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Logique & Analyse 145 (1994), 2340
STRICT FINITI S M AS A V I A BLE A L TE RNA TI V E IN THE
FOUNDATIONS O F M A T H E MA TI C S
*
1. Introduction
Jean Paul Va n BENDEGEM
Whether or not one believes tha t G6d el's theorems shattered Hilbe rt ' s
dream o f p ro vin g ma th ema tics ab s o lu t e ly con siste n t , th e t e r m
"f in it ism" i n h is philo sophy a p plie s t o the me t a le ve l o n ly. Ma ny
philosophers s e e Ernst We lt i [1987] fo r an overvie w h a v e been in
trigued by the possib ility to extend Hilb e rt's f in itis m to the object level.
It seemed and it still seems a neat wa y to escape Goole! a ltoge the r.
Although Gödel h imself had something quite different in min d when he
wrote about " a n extension o f fin itary ma thematics tha t has n o t ye t
been used", the strict finit ist point of view too can be seen as an exten
sion o f finitary mathematics.
Perhaps one w i l l remark that to con sider strict f in itism as an e xte n 
sion is stretching the meaning o f extension a b it, if not a b it too much.
For i t is a ve ry we llk n o wn remark in the literature o f philosophy o f
mathematics that strict f in it is m i s mo st su ita b ly characterized b y the
label " s t rict" . Th e re sultin g mathematics i s so p o or an d weak, th a t
terms such as trivia l, uninteresting and the lik e come t o min d . I t i s
therefore both a philosophically and mathematically interesting question
to ask whether this is necessarily so. The main thesis o f th is paper is
* This paper is essentially an improved version of a paper that was presented at and
published after the First International Symposium on GOdel's Theorems in Paris under a
slightly diff erent title: 
O n a n
e x t e n s i o n
o f
f i n i t a
r y
m a t h e
m a t i c
s
w h
i c
h
h
a
s
m
o
s
t
certainly not yet been used" (see my [1993a1). The meaning of this title is explained in
the paper. More or less the same material has been presented at a lecture at t he
Technical University of Athens, Greece on the invitation of Aristides Eitahas. My thanks
to the audience there, Kostas Gavroglu in particular, f or comments and criticisms. The
same material but in a very early stage of its development, has been presented at the
Ecole Normale Supérieure of Paris in the Séminaire de Philosophie et Mathématiques
organised by Maurice Loi. My thanks to Mauric e Loi and to Yehuda Ray f or the
invitation. Thanks also to Graham Priest for providing me with the material I needed at
precisely the right time and the opportunity to discuss the material at several places in
Australia. Finally, thanks to the "Leuven"group on philosophy of mathematics, Leon
Horsten in particular. A "popular" version of this material was published in my [1992]
and a discussion of the relevance of this approach to Hilbert's program is to be found in
my [1993b]. Applications to physics have been treated in my [1993c] and [1994].
24 JEAN PAUL VAN BENDEGEM
that the an swe r is (mo st e mp hatically) no. A ctually, a stronge r cla im
wi l l be made. I t is po ssible to formulate a strict f in itis t ve rsion o f a
classical in f inite mathematical theory, such that the fo rmer is a p roper
extension o f the latter (without meaning stretching).
Probably one is tempted to think that this is an impossib ility. I t would
be, i f the additional claim we re th a t st rict f in it ist math e matics must
have the same lo g ical basis as cla ssica l mathe matics. Here, p a raco n
sistency enters in to the discussion. Separating the notion o f t riv ia lity
from the notion of consistency, classical log ic can be replaced b y a mo re
generous log ic that provides the additional power to turn strict fin it ism
into a n e xtensio n o f cla ssical math e ma tics. Anyon e f a m ilia r wit h
G6del's (neo)Platonist philosophical ideas will realize that such an ap
