighbourhood functions); we think of H as containing, besides number quantifiers, only ordinary function quantifiers Va, 3a; CSH contains in addition special 'choice' quantifiers Va, 3a for which it is required that in combinations Va3x, Va3(3 the x, resp. (3 depend continuously on a.

The elimination theorems now take the form:

THEOREM. (Kreisel and Troelstra, 1970; Troelstra, 1974). There is a mapping U' defined for formulae not containing choice parameters free such that for suitable H, CSH with H £ CSH

C~ f-U'(A)~A

CSHf-A~Hf-U'(A)

U'(A) == A for A E L[H].

(For certain choices of H, A must be restricted to formulae not containing quantifiers 3a.)

The study of these elimination mappings has led to interesting metamathematical results, such as: EL + Fan theorem (= intuitionistic form of Konig's lemma) is conservative over intuitionistic arithmetic (Troelstra, 1974).

AXIOMS FOR INTUlTlONISTlC MATHEMATICS 67

4.4. Already from traditional intuitionistic literature various striking mathematical consequences of the continuity postulates (primarily CONT 0) are known. The mathematical applications can be roughly divided into two groups

(a) theorems which are actually classically false when we interpret the choice quantifiers as ranging over classical number theoretic functions and apply classical logic; example: all real-valued functions on IR are continuous;

and

(b) theorems which also hold classically, but where the classical proof depends on classical logic, and the constructive proof uses the continuity axioms. Example: one direction of Riemann's permutation theorem: if all permutations of terms of a convergent series L Xn yield convergent series, then L Xn is absolutely convergent.

(For a detailed exposition of the examples see e.g. Troelstra, 1976, § 6.)

Combination of theorems of type (b) with the elimination theorem opens the following, largely unexplored possibility: after establishing a (classical) theorem A in CSH , we obtain a constructive version (in Bishop's sense) u(A) which can be established in H. Another method for converting a classical theorem with a classical proof, into a constructive version is provided by the Dialectica interpretation - the preceding method will be applicable in fewer cases, but may yield better results when applicable. This is clearly a subject for further investigation, and requires a hunt for suitable examples from mathematical practice.

4.5. There are many open problems regarding logical relationships between various forms of continuity schemata. For example, I do not know of an interpretation or model validating CONTo but not CONTI - although there are various models validating CONTo! but not CONT 0 (see e.g. J. R. Moschovakis, 1973; in combination with van Dalen, 1974).

68 A. S. TROELSTRA

Some attention has been given to 'Kripke's schema'

KS 30:[A ~3x(o:x = 0)] (0: not free in A).

KS is consistent with CONTo! (J. R. Moschovakis, 1973) but on the other hand inconsistent with CO NT h at least if we permit A in KS to contain choice parameters: in

V 1330: [V x(f3x = O)~ 3x(o:x = 0)]

0: cannot possibly depend continuously on 13.

5. EXAMPLE III: THE UNIFORMITY PRINCIPLE

5.1. This was first introduced, together with a discussion of its intuitive meaning, by Troelstra (1973 A)3.

We are primarily interested in the following two forms of the uniformity principle for numbers:

UP! VX3!xA(X, x) ~ 3xVXA(X, x)

UP VX3!xA(X, x) ~ 3xVXA(X, x)

and the uniformity principle for functions:

UP t VX30:A(X, 0:) ~ 30:VXA(X, 0:).

5.2. We briefly recapitulate the remarks on the plausibility of these principles from Troelstra (1973 A).

The argument is similar to plausibility arguments for continuity schemata after a rough description of choice sequences. 'VX3x' requires a method producing an x for each X. But sets are much more 'diffuse' than sequences: with sequences it is possible even if they are not given by a law, to approximate them to any desired degree of accuracy by constructing initial segments, whereas for sets we do not need to know even a single element belonging to the sets. Hence, if we


have to assign something so definite (so 'discrete') as a natural number to each set, it can only be done by assigning a number which serves the purpose for all sets.

There is a weakness in this rough argumentation: sets are 'diffuse' only when we think of their extension, but there is also the possibility of constructing an x from the definition of x, in the same way as a natural number assigned to each recursive function may depend on the gOdel number of the function. Now three remarks are relevant here.

(1) In the language of HAS we cannot refer to definitions of sets, only to their extensions. Hence we still expect, as is indeed the case, that HAS + UP is consistent.

