1 Introduction

On the arithmetical content of restricted forms of

comprehension choice and general uniform b oundedness

Ulrich Kohlenbach

BRICS

Department of Computer Science

University of Aarhus

Ny Munkegade Bldg

DK Aarhus C Denmark

kohlenbbricsdk

Abstract

In this pap er the numerical strength of fragments of arithmetical comprehension choice and

general uniform b oundedness is studied systematically These principles are investigated relative

to base systems T in all nite types which are suited to formalize substantial parts of analysis

but nevertheless have provably recursive functionals of low growth We reduce the use of

pro ofs of a large class of formulas to the use of instances instances of these principles in T

of certain arithmetical principles thereby determining faithfully the arithmetical content of the

former This is achieved using the metho d of elimination of Skolem functions for monotone

formulas which was introduced by the author in a previous pap er

As corollaries we obtain new conservation results for fragments of analysis over fragments of

arithmetic which strengthen known purely rstorder conservation results

We also characterize the provably recursive functionals of type of the extensions of T

based on these fragments of arithmetical comprehension choice and uniform b oundedness

Introduction

This pap er studies the numerical strength of fragments of arithmetical comprehension choice and

uniform b oundedness relative to weak base systems formulated in the language of all nite types

which are suited to formalize substantial parts of analysis

In a previous pap er we have introduced a hierarchy G A of systems where the denable

functions corresp ond to the wellknown Grzegorczyk hierarchy These systems extended by the

schema of full quantierfree choice

AC qf x y A x y Y x A x Y x AC qf fAC qf g

The author is indebted to an anonymous referee whose suggestions led to an improved presentation of the results

Basic Research in Computer Science Centre of the Danish National Research Foundation

where A is a quantierfree formula and various nonconstructive analytical axioms having the

form

x y sxz A x y z

including a generalized version of the binary Konigs lemma WKL allow to carry out a great deal

of classical analysis even for n The axioms and ACqf do not contribute to the growth of

extractable uniform b ounds which in the particular case of G A are p olynomials see and

in particular for more information

In contrast to this fragments of arithmetical comprehension and choice as well as generalizations

of our principle of uniform b oundedness from to more complex formulas do contribute

signicantly to the arithmetical strength of the base systems In we developed a general metho d

to calibrate faithfully this contribution and applied it to instances of comprehension and

choice These results were then used in to determine the arithmetical strength of single sequences

of instances of the BolzanoWeierstra theorem for b ounded sequences in IR the Ascolilemma and

others

In this pap er we give a systematic treatment of the whole arithmetical hierarchy for comprehension

choice and uniform b oundedness and determine precisely their arithmetical strength as well as their

provably recursive functionals of type We also consider much more complex formulas to b e

proved in these systems than we did in our previous pap ers

In the following let us discuss now some of the diculties one has to deal with in order to achieve

this goal and which indicate already the type of results one can exp ect For simplicity we restrict

ourselves for the moment to the secondorder system EA AC qf instead of G A ACqf

which we actually are going to consider b elow

EA is an extension of Kalmarelementary arithmetic with number quantiers EA obtained by

adding nary function quantiers for every n and the schema of explicit denition of functions

ED f x f x tx

where t is a number term of EA and x is a tuple of number variables Furthermore EA contains

the schema of quantierfree induction for all quantierfree formulas of EA which may contain

function parameters Finally EA contains constants and their dening equations for all elementary

recursive functionals of type

In EA the schema of quantierfree induction can b e expressed equivalently as a single axiom

QFIA f f x f x f x xf x

instead of f x In Analogously IA is the induction axiom for y y y f x y

k k

rstorder contexts this is replaced by a schema with y y y A x y as induction formulas

Let us consider furthermore the restriction of arithmetical choice to or equivalently to

formulas of LEA which like QFIA can b e expressed as a single secondorder axiom f ACf

Throughout this pap er A B C denote quantierfree formulas We allow b ounded number quantiers

0 0 0

x t x t to o ccur in A B C since they can b e expressed in a quantierfree way using the b ounded

0 0 0 0 0

searchfunctional from G A T denotes the set of all nite types

Since co ding of nite tuples of numbers is available in EA one can in fact restrict oneself to unary function

variables

where

ACf a x y z f a x y z g x z f a x g x z

Now by iteration one easily veries that EA f ACf proves already full arithmetical choice

So in order to prevent the arithmetical hierarchy of choice principles from collapsing we restrict

ourselves to single instances of f ACf which later on are allowed however to dep end on the

parameters of the theorem to b e proved For the moment we forbid completely the o ccurrence of

function parameters in AC ie we consider the schema

AC x y Ax y g x Ax g x

where Ax y is a formula without function parameters

As a starting p oint for the introduction into our general program let us consider now the following

question

What arithmetical statements are provable in EA AC qf AC

A rst observation is that AC proves CA ie

f x f x Ax

formula without function parameters Combined with the axiom QFIA this where Ax is a

yields every function parameterfree instance of IA Hence the rstorder system EA IA is