proach was quite simply impossible f o r him, nothing less than a p h ilo
sophical heresy. Hen ce I mig ht as w e ll talk abou t " a n e xtension o f
fin itary mathematics that Gödel wou ld most d efinitely not have used".
Instead o f presenting the general theory rig h t fro m the start, I will go
through a detailed discussion o f elementary number theory on the se
mantical le v e l. I n paragraph 2 t h e standard cla ssical mo d e l is p re 
sented. I n paragraph 3 the paraconsistent vie w is introdu ced and in 4
the rich f in itist model is formulated with the follo wing properties:
(i) I f c'/' A then f
. " 1 , I f
A
i s
a
v a l i
d
s t a t
e m e n
t
i
n
c l a
ssi
c a l
e l e
m e
n t a
r y
number theory, then A is valid in rich finit ist elementary number theory.
As said before, the cla im is indeed that rich f initist number the ory is an
extension o f classical number theory.
(ii) There exists at least one nonempty subset Fin o f the set o f all f o r
mulas F o f the language of elementary number theory, such that for A in
Fin:
Hci A iff r
• A
Furthermore this subset is easy to characterize.
In a single statement th is means that this part o f classical mathematics
can be rewritten in a st rict finit ist fashion wit h o u t the classical mathe 
matician's (qu ite understandable) wo rry th a t a ll kin d s o f nice results
wi ll disappear in the transition. What the platonist does during the day,
the finit ist can rewrite in th e evening. I n paragraph 5 , I p rese n t t h e
general method due to Graham Priest (altho ugh designed fo r d iffe rent
purposes). Finally, in 6 a philosophical discussion concludes the paper.
It mig h t help the reader to b ette r understand the approach outlined
here, i f some history is told. In previous work, I tried to formulate a type
STRICT FINITISM AS AN ALTERNATIVE FOUNDATION 25
of " strict" fin it isml in compatible with classical mathematics. Classical
theorems turned out false ("There is no largest prime nu mb er"), fin itist
theorems we re classica lly speaking false (" There i s number equal t o
its successo r" ). No t altogether satisfied wi t h t h is re s u lt , I t rie d
(though not ve ry successfully) to f in d strong e r models. F a milia r with
the wo rk on relevant and paraeonsistent lo gic, I did not realize the con
nection there could be between these lo gics and strict finitism. Reading
the papers o f Chris Mortensen, I notice d that paraconsistent and re le
vant mathematical theorie s t yp ica lly h a ve fin it e mode ls. I t seemed
natural to tu rn the question around: is it possible to f in itize theories b y
going paraconsistent? Unfortunately, the models Mortensen prese nted
in his papers, were not what I needed as a st rict fin it ist and i t wa s not
clear to me at the t ime h o w to generate other models. Thanks to a re
sult o f Graham P riest (presented in paragraph 5 ), such a meth od i s
available. It is actually extremely powerful. Thus, this paper can also be
seen as a continuation o f the wo rk o f Mortensen and Priest, not f ro m
the relevantparaconsistent point o f view but from the strict fin itist p e r
spective. A s must be obvious, th is does n o t imp ly that they share the
latter perspective.
2. The classical case
Let PA b e the theory o f (Peano) arithmetic. Th e language o f PA con
sists o f the language o f firstorder predicate logic in it s standard f o rm
with predicates restricte d to " = " (equ a lity) and fun ctions restricte d to
"S" (su ccessor), " + " (a d d ition ) and " . " (mu ltip lica t ion ). " 0 " is the
only constant of the language.
A mo d el M o f P A i s a t ri p le M = /, v
i
) w h e r e N
i s t h e
(standard) do ma in (o f the n a tural n u m b e rs )
2
, / i s a n
i n t e r p r e t a t i o n
function and v
i i s a
v a l u a
t i o n
f u n
c t i o
n
b a
s e
d
o
n
/
,
s a
t i
s f
y i
n g
t
h
e
f
o
l
l
o
w