(2) Quite apart from the restriction to the language of HAS, the possible effect of referring to definitions instead of exten!)ions, becomes considerably less in the case of an V X3!x-assertion, (which implies VX3xVz(A(X, x) & VYVy(Xy ++ Yy) & A(Y, z)~ x = z» hence this makes the preceding plausibility argument more cogent. Compare this to the compatibility of CT 0 and CONT oL

(3) Similarly, in HAS nothing precludes the sets to be 'choice' sets (Le. depending on non-lawlike parameters); and for a choice-set we cannot expect in general to be able to refer to its definition.

5.3. An illustration of remarks 20 and 30 is found in the following theorem and its proof:

THEOREM. CONTo! implies UP! (in HASI).

Proof. Assume VX3!xA(X, x); let X be a set and define

Z", = {x: Xx & 3x(ax = O)}. Then

Va3!xA(Z"" x)

and hence for some x, y

V(3 E (Az.l)yA(Z/3' x).

Apply this to (3 = Ax. 1 and (3' E (h.l)y *(0), then A (X, x), A(0, x) and thus VXA(X, x). •

70 A. S. TROELSTRA

5.4. Now we wish to make a few remarks concerning parametrization principles. The prime example is Kripke's schema KS; in L(HASI) this may be expressed as

VX3a[X ++3x(ax = 0)]

where X ranges over zero-place relations. Or, admitting a choice axiom ACoI :

VX3aVx[Xx++3y(a(x, y) = 0)].

All sets are parametrized by functions with the help of the predicate A(a, x)=3y(a(x, y) = 0).

As we shall demonstrate, CONTo+a parametrization principle implies UP relative to HASI:

THEOREM4 • Assume CONTo and a parametrization principle of the

form

PFA VX3aVx[A(a, x)++Xx].

(PF A: parametrization by functions via A.) Then UP holds.

Proof. (i) We first show that PF A implies a corresponding principle PF B where B satisfies (* also for concatenation of a finite sequence and an infinite one)

(1) VaVn3{3Vx[B(a, x)++B(n * (3, x)].

To see this, we define C(a, (3) ('{3 is a code for a') by

C(a, (3) =def V k[Vx3y > X«{3)kY = ak) /\ V z(z"# ak ~ -,Vx3y > X«{3)kY = z)].

We note that

(2) Va3{3C(a, (3)

(3) C(a, (3)/\ C(a', (3) ~ a = a'

(4) C(a, (3) ~ 3{3'C(a, n * (3').


(2) is immediate if, for any a, we take for {3 the sequence given by ({3)k = Ax.ak for all k. (3) is immediate; (4) is easily verified by choosing {3" such that ({3")kX = ({3)kX for all x> 1 th(n), and {3" En; then {3"= n * (3'.

Now if we put

B({3, x)=3a(C(a, (3)/\A(a, x»

then PF A obviously implies

PFB VX3{3Vx[B({3, x)++Xx]

and also (1) is satisfied because of (4).

(ii) For a B satisfying (1), PF B can be shown to imply UP. Assume

VX3xD(X, x);

by PFB

V{33xD(A.y.B({3, y), x)

and thus with CO NT 0, for some y E K

V{3D(A.y.B({3, y), y({3».

Assume yn# 0 for say (A.x.O)y = n. Since A.y.B(n * (3, y) ranges over all X (by (1», it follows that since V{3D(A.y.B(n*{3, y), yn -'-1),

3xVXD(X, x) •

We contrast this with

THEOREM. Any parametrization principle PF A implies the negation of a suitable instance of UP I

UP I VX3aB(X, a) ~ 3aVXB(X, a).

72 A. S. TROELSTRA

Proof Apply UP! to B(X, a)=def'v'X[A(a, X)BXX], then 3a'v'X'v'x[A(a, X)BXX]; and this implies 'v'XY(X = Y) which is obviously false.

As to the plausibility of UP!: this schema is analogous to UP, and the argument is once again that species are so 'diffuse' when compared with functions that 'v'X3aA(x, a) can only be satisfied by finding an a which does not depend on x such that 'v'XA(X, a).