AC a subsystem of EA AC qf

What is the precise relationship b etween EA AC qf AC and EA IA

It will turn out that the former theory is conservative over the latter for some formulas including

sentences but not for all formulas

IA without some restriction imp osed on That EA AC qf cannot b e conservative over EA

the formulas follows from the following observation

By applying the functional f x maxf i to the function g in AC one obtains the

max

formulas corresp onding instance of the socalled b ounded collection principle for

CP x ay Ax y z x ay z Ax y

where A

CP ie EA AC proves every function parameterfree instance of So EA AC qf

CP is a subsystem of EA AC qf AC

It is wellknown see that there exists an instance A of CP which is not provable in EA

IA On the other hand EA CP is conservative over EA IA by a result due to H

Friedman and implicitly JParisLKirby see eg for details The universal closure of the

instance A of CP can b e shown to b e equivalent to a sentence in EA IA Hence EA

AC qf AC is not conservative over EA IA

3 0 0

The universal closure with resp ect to number parameters a is sup eruous for f ACf since it can b e

0 0

captured by the universal closure f However b elow we consider single instances AC of f ACf where it

1 1

0 0

do es make a dierence Because of the closure wrt arithmetical parameters a a single instance AC contains

a whole sequence of instances of AC

Here is another arithmetical use of AC we can make relative to EA AC qf

As mentioned ab ove CA is a trivial consequence of AC in the presence of classical logic

Now combining CA with AC qf one can easily prove CA and therefore every function

parameterfree instance of IA Hence EA IA is a subsystem of EA AC qf AC

as well even if the functional would not b e included in EA

max

So the arithmetical strength of AC dep ends heavily on the secondorder axioms like QFIA

AC qf and the characterizing axioms for functionals as which are available in the context

max

in which AC is considered

As a sp ecial corollary of the results of this pap er it follows that EA AC qf AC is

k k

conservative over EA IA which implies the result of H Friedman JParisLKirby Furthermore

we show that EA AC qf AC is conservative over EA IA wrt monotone formulas

k k

of arbitrary complexity These results are sensitive to small changes of the base system EA Eg

if we add the primitive recursive functional dened by

f g g f g x f x f g x

it it it

AC and to EA then the Ackermannfunction b ecomes provably total in EA AC qf

the resulting system proves the consistency of EA IA EA AC qf proves the second

order axiom of induction Combined with CA one obtains every function parameterfree

IA which is known to prove the totality of the Ackermannfunction IA Hence EA instance of

as well as the consistency of EA IA is a subsystem of EA AC qf AC

AC proves the totality Using a more involved argument one can show that already EA

of the Ackermann function see chapter of for details on this

So any pro of of conservation of systems based on AC over IA has to take into account

k k

carefully the structure of the functionals of type which are denable in the given system

Things b ecome of course even more complicated for the systems G A ACqf instead of EA

AC qf which we are treating in this pap er

Among other things we show that relative to base systems T G A ACqf the use of

can b e reduced to the use of AC f in a pro of of a formula B f CA f and

k k k

This is true also for B f of arbitrary complexity in the arithmetical hierarchy if B f is monotone

ar ar

in the sense of denition b elow

We also show that the provably recursive functionals of type of G A ACqf WKL

CA AC are just the functionals of these types denable in T k where T is the

k k

fragment of Godels T with recursion up to the type k only

4 0 0

Both asp ects are not taken into account appropriately in where CA and AC are studied systemat

k k

ically for the rst time As a consequence of this theorems and corollaries in are not correct as

stated see and in particular chapter of for a thorough investigation of this matter

These results are used to prove new conservation results for EA CP over EA IA which

k k

strengthen the FriedmanParisKirby result

Finally we consider generalizations UB j of the principle of uniform b oundedness UB

which was studied in In we showed that UB proves already relative to G A AC

qf many imp ortant analytical theorems like Dinis theorem the attainment of the maximum for

d d

f C IR the sequential HeineBorel prop erty for the existence of an inverse function

for every strictly monotone function f C and others but do es not contribute to the growth

of extractable b ounds thereby guaranteeing the extractability of p olynomial b ounds when applied

in the context of G A ACqf

formulas is inconsistent with G A Whereas the straightforward generalization of UB to

already for k our restricted version UB j introduced in the present pap er is consistent

In we implicitly used a sp ecial case of UB j to prove the BolzanoWeierstra principle and

the Ascolilemma and it were these pro ofs which were used to calibrate faithfully the arithmetical

strength of these principles

We show that our results on fragments of arithmetical comprehension and choice mentioned ab ove

UB j f is used in the pro of AC f also CA f remain valid if in addition to

k k k

of B f

Monotone formulas and their Skolem normal forms

In this section we review some of the pro oftheoretic to ols from on which the present pap er is

based and also recall some of the basic concepts and denitions from

The set T of all nite types is dened inductively by

i T and ii T T

Terms which denote a natural number have type Elements of type are functions which map

ob jects of type to ob jects of type

The set P T of pure types is dened by

i P and ii P P

Brackets whose o ccurrences are uniquely determined are often omitted eg we write instead

of Furthermore we write for short instead of Pure types can b e

k k

represented by natural numbers n n The types are so represented

by For arbitrary types T the degree of for short deg is dened by deg

and deg maxdeg deg For pure types the degree is just the number which

represents this type

Description of the theories EG A

A pro oftheoretic treatment of the FriedmanParisKirby result was rst given in However the pro of in

contains gap See for a correction of Siegs pro of Another pro oftheoretic treatment can b e found in