ing conditions:
( / I ) 1 (0 ) = 0 (where O is the number zero in the domain)
(/2) / (S x) = / ( x ) C) 1 (where (D means addition in the model)
( /3) / ( x + y) = / (x) (D 1 (y )
(14) 1 (x . y ) = 1(x) 0 1 (y) (where (g) means multip licatio n in the
1 Se e my [19871
2 No ns t anda rd models are not c ons idered, alt hough t hey do not c omplic at e matters
in any s erious way . The only reason fo r the restriction to standard models is f o r clarity 's
sake.
26 JEAN PAUL VAN BENDEGEM
model)
(15) / ( = ) = f <n, n> I n E NI
(V I )
( V2)
( V3)
( V4)
I)
/
(
x
=
y
)
=
I
i
f
f
<
/
(
x
)
,
i
(
y
)
>
e
/
(
=
)
(or, equivalently, / (x ) = 1(y))
v l
(

A
)
=
1
i
f
f
v
i
(
A
)
=
0
v 1
(A
v
B
)
=
1
i
f
f
v
i
(
A
)
=
1
o
r
v
i
(
B
)
=
1
vja x A (x))= 1 i f f there is an / ' that d iffers f ro m 1 at mo st in
the value of / (x ) such that v
r
( A ( x ) ) = 1 .
A formula A is valid (c l A ) if f for all models M, v
i
( A ) = 1 .
3. Going paraconsistent
In order to prepare the ground f o r the strict fin itist case, one intermedi
ate step is necessary. I n the scheme o f paragraph 2, v
i i s a f u n c t i o n
from the formulas of PA to {0 , 1}. Suppose n o w t hat v
i i s a f u n c t i o n
from th e fo rmula s o f P A to the f o ll o win g set: 1{ 0 }, 10, I I , { 1 }1
3
.
Suppose further that the conditions are replaced by:
(11)(14) as before
(15') 1 + ( = ) = k n , n> I n e 1V)
1  ( = ) = (IN I )( N)\ 1 +(=)
Instead o f talking o f the extension of a predicate, w e n o w t a lk o f it s
positive and its negative extension. Although not relevant at this stage,
it is a crucial step in the next paragraph.
( V
I
' )
l
e
v
/
(
x
.
y
)
i
f
f
<
1
(
x
)
,
1
(
y
)
>
e
1
+
(
=
)
0 E v 1
(x =
y )
i f
f
<
/
(
x
) ,
/
(
y
)
>
E
I
—
(
=
)
(V2') l e v
/
(   A )
i f f
0
e
v
i
( A )
0 e v
i
(
A
)
i
f
f
I
e
v
/
(
A
)
( V3') l e v
/
( A v
B )
i f f
1
e
v
/
( A )
o
r
1
e
v
i
( B
)
0 e v
t
( A
v
B
)
i
f
f
0
e
v
i
(
A
)
a
n
d
O
e
v
t
(
B
)
3 This type of valuation function has been proposed by Priest in a number of papers
and in his [1987] book (see references there).
( V4') l e v
/
( 3 x A
( x ) )
i f f
t h e
r e
i
s
a
n
/
'
t
h
a
t
d i
f f
e r
s
f
r
o
m
I
a
t
m
o
s
t
i
n
the value of / (x ) such that 1 E 1/
1
, ( A ( X ) ) .
0 E (3 x i i ( x ) ) if f for all .1' that differ fro m I at most i n th e
value of / ( x ) , 0 e v
r ( A ( x ) ) .
A formula A is valid ( p c A) iff l e v
i
( A ) f o r a l l
m o d e l s
M .
It is not that hard to see that the follo win g theorem must hold:
Theo re ml.i=pc B iff B .
Proof. I t is sufficie nt to show that th e truth value o f a formula o f the
form x = y can o n ly be (0 ] o r {
I
}
. I f s o ,
i t i s
o b v i o u s
t h a t
( V 2 ' )
,
( 1 7 3
' )
and ( V4') reduce immediate ly to (V2 ), (V3) and ( V4). Suppose then th a t
v
i
(
x
=
y
)
=
(
0
,
I
)
.
T
h
i
s
i
s
t
h
e
c
a
s
e
i
f
<
/
(
x
)
,
i
(
y
)
>
E
/
4
(
.
)
a
n
d
<1 (x), I(y )> E  (=), which is impossible. QED
Thus, although this formulatio n looks mo re complicated than the p re 
ceeding one, it is e n tirely equivalent to it. I n the next step t he va lu e
{0,1} w il l show its importance.
4. Rich f init ism
STRICT FINITISM AS AN ALTERNATIVE FOUNDATI ON 2 7
Starting with a classical model M < N , I, y> as defined i n 3, a new
model M* = < N*, I *, v * 1
. > , c a l l e d
t h e
m o d e l
d e r i v e
d
f r o
m
M ,
wil
l
b
e
constructed that has the follo wing properties:
(i) N * is finite
(ii) if c 1 B then
M* i s precisely the mo d e l th a t satisfies the first o f the two re q uire 
ments of a rich fin itist model mentioned in the introduction of this paper.
N* is the fo llowin g set: HO], [ I ], [2 ], [ L , L C) 1, .. . ]). Unle ss oth er
wise indicated , L i s co nsidered to b e a fixe d number. Th e squ are
bracket notation is meant to clarify how the elements o f N* are relate d
to N. The easiest way is to read [n] as an equivalence class under a
(nonstipulated) equivalence relation, o r as a partition o f N in a f in ite
set o f parts.
The interpretation function I * , derived from I, is defined as follows:
28 JEAN PAUL VAN BENDEGEM
I E ( X = y) i f f < 1 * (4 1*(y)>E /*
0 E i t
i
* • (x
=
)
7 )
i
f
f
<
e (   A) i f f 0
/* (x),
v
t
*
.
(
A
)
/ * ( y ) > E i
* 
( = )
0 E A ) if f I E ( A )
( / I *) / * ( 0 ) = [01
(12 *) / * ( S x )= [/(Sx)1
( /3*) 1 * ( x + y) = [ 1 (x) 1 ( y ) ]
(14 *) 1 *( x . y ) = [ 1 ( . 0 0 1(y)1
(15 *) < 1 * ( 4 1 *(y)> e /* i f f there is a n E= [/(x)1, and there
i s a m e N y ll, such that < n, m > e 1 +(= )
< 1 * ( ) 1 * ( y ) > E 1* —(=) if f there is a tz E [1(x)1, and there
is a t n E [1(y)], such that < n, in > E 1 —(=).
The valuation function v
i
* . s a t i s f i e s
t h e
f o l l o w i
n g
c o n d i t
i o n s :
( 1
7
1 *
)
( I/2*)
( I/3*) I E v
i
*. ( A
v
B )
i f f
1
e
v
i
* .
( A )
o
r
l
e
v
i
* .
( B )
0 e v
i
* . ( A
v
B
)
i
f
f
0
e
4
.
(
A
)
a
n
d
1
E
4
.
(
B
)
( WO') I e v
t
* . ( 3 x A
( x ) )
i f f
t h e
r e
i s
a
n
/
* '
t h
a t
d i
f f
e r
s
f
r
o
m
/
*
a
t
m
o
s
t
in the value of / *( x ) such that I e v
1 ( A ( x ) ) .
0 e v
i
* .
( 3 x
A
(x))
i
f
f
f
o
r
a
l
l
/
*
'
t
h
a
t
d
i
f
f
e
r
f
r
o
m
/
*
a
t
m
o
s
t
i
n
the value of / *(x), 0 v
/
* . , ( A ( x ) ) .
A formula A is r*valid ( t