5.5. Survey of Some Results

This section may be skipped by a reader who is only interested in the main ideas. Information on the consistency of UP!, UP, Upe (restriction of UP to closed instances) is contained in Troelstra (1973A), de Jongh-Smorynksi (1974), van Dalen (1974). Let us introduce some schemata IP hA ~ 3xB) ~ 3xhA ~ B) (x not free in A)

IP! hA ~ 3aB) ~ 3ahA ~ B) (a not free in A)

IPo 'v'x(A VIA) & ('v'xA ~ 3yB) ~ 3y('v'xA ~ B) (y not free in A)

AC-NS 'v'x3XA(x, X) ~ 3Y'v'xA(X, (Y)x).

THEOREM (de Jongh-Smorynski, 1974). (i) HAS + AC-NS + IPo + M + UP! + upe is consistent, and in fact

conservative over HASe + AC-NS (HASe : classical second-order arithmetic) w.r.t. arithmetical sentences. (Similarly if AC-NS is dropped on both sides.)

(ii) HAS! + AC-NS + ACQ1 + UP! + UP~ + IPo+ M is consistent.

THEOREM (van Dalen, 1974). HAS1+KS+CONTo!+CONTg+ UP + ACo! + UP! + UP~ is consistent.

THEOREM (Troelstra, 1973A). (i) HAS + CT 0 + IP 0 + M + UP + AC-NS is consistent

HAS + IP + CT 0 + AC-NS + UP is consistent. (ii) HAS+CONT! + IPo+ M + Up! + AC-NS is consistent

HAS + CONT 1 + IP + AC-NS + UP 1 is consistent. (Here too we can in fact obtain appropriate conservative extension results.)


In the models constructed by de Jongh-Smorynski, and van Dalen, UP t (with parameters) is false.

5.6. As to open problems connected with UP and UP h we have first of all

(i) Problem. Explore mathematical consequences of UP, UP t (not already obtainable by continuity postulates). We mention in this connection that as a result of UP, the sets of natural numbers do not permit a non-trivial apartness relation. (We call an apartness relation # non-tFivial iff 3x3y(x # y).) To see this, let X # Y for some X, Y; by the properties of an apartness relation V Z(X # Z v Y # Z) or equivalently VZ3x«X # Z & x = O)v(Y#Z & x;i' 0)), hence by UP VZ(X # Z) vV Z(Y # Z); each disjunct 'is falsifiable, taking X, Y for Z respectively. From the viewpoint of metamathematics there is a problem of obvious interest.

(ii) Problem. Suppose we introduce 'uniform' species quantification satisfying UP next to ordinary species quantification, Can we define an elimination mapping and prove elimination theorems similar to the choice-quantifier elimination in the preceding example?

5.7. Generalizing UP

It is natural to propose for suitable families <!J A

{X: A(X)}

of sets of natural numbers a relativized version of UP

UP A VX[A(X)~ 3xB(X, x)]~ 3xVX[A(X)~ B(X, x)].

UP A reduces to UP for A(X) == (0 = 0). It is quite obvious that A has to fulfill certain requirements: UP A is obviously false for the family of decidable sets, i.e. for A(X)== Vx(Xx v-,Xx). On the other hand, UP A

is trivially implied by UP for all families <!J A of the form

A(X)==:iYVx[Xx~ YXI\Cx]

(i.e. <!JA is the power set of {x: Cx}) since VX[A(X)~3yB(X, y)] is equivalent to

VY3yB(Yn Ax.CX, y);

74 A. S. TROELSTRA

with UP

3yVYB(YnAx.Cx, y)

hence 3yVX[A(X) ~ B(X, y)] (B in L(HAS)). In Troelstra (1973A), § 4 it is shown that HAS + UP A is consistent if A (X) == -,C(X) for suitable C.

Let us consider the property ''li A is amalgamated', expressed by

Amal (A)==VX E'liA VY E'li A -,VZ E'liA(X,t. Zv y,t. Z).

Parallel to the remark contained in 5.6 (i), it is easy to see that

UP A ~ Amal (A).

Conversely, it seems plausible to propose the general schema

UPo Amal (A) ~ UP A.

This schema cannot properly be termed a generalization of UP, since for the power set of N we cannot prove that it is amalgamated, so we cannot obtain UP as a special case of UP G.