Whereas we generally use the sup erscript to denote the restriction S of a schema S to function parameterfree

instances of S this sup erscript has a dierent meaning in the context of principles of uniform b oundedness Although

this might b e troublesome we wish to stick to the notation for these principles from where they were introduced

Our theories T used in this pap er are based on manysorted classical logic formulated in the lan

guage of all nite types plus the combinators which allow the denition of abstraction

denotes the intuitionistic variant of T T

The systems G A for all n are introduced in to which we refer for details G A

n n

has as primitive relations for ob jects of type the constant functions min max S

successor A A where A is the ith branch of the Ackermann function ie A x y

n i

0 y

y A x y x y A x y x y A x y x functionals of degree where

f x max f f x and is the iteration of A on the f values for i ie f x

i i

x x

Q P

f i We also have a b ounded search functional and b ounded predicative f i f x

i i

recursion provided by recursor constants R where predicative means that recursion is p ossible

only at the type as in the case of the unbounded KleeneFeferman recursors R Moreover

G A contains a quantierfree rule of extensionality QFER

In addition to the dening axioms for the constants of our theories all true sentences having the

form x A x where A is quantierfree and deg are added as axioms By true we refer

to the full settheoretic mo del S In given pro ofs however only very sp ecial universal axioms will

b e used which can b e proved in suitable extensions of our theories Nevertheless we include them

all as axioms in order to emphasize that pro ofs of universal sentences do not contribute to the

growth of extractable b ounds In particular this covers all instances of the schema of quantierfree

induction The main results in this pap er are also valid for the variant of G A where the universal

axioms are replaced by the schema of quantierfree induction The restriction deg has a

technical reason discussed in

G A G A

n 1

n2IN

PA PA are the extensions of G A G A obtained by the addition of the schema of full induction

n n

i i

and all impredicative primitive recursive functionals in the sense of

when the quantierfree rule of extensionality is denotes the theory which results from T ET

i i

replaced by the axioms of extensionality E

x y z x y z x z y

for all nite types x y is dened as z z xz z y z z where

k k k

G R and T denote the sets of all closed terms of EG A and EPA T is the subset of all

n n k

i i

closed terms of T which contain the Godelrecursors R for of degree k only

Denition Between functionals of type we dene relations less or equal and smaj

strongly majorizes by induction on the type

x x x x

x x y x y x y

x smaj x x x

x smaj x y y y smaj y x y smaj x y xy

Remark smaj is a variant of WA Howards relation maj from which is due to For

more details see

b e a formula of G A a are all free variables of A and x y A x y a its Godel functional Let Aa

n D

interpretation see eg for details on Godels functional interpretation We say that a tuple of

closed terms t realizes the monotone functional interpretation of Aa if

t sma j x a y A x a y a x

Monotone functional interpretation which directly extracts a tuple t satisfying from a pro of of

was introduced in See also for details Aa

We next dene what it means for a formula to b e monotone In order to motivate the somewhat

technical denition lets consider the simple case of a formula A y xA y x A is monotone

y y x x A x y A x y

Innermost existential quantiers and outmost universal quantiers are not supp osed to b e monotone

Hence we get the following

Denition Let A LG A be a formula having the form

A u v tuy x y x w A u v y x y x w

k k

x w free t G R and are arbitrary nite where A is quantierfree and contains only u v y

types

A is called arithmetically monotone if

u v tux x x x y y y y

k k k k

x x y y w A u v y x y x w

M onA

i i i i k k

w A u v y x y x w

k k

The Herbrand normal form A of A is dened to be

H k

A u v tuh h y y w

A u v y h y y h y y w where

k k k i

z z

Remark In denition and theorems below one may also have tuples w instead

of w in A where w w w and is arbitrary Also instead of u we may have u

where u u with deg for i q In particular we can consider an innermost u

and an outermost universal number quantier x existential number quantier y as part of w

as part of u So for x and y no monotonicity is required in denition

Here t sma j x means t sma j x

i i

Theorem Let n and G R Then

k n

h monotone G A M onA u v tuh h

i n k

A y uhw A y uh

k k

x y h x where h monotone x x y y h y

j j i i i i i

n o

bAC denotes the schema of bounded Denition Bounded choice bAC

choice

x y Z x Ax y Z Y Z xAx Y x Z bAC Z

In general G A A do es not imply G A A see for a detailed discussion of this phe

n n

nomenon which is in contrast to the rstorder case where the derivability of A follows from that

of A by Herbrands theorem see If however A is monotone then this rule is valid also for