* A ) i f f I
e v
i
s
. ( A )
f o r
a l l
m o d e
l s
M *
.
Lemma. Fo r any formula A, f o r any classical model M a nd its d e rived
model M* , we have:
(a) I f I e v
i
( A ) ,
t h e n
I
E v
i
* . ( A )
„
(b) I f 0 e v
i
( A ) ,
t h e n
0
e v
l
* .
( A ) .
Pro o f (by induction on the length o f the formulas):
(i) B a sis : note f irst th a t if t is a t e rm, t h en i t i s e asy t o s h o w
1*(t )=[1(0 ].
(a) Suppose now that A is x = y and that 1 v
i
( x = y ) . B y
O M ,
I e 1 )
1
(x
=
y )
i f
f
<
/
(
x
)
,
1
(
y
)
>
E
1
+
(
=
)
.
T
h
u
s
,
b
y
(
/
5
*
)
,
t
h
e
r
e
is a n ( = 1(.0)E11(x)1 and a t n (= / 3 1
) )
E [ I
M ] , s u c h
t h a t
<n , m> E + ( = ). Hence, < /* (x), r ( y ) > E /
* 
1

( = ) , i . e . l e
STRICT FINITISM AS AN ALTERNATIVE FOUNDATION 29
v 1
*.
(x
y
)
(b) proceeds along precisely the same lines.
(ii) Indu ction step: three cases have to be distinguished: negation, dis
junction and the existential q uantifier. The three proofs are id e nti
cal, I therefore restrict myself to negation. Le t A be o f the fo rm —
then:
(a) I e v
1
( —
C )
i f
0
E
V
/
(
C
)
then 0 e v
i
* . ( C )
b y
i n d
u c t i
o n
i f 1 e v
i
%
( —
C )
,
hence if l e (— C) then J e v
i
% (— C ) .
(b) proceeds along precisely the same lines. Q E D
The central theorem is, o f course, the fact that:
Theorem 2. if 1c1 B then l= r* B.
Proof. Because of theorem I , it is sufficien t to show that if H
p c B t h e n
Hr* B. Now, if l=pc B then for all y I e ( B ) and, by the lemma, f o r all
derived models M*, for all v
i
* . , l E
( B ) ,
hence
B .
Q E
D
As said in the introduction, there is a ra t h e r easy wa y t o strengthen
theorem 2 to an equivalence. Let Fin be the subset of all formulas o f the
language of PA such that:
A E Fin if f there is no term t occurring in A such that
/ * (t ) = [L, L E K ... ]
Theorem 3. I f B E Fin then 1=cIB i f Wrf B.
Proof. I t is easy to see that f o r i, w it h the exception o f [L , L e i ,
[ i] , i . It then f o llo ws that / * reduces to I and hence 4 . to v
i
• T h u s , i f
l=t* B then B . Together with theorem 2, this shows the equivalence.
QED
Remark 1. Fin is not necessarily the o n ly set that satisfies the equiva
lence. Fo r it is very w e ll possib le t hat a statement A does turn out
uniquely true, even though there is a te rm t occurring in it, such that
/ * (t )= [L, L e i , ...1. Example: if S(L)0 stands f o r SS S O , wit h L oc
currences o f S, then the statement 0 < S(L)0 (wit h the standard in t e r
pretation f o r <), is un iquely true. / * ( 0 ) = [0] = 0 and / * ( S(L)0) =
[L, L EH, . . .]. Therefore, given the classical interpretation function I , no
matter what element k we pick from [ L, LED1, ...1, we w i ll have that 0
30 JEAN PAUL VAN BENDEGEM
< k. Hence 1/;. (0 < S(LO) =111
Remark 2. I t seems quite natural to ask the question whether the above
result extends to the notion o f semantical consequence. In other words,
is it possible to extend theorem 2 to:
If A
I
,
A
2
,
A
n
1
c
l
B
t
h
e
n
A
I
,
A
2
,
A
„
H
r
B
?
The answer is no, as the followin g simple counterexample shows:
S ( +
1
)
0 =
S M
O
r t
S
(
L
)
0
=
S
q
L

0
0
.
In the classical model this holds, because both premise and conclusion
are false. Yet, in a fin ite model with domain N t [ 0 1 , [11 , [ L a i l
[ L L l
, . .
. ] 1 ,
w
e
h
a
v
e
v ;
.
( S
(
1
) 0
=
S
(
L

1
)
0
)
=
t
0
1
,
a
s
r
(
S
(
L
)
0
)
=
[L, Lei, ...] a nd 1*(s(L00).[Le1[=Lel, b ut
v2.(S(L+00 = S(L)0)= to , l b as / * ( S ( L
4  1
) 0 ) = P ( S ( L ) 0 ) =
[ L ,
L E N , .
. . [ .
If we take L e / *( S ( L +
1
)0 ) b o t h
f o r
n
a n d
m ,
t h e
n
< n
,
m
>
=
< L
,
L
>
E I + (=), hence l e v
i
* . ( S ( L +
0
0
,
 
