On uniform weak König's lemma

Annals of Pure and Applied Logic 114 (2002) 103–116www.elsevier.com/locate/apal

On uniform weak K$onig’s lemmaUlrich Kohlenbach

Basic Research in Computer Science (BRICS), funded by the Danish National Research Foundation,Department of Computer Science, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark

Dedicated to Anne S. Troelstra on his 60th birthday

Abstract

The so-called weak K$onig’s lemma WKL asserts the existence of an in0nite path b in anyin0nite binary tree (given by a representing function f). Based on this principle one can for-mulate subsystems of higher-order arithmetic which allow to carry out very substantial partsof classical mathematics but are 20

2-conservative over primitive recursive arithmetic PRA (andeven weaker fragments of arithmetic). In Kohlenbach [10] (J. Symbolic Logic 57 (1992) 1239–1273) we established such conservation results relative to 0nite type extensions PRA! of PRA(together with a quanti0er-free axiom of choice schema which—relative to PRA!—implies theschema of =0

1-induction). In this setting one can consider also a uniform version UWKL ofWKL which asserts the existence of a functional � which selects uniformly in a given in0nitebinary tree f an in0nite path �f of that tree. This uniform version of WKL is of interestin the context of explicit mathematics as developed by S. Feferman. The elimination processin Kohlenbach [10] actually can be used to eliminate even this uniform weak K$onig’s lemmaprovided that PRA! only has a quanti0er-free rule of extensionality QF-ER instead of the fullaxioms (E) of extensionality for all 0nite types. In this paper we show that in the presence of(E), UWKL is much stronger than WKL: whereas WKL remains to be 20

2-conservative overPRA, PRA! + (E) + UWKL contains (and is conservative over) full Peano arithmetic PA. Wealso investigate the proof–theoretic as well as the computational strength of UWKL relative tothe intuitionistic variant of PRA! both with and without the Markov principle. c© 2002 ElsevierScience B.V. All rights reserved.

MSC: 03F10; 03F25; 03F35; 03F50; 03D65

Keywords: K$onig’s lemma; Higher order arithmetic; Functionals of 0nite type; Fan rule;Explicit mathematics

1. Introduction

The binary (so-called ‘weak’) K$onig’s lemma WKL plays an important role in theformulation of mathematically strong but proof–theoretically weak subsystems of anal-ysis. In particular, the fragment (WKL0) of second-order arithmetic which is based on

E-mail address: [email protected] (U. Kohlenbach).

0168-0072/02/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S0168 -0072(01)00077 -X

104 U. Kohlenbach / Annals of Pure and Applied Logic 114 (2002) 103–116

recursive comprehension (with set parameters), =01-induction (with set parameters) and

WKL occurs prominently in the context of reverse mathematics (see [18]). Although(WKL0) allows to carry out a great deal of classical mathematics, it is 20

2-conservativeover primitive recursive arithmetic PRA, as was shown 0rst by H. Friedman using amodel–theoretic argument. In [17] a proof–theoretic argument is given for a variant of(WKL0) which uses function variables instead of set variables. In [10] we establishedvarious conservation results for WKL relative to subsystems of arithmetic in all 0nitetypes. As a special case these results yield that

(1) E-PRA! + QF-AC1;0 + QF-AC0;1 + WKL is 202-conservative over PRA;

where E-PRA! + QF-AC1;0 + QF-AC0;1 + WKL is a 0nite type extension of (WKL0)(see below for a precise de0nition). The proof of this fact relies on a combination ofG$odel’s functional interpretation with elimination of extensionality (see [14]), negativetranslation and Howard’s [8] majorization technique. The 0rst step of the proof reducesthe case with the full axiom of extensionality to a subsystem WE-PRA! + QF-AC1;0 +QF-AC0;1 +WKL which is based on a weaker quanti0er-free rule of extensionality only(see below) which was introduced in Spector [19]. From this system, WKL is theneliminated. This elimination actually eliminates WKL via a strong uniform version ofWKL, called UWKL below, which states the existence of a functional which selectsuniformly in a given in0nite binary tree an in0nite path from that tree. This yields thefollowing conservation result (which is not stated explicitly in [10] but which can beobtained from the proofs in Section 4 of that paper, see below):

(2) WE-PRA! + QF-AC + UWKL is 202-conservative over PRA:

In this weakly extensional context based on a quanti0er-free rule of extensionality ‘+’must be understood in the sense that the axioms QF-AC and UWKL must not be usedin the proof of a premise of an application of the extensionality rule. 1 For WE-PA!

we get the following result (with the same convention on + as above)

(3) WE-PA! + QF-AC + UWKL is conservative over PA; where PA denotes

full 0rst-order Peano arithmetic:

Result (2) is of interest in the context of the so-called explicit mathematics as developedby Feferman (starting with [3]) and further investigated also by A. Cantini, G. J$agerand T. Strahm among others, since the uniform weak K$onig’s lemma UWKL seems tobe a very natural ‘explicit’ formulation of WKL. We have been asked about the status

1 See [10] (where we use a special symbol ‘⊕’ to emphasize this point) for details on this, and [13],where we show that without this restriction weakly extensional systems would violate the deduction theoremalready for closed 20

1-axioms. Actually it is suQcient to impose this restriction on the use of the additionalaxioms for UWKL only.The conservation results in [10] are much more general than the one we mentioned. This makes the proofsmore involved than is needed for the special (20

2-) case relevant here. A corresponding simpli0cation of ourargument has been worked out in [1].

U. Kohlenbach / Annals of Pure and Applied Logic 114 (2002) 103–116 105

of UWKL in the presence of full extensionality. In this note we give a surprisinglysimple answer to this question showing, in particular, that

(4) E-PRA! + QF-AC1;0 + QF-AC0;1 + UWKL contains (and is

conservative over) PA

and

(5) E-PA! + QF-AC1;0 + QF-AC0;1 + UWKL has the same strength as

(201-CA)¡0 :

In the 0nal section, we investigate the status of UWKL in the context of the in-tuitionistic variant E-P(R)A!i of E-P(R)A!. In [12] we have shown that many non-constructive function(al) existence principles can be added to systems like E-PRA!

i

without changing the growth rates of the provable (not only the provably recursive)functions of the system. This is true although the proof–theoretic strength of the result-ing ‘hybrid’ systems is as strong as that of their counterpart with full classical logic.We apply this to UWKL and show that if a sentence ∀x0∃y0A(x; y) is provable in E-PRA!

i + AC + UWKL, then one can construct a primitive recursive bounding function∀x∃y6p(x)A(x; y) (here A is of arbitrary logical complexity). Moreover, this systemis closed under the so-called fan rule. This even holds in the presence of a strongindependence-of-premise principle IP¬ for negated formulas but fails if the Markovprinciple M for numbers is added:

Every �(¡)-recursive function is provably recursive in E-PRA!i + UWKL + M.

2. Preliminaries

The set T of all 0nite types is de0ned inductively by

(i) 0 ∈ T and (ii) �; � ∈ T⇒ �(�) ∈ T:

Terms which denote a natural number have type 0. Elements of type �(�) are functionswhich map objects of type � to objects of type �.

The set P⊂T of pure types is de0ned by

(i) 0 ∈ P and (ii) � ∈ P⇒ 0(�) ∈ P:

Brackets whose occurrences are uniquely determined are often omitted, e.g. we write0(00) instead of 0(0(0)). Furthermore we write for short ��k : : : �1 instead of �(�k) : : :(�1). Pure types can be represented by natural numbers: 0(n) := n + 1. The types0; 00; 0(00); 0(0(00)) : : : are so represented by 0; 1; 2; 3; : : : : For arbitrary types �∈Tthe degree of � (for short deg(�)) is de0ned by deg(0) := 0 and deg(�(�)) := max(deg(�); deg(�) + 1). For pure types the degree is just the number which representsthis type.


The system E-PRA! is formulated in the language of functionals of all 0nite typesand contains ��;�; ��;�; �-combinators for all types (which allows one to de0ne �-abstraction) and all primitive recursive functionals in the sense of Kleene (i.e. primitiverecursion is available only on the type 0). Furthermore, E-PRA! contains the schemaof quanti0er-free induction

QF-IA: A0(0) ∧ ∀x(A0(x)→ A0(x′))→ ∀xA0(x);

where A0 is quanti0er-free, as well as the axioms of extensionality

(E): ∀x�; y�; z��(x =� y → zx =� zy)

for all 0nite types (where for �= 0�k : : : �1; x=� y is de0ned as ∀z�11 ; : : : ; z

�kk (xz1 : : : zk

=0 yz1 : : : zk)). 2 We only include equality =0 between numbers as a primitive predicate.So E-PRA! essentially is PA

!�+ (E), where PA

!� is Feferman’s system from [4].