G A but for very dierent reasons

Theorem Let A be as in thm and be a set of sentences x y sxz G x y z

where s is a closed term of G A and G a quantierfree formula and let A denote the negative

translation of A Then the fol lowing rule holds

G A ACqf A M onA

G A A and

by monotone functional interpretation one can extract a tuple G R such that

G A satises the monotone functional interpretation of A

where fY sx z G x Y x z x y sxz G x y z g In particular the

second conclusion can be proved in G A bAC

The weakened conclusion G A M onA A fol lows already from G A ACqf A

n n

Making arithmetical comprehension monotone

In this section we consider the arithmetical content of instances CA uv of CA which are

k k

used in given pro ofs of sentences u v tu B u v as discussed in the introduction

Denition

f x u g x u u u CAf g x

k k

Here we can use Godels translation or any other of the various negative translations For a systematical

treatment of negative translations see

This last assertion is not stated in the formulation of the theorem in but do es follow immediately from its

pro of

10 0 0

Whether one has here u or u dep ends of course on whether k is even or o dd

k k

Remark There is no need here to incorporate closure under number parameters in the denition

CAf ie by dening of

f l x u CAf l g x g x u u u

k k

since the latter can be reduced to the former relative to G A for n by coding l x together and

applying comprehension without number parameters to this pair

In order to b e able to apply the metho d of elimination of Skolem functions for monotone formulas

from section we follow this strategy

Construct an arithmetical principle A f such that for suitable G R

ar n

G A M onf A f

n ar

CAf and G A AC qf f A f

CA f A f G A f

ar n

CA uv in a given pro of of a monotone sentence u v tu B u v Because of the use of

S 0 0

can b e reduced to the use of A uv where uv uv which in turn by and theorem

can b e reduced to the use of A uv Because of nothing is lost by this reduction

mon

TND f of the It will turn out that the correct principle A f is a monotone version

tertiumnondatur principle for formulas

k k

Denition In the fol lowing m if k is even resp m if k is odd

tertiumnondatur axiom is given by the fol lowing formula where f is a function The

variable of appropriate type

f TND

f x y z y z y y z y z x y

m m m

u v u v u f x u v u v u

m m m

TND f We also need the fol lowing prenex normal form of

TND f

y u v z y u v z y x u

m m m m m m

f x y z y z y f x u v u v u

m m m m m m

The Skolem normal form of TND f is given by

TND f

h h h g g x y v y v y

m m m m

m m

f x y g x y y g x y y v v y

m m m m m

f x h x v h x y y v v v h x y y v v

m m m m m m m

11 0 0

Here and in the following the quantiers y u are only present if k is o dd

m+1 m+1

Remark For n we have coding of nite tuples of xed length available in G A Hence

quantierblocks can be contracted to a single quantier Since in al l of our results we assume that

at least n it is no restriction in the denition above to consider only single quantiers

Lemma For every k IN the fol lowing implication can be proved in G A

CA f f TND f

k k

Pro of

For notational simplicity we conne ourselves to the case k which well shows the general pattern

of the pro of for arbitrary k

pr S

TNDf yields the existence of functions g g h h such that

f x y g x y y g x y y v v f x h x v h x y v v x y v y

in turn yields

y z f x y g x y y z v f x h x v h x y v v x y v

y z f x y g x y y z u v f x h x v u v x y v

y z f x y g x y y z v u v f x h x v u v x y

x y z y z f x y z y z v u v f x h x v u v

and nally

x y z y z f x y z y z v u v f x h x v u v

applied to y h x v g x h x y h x h x g x h x gives

f x h x g x h x h x h x g x h x g x h x h x h x g x h x g x h x x

f x h x g x h x h x h x g x h x v v

We now show

f x h x g x h x h x h x g x h x g x h x h x h x g x h x g x h x x

f x y z y z y z y z

yields the claim of the lemma with

g x xh h g g

f x h x g x h x h x h x g x h x g x h x h x h x g x h x g x h x

Pro of of

xf h h g g implies

f x h x g x h x h x h x g x h x v v

Hence by putting y h x v g x h x

f x h x g x h x y z y z

and therefore

f x h x z y z z y z

v u v f x h x v u v

By this implies

y z y z f x y z y z

xf h h g g implies by

v f x h x g x h x h x h x g x h x v

and therefore

f x h x g x h x u v u v

f x h x g x h x y z y z

By this yields putting y h x

f x h x v u v v u v

and therefore

u v u v f x u v u v

which concludes the pro of of and hence of the lemma

Denition For a formula Aa x x x x of G A where a x x A a

k n

k k

are al l free variables of A which may have arbitrary type we dene

d d

x x x x x x x x x A a x x x A a

k k k

In the following we need a variant M on of M on where monotonicity is required for all number

quantiers compare this with remark

Denition Let Aa x y x y A a x y x y Then

k k

x x y y x x y y M on Aa

k k k k

x x y y A a x y x y A a x y x y

i i i i k k k k

as in the previous denition we have Lemma For A a

G A M on Aa

12 0 0

Here the quantiers x and y may b e empty dummy quantiers

Pro of Trivial

The collection principle is the schema

CP x ay Ax y z x ay z Ax y

for all formulas Ax y

Convention In CP and other axiom schemas which we wil l consider below Ax y may

contain arbitrary parameters besides x y of the language we consider Eg if we write G A