Note that if g; A is such that in each X, Y E g; A the whole collection of Z", defined by

Z'" = {x: (Xx 1\3y(ay = 0)) v (Yx 1\ Vy(ay,t. O))}

belongs to 'li A, and CONT 0 holds, then Amal (A) readily follows (compare the argument in 5.3, and 5.6 (i)). On the other hand, Amal (A) expresses a much stronger property than the assertion that ''li A does not admit an apartness relation. In fact, if we take the 'li A

given by

A (X) == (X = {O}) v (XO 1\ Xl)

then for B(X,y)=(y=OI\-,XI)v(y=II\Xl) we have VXE 'liA3yB(X, y), but obviously not 3yVXE'liA B(X, y), although 'liA does not admit an apartness relation.


6. EXAMPLE IV: SUBCOUNTABILITY OF DISCRETE SETS

6.1. Our next example concerns a schema (indicated by SCDS) which has been investigated hardly at all - its logical relationship to UP and its mathematical consequences are unexplored.

Roughly, the principle of subcountability of Discrete Sets states

"A discrete collection (i.e. equality between elements of the collection is decidable) can be indexed by a subset of the natural numbers".

A stronger formulation requires in addition that the elements of the collection are distinct iff their indices are distinct.

In this generality, the principle is false: if we admit lawless sequences (e.g., Troelstra, 1976, § 2) as legitimate objects, we have e = TI v e ~ TI for lawless sequences e, TI, but we cannot find any predicate A containing (hidden or explicitly) only finitely many lawless parameters such that

Ve3xVyA(x, y, ey)/\ Vxyz[A(x, y, z) ~ 3!uA(x, y, u)].

(A countable model of the theory of lawless sequences does exist, but it is not countable in the language of the theory itself, d. Troelstra, 1970).

For families of sets of natural numbers the principle can be expressed in a weak form

SCDS' VXY(A(X)I\A(Y)~X= YvX~ Y)

~ 3ZV X(A(X) ~ 3x(X = (Z)x»'

Here (Z)x ={y: Zj(x, y)}. In SCDS', the family of sets considered is {X: A(X)}; the premiss assumes this family to be discrete. The conclusion asserts indexing by a subset of natural numbers, but not the stronger assertion that the index is uniquely determined (so as to make the index set itself 'mirror' the discreteness of {X: A(X)}). This stronger assertion can be expressed as

SCDS VXY(A(X)/\A(Y) ~ X = Yv X~ Y)

~ 3ZV X(A(X) ~ 3! x(X = (Z)x»'

76 A. S. TROELSTRA

6.2. In a sense, SCDS and SCDS' start where (a generalization of) UP stops. Under suitable syntactical restrictions on A we can show consistency of UP(A), as noted in 5.7.

Suppose now UP(A) holds'" for a class '€ of predicates A such that (modulo provable equivalence)

A(X)E '€~ A«X)y) E'€

A(X), B(X)E,€~A(X)AB(X)E'€.

The class of negated formulae is such a class '€; and it is provably consistent (see 5.7) to assume UP(A) for such a class.

Assume for an AE'€ VXY(A(X)AA(Y)-+X= YvX~ Y). This is equivalent to VX(A«X)o)AA«Xh) -+ (X)o= (X)1 v (X)o ~ (X)1).

Application of UP(B) for B(X)==A«X)o)AA«X)1) yields

\fXY(A(X)AA(Y) -+ X = Y) v\fXY(A(X)AA(y) -+ X~ Y).

The second case is excluded unless ?; A is empty. The first case expresses that ?Ji A is a singleton. So either ?Ji A is a singleton, or ?Ji A is empty; in both cases SCDS is trivially fulfilled.

6.3. A not entirely trivial instance of SCDS' can be obtained as follows. Assume A(x, X)==-,B, and let

\fXY(3xA(x,X)A3yA(y, Y)-+X= YvX~ Y). Then

VxyXY(A(X,X)AA(y, Y)-+X= YvX~ Y).

With the help of UP(B) for B == A (x, (X)o) A A (y, (Xh) we obtain

Vxy{VXY(A(x, X)AA(y, Y) -+ X = Y)

v\fXY(A(x, X)AA(y, Y) -+ X~ Y)}.

This shows that for any x ?; A is either empty or a singleton. So we can satisfy SCDS' taking for Z

(Z)" ={y: 3X(A(x, X)AXy)}.

AXIOMS FOR INTVTTTONISTIC MATHEMATICS 77

The argument as it stands does not yield SCDS. Of course this is only an example, it should not be hard to find other cases.