E-PA! is the extension of E-PRA! obtained by the addition of the schema of fullinduction and all (impredicative) primitive recursive functionals in the sense of G$odel[6] and coincides with Troelstra’s [20] system (E-HA!)c.

The ‘weakly extensional’ 3 versions WE-PRA! and WE-PA! of these systems resultif we replace the extensionality axioms (E) by a quanti0er-free rule of extensionality(due to Spector [19])

QF-ER:A0 → s =� t

A0 → r[s] =� r[t];

where A0 is quanti0er-free, s�; t�; r[x�]� are arbitrary terms of the system and �; �∈ arearbitrary types.

Note that QF-ER allows one to derive the extensionality axiom for type 0, butalready the extensionality axiom for type-1-arguments, i.e.

∀z2∀x1; y1(x =1 y → zx =0 zy)

is underivable in WE-PA! (see [8]).In the last section of this paper we will also need the intuitionistic versions WE-

PRA!i and E-PRA!

i of WE-PRA! and E-PRA!.The schema of choice is given by

AC�;�: ∀x�∃y�A(x; y)→ ∃Y �(�)∀x�A(x; Yx); AC :=⋃�;�∈T

{AC�;�};

where A is an arbitrary formula.The restriction of AC to quanti0er-free formulas A0 is denoted by QF-AC.

2 We deviate slightly from our notation in [11]. The system denoted by E-PRA! in the present paperresults from the corresponding system in [11] if we replace the universal axioms (9) in the de0nition of thelatter by the schema of quanti0er-free induction.

3 This terminology is due to [20].


Remark 2.1. WE-PRA! + QF-AC0;0 =01-IA; �0

1-CA, where

=01-IA: ∃y0A0(0; y) ∧ ∀x0(∃y0A0(x; y)→ ∃y0A0(x′; y))→ ∀x∃yA0(x; y)

and

�01-CA: ∀x0(∃y0A0(x; y)↔ ∀y0B0(x; y))→ ∃f1∀x(fx = 0↔ ∃yA0(x; y))

with A0; B0 quanti0er-free (parameters of arbitrary types allowed).

So the system RCA0 from reverse mathematics (see [18]) can be viewed as a sub-system of WE-PRA! + QF-AC0;0 by identifying sets X ⊆N with their characteristicfunction.

In the following we use the formal de0nition of the binary (‘weak’) K$onig’s lemmaas given in [21] (see also [22]; here ∗; Tbx; lth(n) refer to the primitive recursive codingof 0nite sequences from [20]):

De�nition 2.2 (Troelstra [21]).

Tf :≡∀n0; m0(f(n ∗ m) =0 0→fn=0 0) ∧ ∀n0; x0(f(n ∗ 〈x〉) =0 0→ x601)

(i.e. T (f) asserts that f represents a binary tree)

T∞(f) :≡T (f) ∧ ∀x0∃n0(lth(n) = x ∧ fn= 0);

(i.e. T∞(f) expresses that f represents an in0nite binary tree),

WKL :≡∀f1(T∞(f)→∃b1∀x0(f(Tbx) = 0)):

De�nition 2.3. The uniform weak K$onig’s lemma UWKL is de0ned as

UWKL :≡ ∃�1(1)∀f1(T∞(f)→ ∀x0(f((�f)x) = 0)):

Instead of the full uniform version UWKL of WKL one can also consider a sequen-tially uniform version WKLseq which asserts the existence of a sequence of in0nitepaths bi in fi for a sequence of in0nite binary trees (fi)i∈N:

De�nition 2.4. WKLseq :≡ ∀f1(0)(·) (∀i0T∞(fi)→ ∃b1(0)

(·) ∀i; x(fi(Tbix) = 0)).

However, WKLseq is (in contrast to UWKL) derivable in E-PRA! + WKL (seeProposition 3.1).

3. Results in the classical case

We 0rst show that WKLseq is not stronger than WKL relative to WE-PRA!:

Proposition 3.1. WE-PRA! WKL→WKLseq.