CP then instances of CP may contain parameters of arbitrary type In EA CP however

k k

where EA denotes rstorder elementary recursive arithmetic instances of CP of course contain

only number parameters

CP is equivalent over many systems eg G A for n to the axiom schema of nite choice

for formulas

FAC x ay Ax y z x a Ax z

for all formulas Ax y with the convention stated ab ove

In the presence of function variables as in G A the schema CP can b e expressed as a single

secondorder axiom f CPf where

l a x ay u u u f l a x y u

CPf

z x ay z u u u f l a x y u

CPf we By incorp orating the universal closure wrt to arithmetical parameters l a in

achieve that the universal closure of every instance of CP which contains only number parameters

can b e written as a sentence CP in G A where is a closed term essentially the characteristic

formula Ax y which will b e of imp ortance b elow function of the quantierfree matrix of the

IAf which we need b elow induction The same is true for the principle of

k k

u u u f l u l

d 0 d

IAf

u u u f l x u x u u u f l x u

k k

u u x u f l x u

Lemma Let Aa G R Aa be as in denition Then for suitable

n l

the fol lowing holds

G A CP a Aa Aa

n i

and

CP a Aa Aa G A

i n

Here and in the fol lowing we use the convention that S is empty ie for an axiom

schema S if k

Pro of Induction on k For k the lemma is trivial So let k

k k Consider

x x x x x x A a Aa

k k

By the induction hypothesis applied to the formula

x x x A a x x

k k

CP a CP can b e considered as instance of we have instances note that instances of

k k

CP a CP as well such that implies relative to G A

i n

k k

x x x A

d d

x x x x x x x x x x x x A a

k k k k

Hence

d d

x x x x x x x A a x x x

k k k k

d d

x x x x x x x x x x x x A a

k k k k

) CP (b a

k 1

d d

x x x x xb x x x x xb x x A a x

k k k

log ic

d d

x x x x x x x x x x x A a

k k k

d d

x x x x x x x x x x x A a x

k k k

In the same way as we shifted x x over x we now move x x over x then p ermute

x x with x move over x and so on until we obtain Aa This requires only instances

CP a or simpler ones of CP which can b e considered a fortiori as instances Putting things

together we have shown that relative to G A

CP a CP a CP a Aa Aa

i j

k k k

i j

and

CP a Aa Aa CP a

j i

k k

j i

which concludes the pro of of the lemma

Since in our main results we assume n or n for the level n of G A we also use for simplicity

G A in the following denition and lemmas although some of them can b e carried out even in

G A

Denition and lemma For m IN let G R be such that

G A f x y z y z y

m m

f xy z y z y y y z z y y z z y y

m m m m m m m m m

f x y z y z y

m m m

We denote f by f

Lemma Let k There are eectively nitely many terms G R such that

CP f CAf CAf G A f

k k k

Pro of The lemma follows from lemma

Denition The monotone tertiumnondatur is given by

mon

TND f

x u y z v u y z v u y x x

m m m m m m

0 0

f x y z y z y f x u v u v u

m m m m m m

S S

mon

0 pr

Lemma G A f TND f TNDf

k k

mon

G A f M on TND f

Pro of follows by putting x x

mon

Follows immediately from the denition of TND f

0 mon

CAf TND f Prop osition G A f

k k

Pro of Lemmas and

Lemma One can construct a G R such that

G A AC qf f CA f CPf

k k

Pro of

Using CA f for a suitable G R one can reduce CPf to CP which is provable in

k k

G A AC qf

Prop osition For a suitable G R one has

mon

G A AC qf f CAf TND f

k k

Pro of Induction on k k easy Let k and lets assume that the prop osition holds for all

m k CP f denote the instances of collection from lemma which are needed

k k

to show

CAf CAf

k k

Let G R b e using lemma such that

G A AC qf CA f CAf CAf

k k k

By the induction hypothesis we have

mon

G A AC qf f TND f CAf

k k

for a suitable G R So by prop osition

S S

mon mon

G A AC qf TND f TND f CAf

k k k

S S

mon mon

Introducing dummy quantiers TND f can b e reduced to TND f

k k

for a suitable G R Furthermore

S S S

mon mon mon

TND h TND f TND g

k k k

for

f x y z if x x

hx y z

z if x x g x y

Hence

S S S

mon mon mon

TND f TND f TND f

k k k

for a suitable G R By and we have

mon

G A AC qf TND f CAf

k k

Lemma Let k and A Then

G A IA x u x x y Ax y u uAx u

Pro of Assume

u x x y Ax y u uAx u

We show by induction on n

Gnux

nu x l th x n x x x x i nu uAx u

i j i i

ij n

i=j

13 0 0 0

Note that two instances CA f CA f can b e co ded together into one instance CA f in G A