6.4. SCDS for sets of sequences

(This section, dealing with SCDS for families of sequences, is a digression which may be skipped by the reader.)

D. van Dalen observed the following

PROPOSITION. Let FIM be the system for intuitionistic analysis of Kleene and Vesley, 1965. Let B(l), n) be defined as

B(l), n) = (l)n = 1/Wm -< n(l)m = 0».

Then the following form of SDCS is consistent relative to FIM:

SDCS* Va/3[Aa AA/3 ~ a = /3 v a~ /3] ~ 3yVa[Aa ~ 3!n(B(y, n)

A a E nA Va' E n(Aa' ~ a = a'»)].

Here A is almost negative (i.e. does not contain v, and 3 only in front of prime formulae). In addition, EL* and FIM and many systems in between are closed under the rule corresponding to SDCS* (i.e. :? replacing the main ~ ) ..

Proof. FIM+GC is consistent (Troelstra, 1973,3.3.11) where GC is the schema (A almost negative)

GC Va[A'a ~ 3/3C(a, /3)] ~ 3yVa[A'a ~ !y I a A C(a, y I a)].

Assume V a/3 [Aa A A/3 ~ a = /3 v a ~ /3] and apply GC to

A'a = A(ita)AA(ha)

qa, f3) = (ita = ha A/30 = O)v(ita~ ha A/30 = 1),

then we find a y' such that

(1) Va/3[Aa A A/3 ~ !y'(a, /3)

A{(y'(a, /3)=0~ a = /3) & (y'(a, /3) ~O~ a~ /3)}]

78 A. S. TROELSTRA

(notations as in Troelstra, 1973). Let y be such that

Va(y'(a, a)=y(a)),

then for B as defined above

Va[Aa ~ 3!n(B(y, n)l\a E nl\ Va'E n(Aa'~ a = a')].

To see this, note that if Aa, then Aa 1\ Aa, hence for the y' of (1) y'(a, a)=O, so y(a)=O; i.e. y(ax) = 1I\Vy<x(y(ay)=O) for a suitable x etc. etc. •

To establish the rule corresponding to SCDS*, one should use closure of FIM* under the rule GCR corresponding to GC (cf. Troelstra, 1973, 3.7.9); in other respects the argument is similar.

Parallel to van Dalen's observation, we can establish the following more general, but weaker result:

PROPOSITION. For any Aa, there is a predicate C(y, a, m), such that the following schema

SCDS'* Vaf3[Aa 1\ Af3 ~ a = f3 va¥- f3]

~ 3yVa[Aa ~ 3m(C(y, a, m}1\ Va'(C(y, a', m)~ a = a'))]

is consistent relative to FIM.

Proof. We argue in FIM + GC, and assume results and notations in Troelstra (1973, §3.3). Assume Vaf3[AaI\Af3~a=f3va¥-f3]. In FIM+GC

3 S(Srl(Aa 1\ Af3))++ Aa 1\ Af3,

where Sr1(Aa 1\ Af3) is almost negative. Now apply GC to Vaf3 S [Srl(Aa 1\ Af3) ~ a = f3 va¥- f3], then we find a y such that

(1) Vaf3 S[Srl(Aa I\Af3) ~ !y(8, a, f3)

1\ (y(8, a, f3) = 0 ~ a = f3) 1\ (y(8, a, f3) = 1 ~ a¥- f3)].


L b d· f N 3 N' h . ·3 ·3·3 d I k 3 et V3 e a co mg 0 onto, WIt mverses]1,]2,]3; an et i,

i = 1, 2, 3 be such that

k~O=O, i = 1,2,3. We put

Assume Aa; then for some 5, 5rl(AaI\Aa), and thus 'Y(5, a, a)=O, i.e. for a suitable m C( 'Y, a, m). If also C( 'Y, a', m) it follows that for some 5' 5'r1(Aa'l\ Aa'}, v3(5', a', a') Em, 'Ym = 1; and thus 5,5' E kim. For 5"= j(it5,iz5') it follows that 5"E kim, 5"r1(AaI\Aa'); also a, a'E k3 k3 H (5:" ') d (5:" ') - O' -' 2m = 3m. ence V3 u , a, a Em, an so 'Y u , a, a ,- , I.e. a - a (since 5'r1(Aa'l\ Aa') implies Aa'). •

6.5. It is easy to see that SCDS' also refutes certain consequences of classical logic: take for A (X) == V YZ( Y = Z v Y ~ Z). Then obviously A (X) & A (y) ""+ X = Y v X ~ Y. The conclusion of SCDS' is obviously false if we assume A(X) for some X, because then VYZ(Y=Zv Y~ Z); hence VXA(X), but then for some Z VX3x(X = (Z)x) which is easily refuted by a diagonal argument.