Proof.4 Let f1(0)(·) be such that ∀iT∞(fi). Using the Cantor pairing function j we

de0ne

f(n) ={

00 if ∀i((n)i 6 1) ∧ ∀i; k(j(i; k) 6 lth(n)→ fi(�l:(n)j(i;l)(k)) = 0);10 otherwise:

Since ‘∀i’ and ‘∀i; k’ can be bounded primitive recursively in n; f can be uniformlyde0ned in f by a closed term of WE-PRA!. It is easy to show (using basic propertiesof j) that T∞(f). Hence WKL yields a function b 6 �x:1 such that ∀x(f(Tbx) = 0).This implies (using again basic properties of j) that

∀i; x(fi(�l:b(j(i; l))(x)) = 0)

and so �i; l:b(j(i; l)) satis0es WKLseq.

We now switch to the uniform weak K$onig’s lemma, UWKL, which is not deriv-able in E-PA! + QF-AC + WKL already for continuity reasons: the type structureECF of all extensional continuous functionals as de0ned in [20] forms a model ofE-PA! + QF-AC + WKL (see [20, 2.6.5, 2.6.20]), whereas UWKL implies the exis-tence of a noncontinuous functional (see the proof of Proposition 3.4). Nevertheless,for the weakly extensional systems WE-PRA! and WE-PA! we have the followingconservation results for UWKL:

Theorem 3.2. (1) WE-PRA! + QF-AC + UWKL is 202-conservative over PRA.

(2) WE-PA! + QF-AC + UWKL is conservative over PA.(Here; again; + must be understood in the sense of (2) in Section 1).

Proof. (1) In [10, (4:2)–(4:7)], we constructed a primitive recursive functional

f1; g1 �→ 'fg := ( fg) such that

(1) WE-PRA! ∀f; g T∞('fg)

and

(2) WE-PRA! ∀f(T∞(f)→ ∃g(f =1 'fg)):

By the proof of Theorem 4:8 in [10] (and the fact that WE-PRA! is 202-conservative

over PRA), it follows that

WE-PRA! + QF-AC + UWKL∗ is 202-conservative over PRA;

where

UWKL∗ := ∃B∀f; g; x(('fg)((Bfg)x) =0 0):

4 See also the proof of Theorem IV:1:8 in [18] where a similar argument is used.


It remains to show that

WE-PRA! UWKL∗ → UWKL:

The proof of (2) in [10, (4:7)] shows that g can be primitive recursively de0ned in fas

f(x) :={

min n6 11x[lth(n) = x ∧ f(n) = 0] if such an n exists;00 otherwise:

Thus for (f := '(f; f)

(2)′ WE-PRA! ∀f(T∞(f)→ f =1 (f):

De0ne �f :=B(f; f) for B satisfying UWKL∗. Then

∀x(((f)((�f)x) =0 0)

and so for f such that T∞(f) (which implies f=1 (f)

∀x(f((�f)x) =0 0);

i.e. � satis0es UWKL.(2) As in (1) we obtain from the proof of 4.8 in [10] that

WE-PA! + QF-AC + UWKL is ∀x�∃y0A0(x; y)-conservative over WE-PA!;

where x� is a tuple of variables of type levels 6 1; A0 is quanti0er-free and containsonly x; y as free variables. Now let A be a sentence in the language of PA which canbe assumed to be in prenex normal form and assume that

WE-PA! + QF-AC + UWKL A:Then a fortiori

WE-PA! + QF-AC + UWKL AH;

where AH is the Herbrand normal form of A. By the conservation result just mentionedwe get

WE-PA! AH

and therefore by [9, Theorem 4:1]

PA A:

Remark 3.3. The passage from the provability of AH to that of A used in the proof of(2) above does not apply to WE-PRA! and PRA (see [9] for a counterexample). Indeed,already WE-PRA! + QF-AC0;0 is not 20

3-conservative over PRA: the former theoryproves the schema of =0

1-collection =01-CP, but it is known that there are instances


of =01-CP (which always can be prenexed as 20

3-sentences) 5 which are unprovable inPRA (see [16]).

We now show that the picture changes completely if we consider the systemsE-PRA! and E-PA! with full extensionality instead of WE-PRA!, WE-PA!. This phe-nomenon is due to the following

Proposition 3.4. E-PRA! UWKL↔∃’2∀f1(’f=0 0↔∃x0(fx=0 0)).

Proof. (1) ‘→’: We 0rst show that any � satisfying UWKL is—provably in E-PRA!—(eWectively) discontinuous, 6 i.e.