1 2 3 2

k k k

For n x this obviously is contradictory and so is proved

n applied to u yields an x x such that Ax and y Ax y is now

satised by taking x hx i u y

n n Let u x b e such that is satised for n By there exists an x x such that

y Ax y and u uAx u By we have i nu uAx u Hence

n n n n i

and so ub maxu y xb x hx i satisfy Gn ub xb i n x x

n n i n

It remains to show that u xG n u x is equivalent to a formula

Using CP u uAx u can b e shown to b e equivalent to a formula Since CP

k k k

follows from IA the whole pro of can b e carried out in G A IA

k k

mon

In contrast to TNDf its monotone version TND f do es not hold logically However it

k k

can b e proved using induction More precisely the following prop osition holds

Prop osition Let k There are nitely many instances IA f such that

mon

IA f TND f G A f

k k

IA f which prove relatively to Pro of By the pro of of lemma there are instances

G A

xu x x y z y z y f x y z y z y

m m m m m m

u u v u v u f x u v u v u

m m m m m m

and therefore by the denition of f which makes

u u v u v u f x u v u v u monotone wrt u

m m m m m m

xu x x y z y z y f x y z y z y

m m m m m m

v u v u f x u v u v u

m m m m m m

which is equivalent to

y y z y f x y z y z y xu y x xz

m m m m m m

v u v u f x u v u v u

m m m m m m

By a suitable instance of CP and the monotonicity of wrt z one can shift x x

mon

over z Now one continues in this way until one obtains TND f which needs only suitable

CP All the instances CP with l k which can b e considered as instances of instances of

k l

of CP used follow from suitable instances of IA

k k

mon

Corollary G A f CA f TND f for a suitable G R

k k

Conservation results for AC(f ) and CA(f g )

k k

CA uv and even We are now ready to determine the arithmetical content of instances

k k

AC uv and CA uv in pro ofs of monotone sentences and without monotonicity assumption

if the logical complexity is restricted It turns out that this content is given by certain instances of

mon

TND

Denition

l x y u u u f l x y u

ACf

g x u u u f l x g x u

l x u u u f l x u

CAf g

v v v g l x v

h x hx u u u f l x u

CAf CAj f j f for the projection functions j G R

k k i

Lemma Let k IN Then for suitable G R

ACf CA f G A AC qf f

k k

G A AC qf f CA f CAf

k k

Pro of Obvious

Below we also need a certain nonstandard axiom F

z n k y k F y y y k z n z i y k i k

i n

where for z z nk z k if k n and otherwise

F do es not hold in the full settheoretic typestructure but can b e eliminated from pro ofs of

monotone sentences in our theories This axiom was introduced and studied in and implies the

b oundedness which was mentioned in the introduction and which will b e principle of uniform

generalized in section b elow

b w Prop osition Let n k and B u v tua b a B be a sentence in

l l

LG A where B is quantierfree and t G R Let G R of suitable types and

n n n

a set of sentences having the form x y sxz A A quantierfree s G R Then for a

suitable G R the fol lowing holds

G A ACqf

w B b AC uv a b a CA uv u v tu

l l k k

then

mon

G A M onB u v tu TND uv a b a b w B

k l l

and in particular

G A IA M onB u v tua b a b w B

maxn

k l l

In the assumption of the rule the theory G A ACqf can be strengthened to

G A ACqf F Then in the rst conclusion G A must be replaced by G A

n n

maxn

Pro of By lemma prop osition and the fact that two instances of CA can b e co ded

together into a single instance of CA there is a G R such that

mon

G A AC qf u v tu TND uv CA uv AC uv

k k k

So the assumption of the rule implies

mon

G A ACqf u v tu TND uv a b a b w B

k l l

By lemma the prenexation

mon

uv B TND A u v tuxu y z v a b w

mon

A u v tu TND uv a b a b w B

k l l

is monotone if B is

G A M onB M onA

Now implies

G A ACqf A

and therefore using theorem

G A M onB A ie

G A M onB A

The second part of the claim in the prop osition now follows from prop osition

Here means that F must not b e used in the pro of of the premise of an application of the quantierfree rule

of extensionality QFER G A satises the deduction theorem wrt but not wrt

15 pr

Note that A is not completely in prenex normal form b ecause of the universal quantiers hidden in v tu

However it has the form required in theorem used b elow

16 mon 0 mon

TND denotes the quantierfree matrix of some prenex normal form of TND

The pro of ab ove can b e combined with the elimination pro cedure for F given in thm

yielding the claim ab out adding F

The following corollary in particular states for and v tu nonexistent that the

AC relative to CA and provably recursive function als of type of xed instances of

k k

the base system G A ACqf are denable in the fragment T of Godels T

1 k

Corollary Let k and G R Then the fol lowing rule holds

G A ACqf u v tu CA uv AC uv w B u v w

k k

T such that

PA u v tuw u B u v w

Again we may strengthen the theory in the assumption of the rule above by F

Pro of The corollary follows from prop osition by observing that the condition

M onu v tuw B is empty and using the fact that G A IA has a monotone

by terms functional interpretation as developed in via negative translation in PA