6.6. As noted by Friedman (1975) SSDC is a special case of the assertion: for every metric space there is a subcountable set of points dense in the space. This is seen by regarding {X: A(X)} with A (X) & A (Y) ""+ X = Y v X ~ Y as a space with a discrete metric (p(X, Y)= 1 if X~ Y, 0 otherwise); it follows that {X: A(X)} must be subcountable.

6.7. As to the intuitive plausibility arguments for SCDS (and similarly for the assertion on metric spacess) we can hardly say more than that all examples of discrete collections we know are in fact subcountable (if we bypass lawless sequences etc.).

7. EXAMPLE V: GENERALIZATIONS OF SCDS

7.1. The idea behind UP was: a mapping of something very diffuse (sets) into something very discrete (the natural numbers) must be

80 A. S. TROELSTRA

constant; and for UP t : a mapping of something very diffuse into something separable is constant. For SCDS': a discrete subfamily of something very diffuse is subcountable. The analogy might now suggest:

SCDS~ V XY[A(X) 1\ A( Y) ~ 3a{(Vx(ax = 0) ~ X = Y)

1\ (-lVx(ax = 0) ~ X f= Y)}] ~ 3ZVX[A(X) ~ 3a(X = (Z)a)]

For the special case where {X: A(X)} can be provided with a metric the premiss is true; the hypothesis that every metric space has a subcountable set of points dense in the space then justifies the conclusion. (Let {Xi: i E f}, f £; N, be dense in {X: A(X)}, and let p be a metricon{X:A(X)}; then X(a, (3)=3X(lim p(Xam X)=OI\A(X)1\ Xj3 1\

n ..... oo

Vn(an E f» satisfies the conclusion.) However, the premise of SCDS~ might hold also in cases where {X: A(X)} cannot be provided with a metric, at least this possibility is not excluded.

7.2. Note that reinforcing 3a to 3!a does not give us an acceptable strengthening of SCDS~ to SCDS, since e.g. IR, a connected space, can be represented as {X: A(X)}; and 3!a would imply a mapping into (totally disconnected) Baire space, in strong contrast with continuity theorems. The following weakened variant seems to be more plausible:

SCDS t VXY[A(X) 1\ A(Y) ~

3a{(Vx(ax=0)~X= Y)I\(-,Vx(ax=O}~Xf= Y)}l ~3'YZVX[A(X)~

3a{(Z)a = XI\ Va/3«Z)a = (Z)j3 ~Vx('Y(ax, ~x) = O»)}].

8. RULES CORRESPONDING TO THE AXIOMS

8.1. The various axioms mentioned of varying degrees of plausibility, suggest that we should look, for familiar systems, for closure under the corresponding rule (where the main implication is replaced by metamathematical ~: when the premise is derivable, then so is the conclusion). In the case of the first three examples, this means closure


under Church's rule, Continuity rule, and the Uniformity rule. Closure under these rules has indeed been established for many systems; for some examples see Troelstra (1973, 1973A) and de Jongh-Smorynski (1974).

As to SCDS, for the usual systems (e.g. HAS) we cannot possibly expect to establish the corresponding rule:

SCDS-Rule

f-VXY(A(X) & A(Y)~ X= Yv X~ Y) ~f-3ZVX(A(X) ~ 3! x«Z)x = X)).

For if we take

A(X)=VYZ(Y= Zv Y~ Z)

then the premise is certainly provable in HAS, but the conclusion can only be true if -,3XA (X), i.e. we should be able to show -,V XY(X = Y v X ~ Y) which is impossible since HAS £; HASe (i.e. HAS is consistent with classical logic).