E-PRA!

∀�1(1)(∀f1(T∞(f)→ ∀x0(f((�f)x) =0 0))

→ ∃g1(0)(·) ; g

1(T∞(g) ∧ ∀iT∞(gi) ∧ ∀i∀j ¿ i(gj(i) =0 g(i))∧∀i; j(�(gi; 0) = �(gj; 0) �= �(g; 0))))

and, moreover, g(·); g can be computed uniformly in � by closed terms of E-PRA!.De0ne g primitive recursively such that

g(k) ={

0 if ∀m ¡ lth(k)((k)m = 0) ∨ ∀m ¡ lth(k)((k)m = 1);1 otherwise:

g represents a tree with two in0nite paths, corresponding to an in0nite sequence of 0’sand an in0nite sequence of 1’s. So it is clear that (provably in E-PRA!) T∞(g). Nowlet �1(1) be such that

∀f1(T∞(f)→ ∀x(f((�f)x) =0 0)):

Case 1: �(g; 0) = 0. De0ne a primitive recursive function �i; k:gi(k) such that

gi(k) =

0 if [lth(k) 6 i ∧ ∀m ¡ lth(k)((k)m = 0)]∨[∀m ¡ lth(k)((k)m = 1)];

1 otherwise:

gi represents the same tree as g except that the left branch has been truncated at level i.So again we easily verify within E-PRA! that ∀iT∞(gi). From the construction of giand g it is clear that

∀k∀l¿ lth(k)(gl(k) = g(k)):

Since our coding has the property that lth(k) 6 k, we get

∀k∀l¿ k(gl(k) = g(k)):

5 Here PRA is understood not as a quanti0er-free theory but with full 0rst-order predicate logic.6 The term ‘eWectively discontinuous’ is due to [7] on which we rely in the second part of our proof.


Since �x:1 is the only in0nite path of the binary tree represented by gi, it follows that

∀i(�(gi; 0) = 1);

Case 2: �(g; 0) = 1. The proof is analogous to case 1 with

gi(k) =

0 if [lth(k) 6 i ∧ ∀m ¡ lth(k)((k)m = 1)]∨[∀m ¡ lth(k)((k)m = 0)];

1 otherwise:

This 0nishes the proof of the discontinuity of �. We now show—using anargument from [7] known as ‘Grilliot’s trick’ 7 —that the functional ’2 de0ned by(+)∀f1(’f=0 0↔∃x(fx=0 0)) can be de0ned primitive recursively in � in such away that (+) holds provably in E-PRA!:

We can construct a closed term t1(1) of E-PRA! such that (provably in E-PRA!) wehave

thi ={gj(i) for the least j ¡ i such that h(j) ¿ 0; if such a j existsgi(i) otherwise:

Together with ∀i∀j ¿ i(gj(i) = gi(i)) this yields

∃j(h(j) ¿ 0)→ th =1 gj for the least such j

and together with ∀i(gi(i) = g(i))

∀j(h(j) = 0)→ th =1 g:

Hence using the extensionality axiom for type-2-functionals we get 8

∀j(h(j) = 0)↔ �(th; 0) =0 �(g; 0):

So ’ := �h1:sg ◦ |�(t(sg ◦ h); 0) − �(g; 0)| where sg(x) := 0 for x �= 0 and sg(x) := 1otherwise, does the job.

We now combine the two constructions of ’ corresponding to the two cases aboveinto a single functional which de0nes ’ primitive recursively in �: Let + be a closedterm such that

E-PRA! ∀x0((x =0 0→ +x =1(1) t) ∧ (x �= 0→ +x =1(1) t));

where t is de0ned as above with gi from case 1 whereas t is de0ned analogouslybut with gi as in case 2. Then de0ne ’ := �h1:sg ◦ |�((+(�(g; 0))(sg ◦ h); 0)−�(g; 0)|.(2) ‘←’: Primitive recursively in ’ one can easily compute a functional � which selects

7 This argument plays an important role in the context of the Kleene=Kreisel countable functionals. See[15], whose formulation of it we adopt here.

8 It is the direction ‘→’ which needs (E). The direction ‘←’ can be shown using only QF-ER, since freevariables are allowed to occur in premises A0 of QF-ER.


an in0nite branch of an in0nite binary tree (for example, the leftmost in0nite branch,in particular).