IA has a functional T The latter follows from the pro of that the negative translation of

interpretation in T provable in a subsystem of PA as given in and the fact that every

closed term of T can b e ma jorized in the sense of denition by a suitable term in T

k k

which follows from Howards pro of of this fact for full T as given in

sentence Corollary Let n and A be a

If EG A AC qf CA AC WKL A

k k

then G A IA M onA A

Pro of

Using the deduction theorem for EG A the fact that EG A AC qf F proves WKL see

and the existence of characteristic terms G R for quantierfree formulas of EG A the

n n

assumption implies

l l

AC A CA EG A AC qf F

j i n

k k

j i

for certain terms G R corresp onding to the universal closures of the instances of CA

i j n

AC used in the pro of and

For suitable G R we have

CA CA G A

i n

k k

and

AC G A AC

j n

k k

Together with elimination of extensionality see eg we obtain

G A AC qf F CA AC A

k k

The conclusion now follows from prop osition

Lemma Let u v tu Au v be a sentence with Au v Then one can construct a

sentence u v tuAu v with Au v such that

G A M on u v tuAu v

CP uv Au v Au v G A u v tu

i n

Au v Au v CP uv G A u v tu

i n

where G R are suitable terms

i j n

Pro of Lemmas

Corollary Let n u v tu Au v be a sentence in G A with Au v

t G R and G R of suitable types Then the fol lowing rule holds

n n

AC uv Au v CA uv If G A ACqf u v tu

k k

then G A IA u v tu Au v

We may strengthen the theory in the assumption of the rule above by F

Pro of

AC uv CP uv follows from a corresp onding instance Let A b e as in lemma

i i

k k

of AC which can b e considered as an instance AC uv of AC All these instances

k k k

b b

AC uv AC uv into a single instance AC uv i l can b e combined with

k k k

Hence the assumption of the corollary yields

G A ACqf u v tu CA uv AC uv Au v

k k

The conclusion now follows from prop osition lemma and the fact that

G A IA CP

k k

Corollary For n EG A AC qf CA AC WKL is conservative

k k

wrt sentences over G A IA

k k

Pro of The corollary follows from the pro ofs of corollary and corollary

Remark Corollary is optimal in the following sense For every k there is a sentence A

such that

G A AC A but G A IA A

k k

Pro of There is a rstorder instance A ie without parameters of types of FAC which

do es not follow from IA relative to eg G A see It is clear that G A AC A

k k

Since the universal closure of A can b e shown to b e equivalent to a sentence in G A IA

k k

and hence in G A AC the claim follows

Corollary Let u v tu Au v be a sentence with Au v Then for n the

fol lowing rule holds

If G A ACqf u v tu CA uv AC uv Au v

k k

then G A CP u v tu Au v

We may strengthen the theory in the assumption of the rule above by F

Pro of The corollary follows analogously to the pro of of corollary using lemma for k

instead of k and the wellknown fact see eg that G A CP IA

k k

Corollary For n EG A AC qf CA AC WKL is conservative

k k

wrt sentences over G A CP

k k

Pro of The corollary follows from corollary analogously to the pro of of corollary

Let EA b e Kalmarelementary arithmetic EA with number quantiers and let us consider the

of G A where the arbitrary true universal axioms from its denition in are variant G A

n n

replaced by the schema of quantierfree induction with arbitrary parameters only The results

ab ove also hold for G A since no other universal axioms from were used EA can b e considered

as a subsystem of G A and the latter is conservative over the former Hence we obtain the following

corollaries for EA

Corollary Let A be an arbitrary sentence of EA Then the fol lowing rule holds

EA CP A EA IA M onA A

k k

In particular we have the following

Corollary Let A A be sentences from EA such that

EA CP A A

EA IA A A and

EA IA M onA

Then EA CP A implies EA IA A

k k

Combined with lemma we nally obtain

Corollary ParisKirby H Friedman

EA CP is conservative over EA IA

k k k

Or equivalently the secondorder axiom of quantierfree induction

Generalized principles of uniform b oundedness and their

arithmetical content

In the following we dene a generalization of the principle of uniform b oundedness UB which

was studied in

y k x y k z Ax y k z k x n

UB V

xi y k i z k A x n y k z

i n

where A l A l is a purely existential formula

UB follows from F relative to G A AC qf for n

In G A UB and hence in G A F AC qf one can give very short and p erspicuous

pro ofs of various imp ortant analytical theorems like

Every p ointwise continuous function f IR is uniformly continuous

d d

The attainment of the maximum value of f C IR on

The sequential form of the HeineBorel covering prop erty for

Dinis theorem

The existence of a uniformly continuous inverse function for every strictly increasing continuous

function f IR

Since F do es not contribute to the growth of extractable b ounds one can extract p olynomial b ounds

from pro ofs in G A UB ACqf

UB to formulas is not consistent with G A Whereas the straightforward generalization of