 S ( L )
0 ) .
B u t
i f
w e
t a
k e
L
e
1 * ( S (
1
+
1
) 0 )
for n and L EDle / * ( S (
1
) 0 ) f o r
m
then
< n ,
m >
=
< L
,
L E D
1 >
c
h en ce 0 E 1);• ( S
( L + 1 )
0 = S
( L
) 0 ) .
Note, however, that the fo llo wing does hold:
r
•
(
s
(
L
+
)
0
,
s
(
'
4
0
)
(
s
(
L
)
0
,
s
(
L

1
)
0
)
where A D B stands for  A y B.. Furthermore, if all A, and B belong to
Fin , then the extension o f theorem 3 holds:
If A
I
,
A
2
,
A
n
1
c
1
B
t
h
e
n
A
l
,
A
2
,
r
*
B
.
5. Beyond elementary number theory?
The natural question to ask is whether this method wi l l wo rk fo r other
mathematical theories? There are two answers t o th is question. T h e
first is quite simp ly to observe that it has been done. In fact, Mortensen
in his papers [1988] and [1990] has done precisely that. I repeat that in
this sense, fro m the technical po int o f view, not much new is offered in
this paper. Wh y then did I not simp ly state Mortensen's results? B e 
cause, and this is th e second re ply, I have f o llowe d a diffe re n t route
STRICT FINIT1SM AS AN ALTERNATIVE FOUNDATION 31
leading to the same results. As a matter o f fact, wha t I have done i n
this paper is to apply a method of Priest that is fa r more general. With
just on e rest rictio n , i t i s po ssib le t o fin it iz e a lmo s t a n y the o ry.
Stepswise, with out presenting a ll the details, it goes like this:
(i) Take any firstorder theory T satisfying the o n ly restriction that
the number of predicates of T is finite. Let M be a model o f T.
(ii) Re formulate T in a paraconsistent fashion, e xtending th e truth
values to (101, {0 ,1}, 1 I I instead o f (0,11.
(iii) I f the models of M are infin ite, define an equivalence relation R
over the domain D o f M, such th a t DI R is finite. O r, equiva 
lently, define a partition in a finite set of parts of the domain D
of M. Let the resulting model be MIR or M*.
(iv ) The model M I R o r M* is a finite paraconsistent model o f the
given firstorder theory T such th at va lid it y is extended. Thus
MIR is a strict and rich fin it ist extension of M.
The mo st importa n t th in g t o note is th a t any equivalence rela tio n o r
partition will do. This leaves room fo r an almost unlimited n umber o f
possibilities. Indeed, Mortensen instead o f using N* , preferred to wo rk
with Mmo d n. Howeve r, f o r the st rict f in itist, the n ontrivia l p ro b le m
that remains, is to fin d an R such that the resultin g model deserves to
be called a " n a tural" mode l. Th at a model i s " n a t u ra l" is indicated,
e.g., b y the existence o f theorems such as theorem 3. Mo re ge n erally,
borrowing some terminology o f the structuralist school in philosophy o f
science, i f there is a set o f intended applications of a mathematical th e
ory, say i n p hysics, then both the orig ina l infin ite theory and its rich
finitist extension should agree on this set. In mo re mundane language
and for the case of PA, this means that if we count with small, accessi
ble (n o t in the settheoretic sense o f the term, o f cou rse) o r fea sible
numbers, then both theories should agree. I f , e.g. the truthvalue o f 1 ,
8 were 0 , 1 1 , the n I wo u ld not co nsid er th is a " n a tu ra l" model. A s
shown in th is paper, f o r PA such models exist.
The extension to the integers is a rather du ll exercise. Apart f ro m e x
isting proposals, it is obvious that the most " natura l" partition o f Z, is:
Z*  ( L + 1 ) , [  2 ] , [ I ] , [0 ] , [1 ] , [2 ] , [ L , LE1311, ...][.
However, entering the domain of the rationals, things become different.
One possible finite partition of Q that leads to a "natural" mo del is this
(here the square brackets refer to open or closed intervals):
(a) Q
L
=
f x
I
X
E
Q
&
X
L
J
,
Q
_
L
=
r
x
I
x
E
Q
8
c
.
x

1
,
1
(b) Q
d
=
{ {
,
o
/
q
l
p
i
g
Q
&


L
<
p
,
<
#
0
}
,
32 JEAN PAUL VAN BENDEGEM
(c)
a
n
d
=
{
l
x
,
y
[
x
,
y
E
Q
&
(
x
,
y
E
Q
d
o
r
x
=

L
o
r
y
=
L
)
&  (az)(x < z < Yckz e Qd)/•
A simp le example may illustra t e th is rath e r complicated description.
Take L = 3. Then Q is split up in:
(a) a ll fractions large r than o r equal to 3 (Q , ) a n d a ll fractions
smaller than or equal to 3 ( Q _
3
),
(b) t h e set o f determined fractions:
Qd = t