Corollary to the proof of Proposition 3.4. One can construct closed terms t1; t2 of E-PRA! such that

E-PRA! { ∀�1(1)(∀f1(T∞(f)→ ∀x0(f((�f)x) = 0))

→ ∀f1((t1�)f =0 0↔ ∃x(fx = 0)))

and

WE-PRA! { ∀’2(∀f1(’f = 0↔ ∃x(fx = 0))

→ ∀f1(T∞(f)→ ∀x0(f((t2’f)x) = 0))):

Corollary 3.5. E-PRA! + QF-AC1;0 UWKL↔∃-2∀f1(∃x0(fx= 0)→ f(-f) = 0).

Proof. The existence of - obviously implies the existence of ’ in Proposition 3.4 andhence of �. For the other direction we only have to observe that the existence of ’implies the existence of - be applying QF-AC1;0 to

∀f∃x(’(f) = 0→ fx = 0):

Remark 3.6. In contrast to the corollary to the proof of Proposition 3.4 above thereexists no closed term t in E-PRA! which computes - in �, i.e.

S! �|= ∀�1(1)(∀f1(T∞(f)→ ∀n0(f(�f)n = 0))

→∀f1(∃x(fx = 0)→ f(t�f) = 0))

for every closed term t (of appropriate type) of E-PRA!, since—by [8]—every suchterm has a majorant t∗; � is majorized by �f1; x0:1 and so - would have a majorant�fM :t∗(11(1); fM ) (where fM (x) := max(f0; : : : ; fx)), which contradicts the easy ob-servation that - has not even a majorant in S! (here S! denotes the full set–theoretictype structure).

Theorem 3.7. (1) E-PRA! + UWKL contains Peano arithmetic PA.(2) E-PRA! + QF-AC1;0 + QF-AC0;1 + UWKL is conservative over PA.(3) E-PA! + QF-AC1;0 + QF-AC0;1 + UWKL proves the consistency of PA and has

the same proof-theoretic strength as (and is 212-conservative over) the second order

system (201-CA)¡0 .

Proof. (1) Using ’ from Proposition 3.4 one easily gets characteristic functions for allarithmetical formulas A(x). By applying the quanti0er-free induction axiom of E-PRA!

to them, one obtains every arithmetical instance of induction.(2) This follows from Corollary 3.5 and the conservation of E-PRA! + QF-AC1;0 +

QF-AC0;1 + - over PA, which is due to [4] (note that the usual elimination of


extensionality procedure—which applies to the existence of - but not to UWKL—yields a reduction of E-PRA! + QF-AC1;0 + QF-AC0;1 + - to its variant where theextensionality axioms for types ¿0 are dropped, see [14] for details on this).

(3) follows from [4,5] using again Corollary 3.5 above and elimination of exten-sionality.

Remark 3.8. (1) The functionals ’ and - from Proposition 3.4 and Corollary 3.5provide uniform versions (in the same sense in which UWKL is a uniform version ofWKL) of

(1) 201-CA: ∀f∃g∀x0(g(x) =0 0↔ ∃y0(f(x; y) =0 0))

respectively of

(2) 201-CA: ∀f∃g∀x0; z0(f(x; gx) =0 0 ∨ f(x; z) �= 0);

but yet ’; - are not stronger than (1), (2) relative to E-PRA! (but only relative to E-PA!) as Feferman’s results cited in the proof above show. The reason for this is thatE-PRA! is too weak to iterate ’ or - uniformly since this would require a primitiverecursion of type level 1. In contrast to this fact, UWKL is stronger than WKL alreadyrelative to E-PRA!.

(2) One might ask whether UWKL becomes weaker if we allow �1(1) to be a partialfunctional which is required to be de0ned only on those functions f which representan in0nite binary tree. However, the construction ( (used in the proof of Theorem 3.2)such that

(1) ∀f1T∞((f)

and

(2) ∀f1(T∞(f)→ (f =1 f)

shows that any such partial � could be easily extended to a total one.

4. Results in the intuitionistic case

We 0rst show that in the intuitionistic context of E-PRA!i , UWKL is proof–theoret-

ically as strong as in the classical context but does not contribute to the growth ofprovable function(al)s. In the previous section we have seen that in sharp contrastto this, UWKL does contribute to the growth of provably recursive functions whenclassical logic is allowed. In Proposition 4.8 below we observe that this already happensin the presence of the Markov principle.