UB not hold in the full see the following restricted form is although it do es like

settheoretic type structure

Denition Let k

y a k x y k z Ag x y k z k z a

UB j g

k x y k l z k Ag x l y k z k z a

where Ag v k z a u u u g v k z a u

k k

j t UB where t G R such that tv k z a v Remark G A UB

In we have shown that every single sequence of instances of the BolzanoWeierstra principle

for b ounded sequences in IR and of the Ascolilemma in the sense of follows from suitable

instances of UB j and used this to calibrate precisely the contribution of such instances to

the growth of extractable b ounds This indicates the mathematical relevance of our generalized

principles of uniform b oundedness

Prop osition Let n k For suitable G R we have

G A AC qf F CA g UB j g

k k

where g is a free function variable

Pro of

For a suitable G R CA g yields the existence of a function h such that

v k z a hv k z a Ag v k z a

where A is as in denition Using h the assumption of UB j g can b e expressed as

hx y k z k z a k x y k z

By UB which follows from F and AC qf relative to G A see this yields

k x y k l z k h x l y k z k z a

and hence

x l y k z k z a k x y k l z k A g

Using prop osition we can strengthen prop osition and corollary to

Theorem Let n k and B u v tua b a b w B be a sentence in

l l

LG A where B is quantierfree and t G R Let G R of suitable types and

n n n

a set of sentences having the form x y sxz A A quantierfree s G R Then for

a suitable G R the fol lowing holds

G A ACqf

u v tu CA uv AC uv UB j uv a b a b w B

k k k l l

then

mon

w B b a b TND uv a G A M onB u v tu

l l k

and in particular

w B b a b IA M onB u v tua G A

l l k

In the assumption of the rule the theory G A ACqf can be strengthened to

G A ACqf F

The following corollary implies for and v tu b eing a dummy quantier that

the provably recursive functionals of type of xed instances of UB j relative to the base

system G A ACqf are denable in the fragment T of Godels T

1 k

Corollary Let k and G R Then the fol lowing rule holds

G A ACqf

u v tu CA uv AC uv UB j uv w B u v w

k k k

T such that

PA u v tuw u B u v w

Again we may strengthen the theory in the assumption of the rule above by F

We now show that CAf in fact is implied by suitable instances of UB j

k k

Prop osition Let n k For suitable G R we have

UB j f CAf G A

i n

k k

where f is a free function variable

Pro of Induction on k k CAf is logically equivalent to

g x y z g x f x y f x z g x

and hence to

g x y z g x f x y f x z g x

For a suitable G R UB j f yields the equivalence of and

n g x y nz g x f x y f x z g x

n g x n g x y nf x y z f x z g x

Dene

if y nf x y

g x

otherwise

Let k k k

CAf is equivalent to

g x y z g x u u u f x y u

g x u u u f x z u

By induction hypothesis there exists an instance UB j f which can b e considered as an

instance UB j f which implies relative to G A CAf and hence the existence of

k k

an h such that

x a hx a u u u f x a u

By UB j f for a suitable applied to the negation of CAf is equivalent to

k k

ng x n g x y nu u u f x y u

g x z u u u f x z u

which is satised by

if y nhx y

g x

otherwise

Corollary For n k the fol lowing holds

G A g UB j g g CAg

n n

UB j g G A g UB j g g

Pro of By prop osition g UB j g implies f CAf and hence f CAf by iter

ation

follows from and the pro of of prop osition

Let B b e the typebar recursor constant of equality rank ie B is characterized by the

axioms

x y n n B xz uny z ny

y n n B xz uny u D B xz un y n D ny x

where u is of type and

y k if k n

y n D k

D if k n

otherwise

Denition The schema of dependent choice of type for arithmetical formulas is given by

z x z Az z z z DC x y Ax y x z

with arbitrary parameters where A

Prop osition Let n k B u v w be a quantierfree formula of G A containing only

u v w free t G R Then the fol lowing rule holds

G A ACqf g UB j g u v tuw B u v w

G R B such that

G A BR DC u v tuw u B u v w

can be written as a closed term of T ie it is a primitive recursive functional in the sense of

Godel such that PA BR

Moreover if and for the types in and if S j and then

u B u v w S j u v tuw

Pro of By prop osition and corollary one has

G A AC qf g CAg F g UB j g

Hence the assumption of the rule to b e proved yields

G A ACqf g CAg F u v tuw B u v w

From the work of Sp ector it follows that G A ACqf g CAg has via negative trans

lation a Godel functional interpretation in G A BR by terms G R B In it is

n n

shown that the type structure M of the socalled strongly ma jorizable functionals forms a mo del

of full bar recursion From the pro of of this fact restricted to typebar recursion one obtains the

construction of a term B G R B such that

sma j B G A BR DC B

where sma j is the corresp onding syntactic notion of strong ma jorization as dened in denition

Therefore the pro of of the fact that the negative translation of G A ACqf has

a monotone functional interpretation in the sense of in G A by terms in G R see

n n

extends to G A ACqf g CAg yielding a monotone functional interpretation via

DC by terms in G R B This has the negative translation in G A BR

n n

consequence that as in the case of G A ACqf see the pro of of theorem in we

can eliminate F from the pro of of uv tuw B and extract a uniform b ound on w which

now of course is only in G R B instead of G R and its verication can b e carried out in

n n

G A BR DC

By prop osition it follows since deg that can b e written as a primitive recursive

functional such that PA BR

The nal claim follows using again the mo del M Since M S M S and M S the

assumption S j implies M j and therefore since M j bAC see M j From

DC Therefore it follows that M j PA BR

u B u v w M j u v tuw

and hence since

u B u v w S j u v tuw

Corollary The provably recursive functionals of type of g UB j g relative to

G A ACqf are denable in T

Remark Because of corollary PA is a subsystem of G A ACqfg UB j g

Hence corollary is optimal

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