8.2. Only by reinforcing HAS we can expect the rule to hold; in our example, HAS needs to be reinforced by -,V XY(X = Yv X ~ Y). A similar remark applies, mutatis mutandis, to rules corresponding to: every metric space has a subcountable set dense in the space. (Requiring A(X) to be non-empty is not sufficient as a restriction: take e.g. A(X)=[X={2n:nEN}v(X£;{2n+l:nEN} & VYZ(Y=Zv

y~ Z»)].)

9. COMPLETENESS OF INTUITIONISTIC PREDICATE LOGIC

9.1. For intuitionistic propositional logic, and certain fragments of predicate logic, we can establish completeness relative to validity in structures, if we permit the domains and relations of those structures to contain lawless sequences (Troelstra, 1976, § 7). The usefulness of such results is limited, since lawless sequences as such do not play an important role in (intuitionistic) mathematical practice. So we are also interested in completeness results for structures where the parameters

82 A. S. TROELSTRA

range over other classes of objects with properties more relevant to mathematical practice.

9.2. The assumption of CTo yields incompleteness (Kreisel; see van Dalen (1973». Similarly, the usual continuity axioms (e.g. CONTt), although they can be used, just as CTo, to refute many well-known theorems from classical predicate logic, are not sufficient to refute even all unprovable propositional schemata. For example, Kleene (1965) noted that in his formalization of intuitionistic analysis

IPv (-,A ~ BvC)~ (-,A ~ B)v(-,A ~ C)

is special-realizable, hence not provably refutable. This holds even in case we assume \fa-,-ax E R(a = {x}) (R set of the Godel numbers of total recursive functions) and CTo; we refrain from giving a proof here.

Similarly, UP+ IPv is consistent with HAS (Troelstra, 1973A, § 4). Thus we are led to ask for significant (sub-)classes of formulae of intuitionistic predicate logic for which the axioms mentioned in our examples yield completeness; this problem is still wide open.

10. FINAL REMARKS

10.1. The examples presented are by no means the only ones which have been mentioned or proposed; for variants of the continuity axioms (local continuity) see e.g. Beeson (B).

10.2. Our examples are all of the same general type; they consist of an implication where the premise because of the constructive ('strong') interpretation of 3 and \f3 permits a conclusion which is stronger than in the classical case.

10.3. One should not overestimate the impact of the axioms in our examples on the existing practice of constructive mathematics. For example, since in practice the theory of metric spaces is restricted to separable ones, one should not expect a great impact of the axiom that all complete metric spaces are separable. No doubt a lot of inventiveness and imagination will be needed to find areas of mathematics


where these axioms do have an impact (compare this with the fact that in classical mathematics one hardly ever uses strong forms of impredicative comprehension).

Mathematical Institute, University of Amsterdam

NOTES

1 These properties are in themselves neither necessary nor sufficient to guarantee the 'constructive' character of a theory; see Troelstra (1973A, 1.8). 2 We refrained from discussing e.g. lawless sequences, for which an informally rigorous justification of the axioms can be given (Troelstra, 1976, § 2), but concentrated on 'plausible' axiom schemata. 3 The corresponding u~iformity rule for intuitionistic predicate logic had already attracted attention before - see e.g. Kreisel (1971, p. 145). 4 This generalizes a remark by van Dalen (1974A, 1975A). 5 This possible new axiom seems to have suggested itself to a number of people, and may perhaps be said to belong to 'folklore'. I do not know where it was first formulated explicitly; in any case, the Appendix A of Bishop (1967) is very suggestive in this respect.

BIBLIOGRAPHY

Beeson, M. J.: 1972, Metamathematics of Constructive Theories of Effective Operations, Thesis, Stanford University. (See also Beeson, 1975.)

Beeson, M. J.: 1975, 'The Nonderivability in Intuitionistic Formal Systems of Theorems on the Continuity of Effective Operations', lourn. Symbolic Logic 40, 321-346.

Beeson, M. J.: A, 'The Unprovability in Constructive formal Systems of the Continuity of Effective Operations on the Reals'. To appear.