De�nition 4.1. (1) Independence-of-premise principle for negated formulas:

IP¬ : (¬A→ ∃x�B)→ ∃x�(¬A→ B);


where A; B are arbitrary formulas, x not free in A and � arbitrary.(2) Markov principle for numbers:

M : ¬¬∃x0A0 → ∃x0A0;

where A0 is quanti0er-free.

Proposition 4.2. E-PRA!i UWKL↔ ∃’2∀f1(’f=0 0↔ ∀x0(fx=0 0)):

Proof. The proposition follows from the corollary to the proof of Proposition 3.4by negative translation and some easy intuitionistic reasoning (using the decidabilityof =0).

As a corollary we get the following strengthened version of 3.4:

Corollary 4.3. E-PRA!i + MUWKL↔ ∃’2∀f1(’f=0 0↔ ∃x0(fx =0 0)):

Theorem 4.4. E-PRA!i +UWKL has the same proof–theoretic strength as its classical

version E-PRA! + UWKL (which by Theorem 3:7 is that of PA).

Proof. The theorem follows by negative translation using easy intuitionistic reasoningand the decidability of =0.

In contrast to this we have the following result that UWKL does not contribute tothe growth of provable function(al)s relative to E-PRA!

i .

Theorem 4.5. Let A(u�; v�; w�) be an arbitrary formula containing only u; v; w as freevariables; � 6 1; � 6 2 and t a closed term (of suitable type). Then the followingrule holds for Ti := E-PRA!

i + IP¬ + AC + UWKL:

Ti ∀u�∀v6� tu∃w�A(u; v; w)

then one can extract a closed term � of E-PRA!i s:t:

Ti ∀u�∀v6� tu∃w 6� �(u)A(u; v; w);

where x1 60(�k ):::(�1) x2 :≡ ∀y�11 ; : : : ; y

�kk (x1y60 x2y).

Similarly for E-PRA!i replaced by E-PA!

i in the de8nition of Ti. Then � is aclosed term of E-PA!

i ; i.e. a primitive recursive functional in the sense of G:odel’s T .

Proof. This result follows from Theorem 3:3 of [12] and the fact that UWKL isequivalent (relative to E-PRA!

i ) to an axiom having the form ∀x2(C→∃y64 sx¬B)(as required in that theorem): take ‘∀x’ as dummy quanti0er, s := �x:1; C :≡ (0 = 0)and notice that the formula ∀f1(T∞(f)→∀x0(f((�f)x) =0 0)) is intuitionisticallyequivalent to its double negated version since =0 is decidable.


Corollary 4.6. Let Ti be as in Theorem 4:5. Then Ti is closed under the fan rule;i.e.

Ti ∀f 61 g∃n0A(f; n)⇒Ti ∃m0∀f 61 g∃n6 mA(f; n);

where A contains only free variables of type 6 1.This result also holds for E-PA!

i instead of E-PRA!i in the de8nition of Ti and for

the systems where one or both of the principles AC and IP¬ are omitted from Ti.

Corollary 4.7. Let Ti be as in Theorem 4.5 and A(f1; n0) be an arbitrary formulaof E-PRA!

i containing only f1; n0 as free variables. Then the following rule holds:

Ti ∀f∃nA(f; n)

⇒ ∃ a primitive recursive (in the sense of Kleene) functional X s:t:

Ti ∀f∃n6 �(f)A(f; n):

In particular: for A(m0; n0) instead of A(f; n), �m is a primitive recursive functionin m.

Proof. This follows as a special case from Theorem 4.5 using the well-known factthat the functions de0nable by closed terms of E-PRA!

i are just the ordinary primitiverecursive functions.

In contrast to this, the addition of the Markov principle M yields the same provablerecursive functions than in the classical case:

Proposition 4.8. Every �(¡0)-recursive function is provably recursive in E-PRA!i +

M + UWKL.

Proof. The proposition follows from Theorem 3.7, the well-known fact that every�(¡0)-recursive function is provably recursive in PA and an application of negativetranslation which yields that E-PRA!+UWKL is 20

2-conservative over E-PRA!i +M+

UWKL.

Acknowledgements

This paper was prompted by discussions the author has had with Gerhard J$ager andThomas Strahm who asked him about the status of the uniform weak K$onig’s lemmain a fully extensional (classical) context.

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On uniform weak König's lemma