2 ,
 1
,
 1
/
2 ,
0
,
1
/
2
,
1
,
2
1
,
(c) a ll inbetween in tervals, i.e. 0 1,1/2[,
]1/2,0[, ] 0 ,I / 2 [ , ]1/2,1[, 11,21, 12,3[I.
There are two senses in wh ich this partition can be considered natural.
First, n ote that, i f a ll t e rms are inte rpreted in Q
d
, w e d o h a v e
t h a t
4. ((3 y )(x . y =1 )) =111. Fo r, i f r ( x ) = [ p i g ] e Q
d
, t h e n 1 * ( y ) =
E q l p l ,
but that i s an element o f Q
d a s w e l l ,
t h u s
t h e
t e r m
( p l q ) .
( q /
p )
i
s
uniquely determined, namely 1. Thus if, say, L > 10, then (1/7).7 = I
holds in t h e strict fin it ist th e o ry. No t e that this approach avoids a l l
mention of roundingoff criteria, approximate results, and the like. The
second sense is related to N and Z. I f A and B are two partitions then B
is an extension o f A (A ext B) if f f or every element a E A , there is an
element b e B such that a c b. Clearly, we have N* ext Z*. Somewhat
less cle a rly (bu t immedia te ly cle a r f ro m the e xamp le ): Z * e x t
However, it is ce rta inly (and unfortunately) not the case that th is Q * is
the on ly partitio n that satisfies Z * ext Q * . A d d itional crite ria are re 
quired. A good reason to reject a partition proposal is i f it turns out that
too many statements are truefalse. Thu s, wh at might appear to be the
most "natural" pa rtition o f Q, namely
Q"=I[x , y [lx =k .3 &y =( k+ 1 ) .8 8 c L k <L  1 1 u
1
1

0
o
,

L
.
5
[
1
u
t
[
L
c
5
,
+
.
[
1
,
w
h
e
r
e
8
i
s
a
n
u
m
b
e
r
c
l
o
s
e
t
o
0
,
actually is not. Usin g this mode l, f o r every x, v ; • (x . x
1 = 1 ) = 1 0 , 1 1 .
Likewise f o r every choice of x, v;.(x + 0 = x)= 1 0 , 11 including the in te
ger values o f x. In Z* (and in N * ) these statements we re exclusive ly
true. Hence, the f irs t mode l presented i s a b e tter candidate than the
second one.
Although the full details w ill be presented in papers to follo w, inclu d 
ing strict and rich finitist geometry  see my [1987a] f or an in tuitive ap
STRICT FINITISM AS AN ALTERNATIVE FOUNDATION 33
proach  let me ju st mention that introducing (some) irrationals is quite
easily achieved.
Suppose one would like to have A ri . Replace the interval I x , E Qind
that has A f f as a member by ]x, AU,[ and 1V2, y[ and add [ N/] as a
member to a I I t is easy to sh o w that 1/1.((ax)(x = A ff ))

= P l . T o c a p 
ture in a strict fin itist theory the notion o f an irrational is not excluded,
strange though it may appear at first sight (fro m the fin itist vie wp o in t,
that is). The next step is the development o f analysis. Mortensen in his
[1990] has already developed a paraconsistent diffe rential ca lculus thus
showing, once again, that for the st rict finitist, the prob lem reduces to
find " natura l" models.
6. At what p rice rich finitism?
6.1. Fro m the classical point o f vie w, the most obviou s and "heaviest"
price to pay is to give up consistency. Perhaps one is impressed by the
idea that n o nt rivia l strict fin it ist math ematics exists, bu t one is no t
willin g to give up consistency.
A general argument to the contrary, is this: the wh o le idea o f con sis
tency proofs started with Hilbert's problem how to control the in troduc
tion o f ideal elements in a mathematical theory. Typica lly, these id e a l
elements had to do with infinity. A s long as everything was finite, there
was no problem. Hence, as all models are all fin ite to start wit h , co n sis
tency is not o f prime importance any more. Rather trivia lit y is the key
issue. We do no t want t rivia l models, say mod els with a ll state me n ts
truefalse. Thus, it seems obvious that consistency and trivia lit y should
be co nsidered separate concep ts. T h in kin g w i t h in a paraco nsistent
framework does precisely that. B y dropping the ex falso, a th eory does
not become t rivia l then inconsistent. B u t classica l log ic is to a large
extent ide ntified b y the ex falso. It the refore seems unavoidable th a t
strict f init ism should go hand in hand wit h paraconsistent logic.
A more specific argument has to do wit h the paradoxical nature of the
very id ea o f a largest numbe r (o r numeral) L . The paradox i s q u ite
familiar, of course: i f L is the largest number, wha t prevents me fro m
writing down the next one, namely L91? Apparently nothing, as I have
just done p recisely that. The strict fin itist's answer is: the re is n o rea l
paradox as LED I = L. In oth e r words, a lthoug h the q uestion can be
asked: " Wh a t is th e re su lt if 1 i s added to L ?" , the a n swe r is q u ite
simply: "The number L". Note that in the question itse lf, only numerals
smaller or equal to L , are mentioned. Pleasing though this solution may
seem, it proba bly w i l l not satisfy the antif in itist. He o r she wil l argue
34 JEAN PAUL VAN BENDEGEM
that, perhaps the set o f numbers o r numerals is fin it e , b u t, apparently,
the number o f operations, say additions, I must be able t o t a lk about,
must be larger. Therefore, the cardin ality o f the set o f a ll additions is
larger than L. Hence, L is not the limit, and the paradox is restored.
In my [1987], I opted fo r a conve ntionalist solution: as i t does not
make sense to ta l k a bout the largest numb e r, I accepted in ste ad a
provisional largest number in f u ll knowledge o f the fact that larger finite
numerals a re imagin a b le. I n t h e appro ach o utlin e d here , a n o th er
solution is p ossible: th e large st n u mb e r is an inherently p aradoxical
id ea
4
.
It
i s
e
a
s
y
t
o
s
h
o
w
t
h
a
t
i
f
/
*
(
x
)
=
I
*
(
y
)
=
[
L
,
L
E
)
1
,
.
.
.
]
,
t
h
e
n
b
o
t
h
1 E V
*
( X
=
y
)
a
n
d
0
E
/
*
(
X
=
y
)
,
h
e
n
c
e
t
h
e
t
r
u
t
h