Beeson, M. J.: B, 'Principles of Continuous Choice and Continuity of Functions in Formal Systems for Constructive Mathematics'. To appear. ,,-

Bishop, E.: 1967, Foundations of Constructive Analysis, McGraw Hi\I, New York. Ceitin, G. S.: 1959, 'Algorithmic Operators in Constructive Complete Separable Metric

Spaces' (Russian), Doklady Akad. Nauk 128, 49-52. Dalen, D. van: 1973, 'Lectures on Intuitionism', in A. R. D. Mathias, H. Rogers (eds.),

Cambridge Summer School in Mathematical Logic, Springer-Verlag, Berlin, pp. 1-94. Dalen, D. van: 1974A, 'Choice Sequences in Beth models', Notes on Logic and

Computer Science 20, Dept. of Mathematics, Rijksuniversiteit, Utrecht (The Netherlands). Revised in 'An Interpretation of Intuitionistic Analysis', to appear in Annals of Math. Logic.

Dalen, D. van: 1975, 'Experiments in Lawlessness', Notes on Logic and C()mputer Science 28, Dept. of Mathematics, Rijksuniversiteit, Utrecht (The Netherlands). Revised in 'An Interpretation of Intuitionistic Analysis', to appear in Annals of Math. Logic.

84 A. S. TROELSTRA

Dalen, D. van: 1975A, 'The Use of Kripke's Schema as a Reduction Principle', Preprint 11, Dept. of Mathematics, University of Utrecht.

Dragalin, A G.: 1973, 'Constructive Mathematics and Models of Intuitionistic Theories', in P. Suppes, L. Henkin, Gr. C. Moisil, A Joja (eds.), Logic, Methodology and Philosophy of Science IV, North-Holland Pub!. Co., Amsterdam, 111-128.

Friedman, H.: 1975, 'Set Theoretic Foundations for Constructive Analysis and the Hilbert Program, Manuscript, Dept. of Mathematics, State University of New York at Buffalo, Buffalo, N.Y.

Jongh, D. H. J. de and Smorynski, C. A: 1974, 'Kripke Models and the Theory of Species'. Report 74-03, Dept. of Mathematics, Univ. of Amsterdam. Appeared in: Annals of Mathematical Logic 9 (1976), 157-186.

Kleene, S. c.: 1965, 'Logical Calculus and Realizability', Acta Philosophica Fennica 18, 71-80.

Kleene, S. C. and Vesley, R. E.: 1965, The Foundations of Intuitionistic Mathematics, Especially in Relation to Recursive Functions, N~rth-Holland Pub!. Co., Amsterdam.

Kreisel, G.: 1968, 'Lawless Sequences of Natural Numbers', Compositio Math. 20, 222-248.

Kreisel, G.: 1971, 'A Survey of Proof Theory II', in J.-E. Fenstad (ed.) Proceedings of the Second Scandinavian Logic Symposium, North-Holland Pub!. Co., Amsterdam, pp. 109-170.

Kreisel, G.: 1972, 'Which Number-Theoretic Problems can be Solved in Recursive Progressions on Ill-Paths Through 01', J. Symbolic Logic 37,311-344.

Kreisel, G. and Troelstra, A. S.: 1970, 'Formal Systems for Some Branches of Intuitionistic Analysis', Annals of Math. Logic 1,229-387.

Moschovakis, J. R.: 1973, 'A Topological Interpretation of Second-order Intuitionistic Arithmetic', Compositio Math. 26, 261-275.

Moschovakis, Y. N.: 1964, 'Recursive Metric Spaces', Fund. Math. 55, 215-238. Orevkov, V. P.: 1963, 'A Constructive Map of the Square into itself, which Move~ every

Constructive Point' (Russian), Doklady Akad. Nauk SSSR 152, 55-58; translated in Soviet Mathematics 4 (1963), 1253-1256.

Orevkov, V. P.: 1964, 'On Constructive Mappings of a Circle into itself' (Russian), Trudy Mat. Inst. Steklov 72, pp. 437-461; translated in Translations AMS 100, 69-100.

Troelstra, AS.: 1969, Principles of Intuitionism, Springer-Verlag, Berlin. Troelstra, A S. (ed.): 1973, Metamathematical Investigation of Intuitionistic Arithmetic

and Analysis, Springer-Verlag, Berlin. Troelstra, AS.: 1973A, 'Notes on Intuitionistic Second-Order Arithmetic', in A R. D.

Mathias, H. Rogers (cds.), Cambridge Summer School in Mathematical Logic, Springer-Verlag, Berlin, 171-205.

Troelstra, AS.: 1976, Choice Sequences, a Chapter of Intuitionistic Mathematics, Oxford University Press, Oxford. To appear.

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