v
a
l
u
e
o
f
x
=
y
i
s
{
0
,
1
}
.
Thus, i f [L, L e 1, ...] is identified as the largest element of N * (a qu ite
natural suggestion), then the rich fin itis t mode l te lls us that it i s both
true and false that the largest natural number is equal to itself.
Another objection mig h t be raised. If one cla ims th at L = LED 1 is
truefalse, wh y then does it n o t f o l lo w th at 0 S 0 (o r 0 = 1 ) i s
truefalse? Wha t is wrong wit h the f o llo wing argument? Finit istica lly
speaking, as the st rict fin itis t th eory extends the classical th e o ry, th e
wellknown P Aaxiom must hold:
(V x)(Vy)((Sx = Sy) D (x = y))
Instantiate x by S(L)0 and y by S ( L
 1
) 0 , a n d ( * )
b e c o m e s :
(*
)
(SS MO SS(L00)D (smo = s(L00) (**)
(or, if a slight abuse of language is allowed for,
(L +1 = L )D(i . , = L 1 ). )
Repeat the argument, and 0 = SO) is derivable. The answer is: because,
as we already indicated, th e strict f in it ist th e o ry does n ot extend th e
notion of semantical consequence. I t is sufficien t to explicate the phrase
"repeat the argument" in the reasoning above. To arrive at the conclu
sion 0 = SO, one needs modus ponens. Thus, the first step wil l be:
SS(L)0 = S S (
1

1
) 0 , ( S S
( L ) 0
=
s s
( L  0
0 )
D
( s
( L
) 0
=
s
( L
 1
)
0 )
S(L)0 = S (L
1
)0
r•
The first premise is truefalse as has been shown. Likewise, the second
premise is tru ef a lse. I t i s s u ff i c i e n t t o n o t e t h a t v
i
* . ( S S ( L ) 0 =
4 I n Priest's terminology , the largest finite numb er wo uld qualify as a dialetheia (see
Priest [19871, pp. 39).
STRICT FINITISM AS AN ALTERNATIVE FOUNDATION 35
S S ( L

) 0 1
= 1
0 ,
1
1
a
n
d
v
; .
(
S
( '
4
0
=
S
(
L

0
0
)
=
1
0
1
.
H
o
w
e
v
e
r
,
t
h
e
c
o
n
c
l
u

sion S(L)0 = S ( L
 1
) 0 i s
e x c l u s i
v e l y
f a l s
e .
H e n
c e ,
t h
e r
e
i
s
a
c
a
s
e
s
u
c
h
that the premises are true and the conclusion false. Thu s the conclusion
is not a semantical consequence o f the p remises, and th e repetition
breaks down.
6.2. One mig h t object that the use o f the truthvalue {0 , 1 } is a t ric k
needed to prove the richness o f the fin it ist theory. Suppose that instead
of the set o f truthvalues I {0 }, {0 , 1 }, ( I I ) we had used the set {0, 1/2,
1}, thu s o b t a in ing a f a mi lia r three value d l o g ic . Sta tements wi t h
truthvalue {0 , 1}, now have truthvalue 1/2. Instead o f truefalse, the y
are n ow undecided. Does not this reflect more closely the finitist's atti
tude? A ctu ally, as has been shown b y Priest, the threevalued lo g ic is
equivalent to the paraconsistent lo g ic, i f valid ity is extended to 1 /2. A
formula A is valid if f for all models M, v(A ) = 1 o r 1/2. Thus, the problem
really comes down to this: are there any arguments fo r the fin itist to ac
cept a statement that is undecided, in some cases as valid? Formulated
thus, the answer is yes. The undecided cases are precisely those cases
that a re n o t acce ssible t o the s t rict f in it i st (sema n t ica lly speaking),
thus they should not p la y any part in the determination whether a sen 
tence is valid o r not. In a way, the choice o f {0,1} is much clearer than
1/2. " Un decided" does n ot preclude that at a la t er stage, i t e ith er be 
comes " t ru e" o r "false". The value {0 , 1 } , however, i s a determined
value.
6.3. Let me try ye t another argument to defend the use of paraconsis
tent logic as the underlying logic o f strict fin itism. Th ere is something
paradoxical about the fact that, on the one hand, sema n tically, e very
thing is stric tly fin ite , wh ile, on the other hand, syn tactically, the strict
finitist and the cla ssica l mathe matician are speaking t h e same la n 
guage. Thu s, both are ab le t o asse rt t he sta tement S (L )0 = S(L+1)0.
They will , however, speak about diffe rent models, as f o r the infin itist,
the statement is false, whereas f o r the st rict finit ist i t is truefalse.
These considerations  as Priest remarks in h is [1991 ] paper  mu st
remind one o f the Lfte n h eimS ko lem t h eo rems
5
. A c t u a l l y , t h e
c o r r e 
spondence is qu ite strong. A s an example, think about the notion o f a
largest p rime number. Although on the syntactical level, there will be a
proof of the statement th at th ere is n o largest p rime number, in th e
model there def in itely is. B u t this semantical fa ct is no t expressible in
the the ory itse lf. It is reflected howeve r in th e f a ct that the theo rem
5 Dav id Isles has developed a s im ila r idea (though not in a paraconsistent f ra mework )
in his [1994] . Hi s w ork is closer to t he origina l work of Ye s s eninVolpin , on e of the
founding fathers of modern strict f initis m. See \Mein [1987] f or details.
36 JEAN PAUL VAN BENDEGEM
"There is no largest prime number" w ill turn out to be truefalse in any
particular model M. Th is can b e seen a s f o llo ws. Co n sid er the t w o
statements
(Vx)(S (L )0+ x +SO= S (40 + x) and
(Vx)(S (L )0+ x +SO S( L )0 + x).
Both are truefalse. In the f irst case, a ll numerals above L collapse, and
thus all prime numbers among them collapse as well, hence there is a
largest p rime numbe r. Th e theorem is false. B u t in th e second case,
everything looks q uite classical  a ll numbers la rger than o r equal to L
are d istin ct thus th e th eorem is true . Give n Priest's ge neral me thod
outlined i n pa ragraph 5 o f t h is p a p er, t h e f in it is t a n a lo g u e o f
L6wenheimSkolem theorem is: Given a paraconsistent firstord e r the 
ory T with a fin it e n u mber o f predicates, i f the t h e o ry h as mo d e ls
(countable or uncountable), it has finite models.
7. Strict fin itism witho u t cla ssical mathematics in the background.
The strongest criticism imaginable is no doubt this: granted that a form
of strict fin i t is m i s possib le , i t i s s t i l l t h e case t h a t t o b e a ble t o
formulate i t , yo u ne e d cla s sic a l stan d a rd ma th e ma tic s i n t h e
background. To be more precise, the domain N* o f the strict fin ite model
M* is a partition of a classical domain N of a classical model M. What i f
these are not available? Or, if you like , should strict finitism not have its
own proper foundations? The answer is yes.
Suppose that a strict finitist starts with a limit e d domain NJ of natural
numbers (o r numerals), Nf = 10, 1, 2, L ) . Let us, f or the moment, ig
nore the problem o f there being a largest number o r numeral L and h o w
the strict finitist is supposed to find it
6
. S u p p o s e
f u r t h e r
t h a t
t h e
f i n i t i s t
wants to do mathematics over NJ. He or she defines a successor fu nc 
lion, addition and mu ltiplica tion over NJ, (possib ly, though n o t neces
sarily) in the follo win g way:
(D I ) Succ: N f —NJ, such that: i f n < L, then Succ(n) = m
(whatever m is) and if n = L, then Succ(n) = L.
(D2) Eli: N J x Nf N J , such that: n EDO n
6 My [1987] focuses mainly on this type of problem, especially the first chapters where
an art if ic ial mathematician, the sheetmathematician, is introduced (or shemath f or
short).
(D3) 0 :
Note that in (D2) and (D3), it is unnecessary to add the limi t L. Th is is
taken care of by (DI ). Example: suppose that N f = {0, I , 2 }, i.e. L = 2,
and that the result o f 2 (134 2 is asked for. Th e n 2 e 2 = 2 ED Succ(1) =
Succ(2 G) I ) = S u c c (2 e su cc ( 0 )) , S u cc(S u cc(2 e o ) ) .
Succ(Succ(2)) = Succ(2) = 2.
Given su ch a set N f with the appropriate functio ns, the language o f
classical mathematics can be interpreted as follows:
M f = <Nf, I f v f If > i s a strict f in itist model, such that:
(11f) lf (0 ) = 0
(12f) I f (S x) = Succ(if(x)) if ! fix) is defined
= L, in all other cases
(13f) I f (x + y) = I f (x) C)f (y ), if If (x) and I f (y) are defined
= L, in a ll other cases
(14f) I f (x.y) = I f(x) 0 I f(y ), if I f(x) and I f (y ) are defined
= L, in all other cases
(15l) I f FM = f <If (x), I f (y)> I I f (x) = I f ( y))
I f (=) = ( (Nf x N f ) \ I »  (=)) L.) t<L , L >)
The valuation function v f
i f ( w i t h
r a n g e 1
{ 0 ) ,
( 0 , 1
} ,
(
I
)
1 )
s a t i s
f i e s
t h
e
following conditions:
(V1D
(V21)
(V3f)
(V4f)
STRICT FINITISM AS AN ALTERNATIVE FOUNDATION 37
n eSucc(m) = Succ(n e m)
Nf x N f —› Nf such that: n 0 0 = 0
n 0 Succ(m) = (n 0 m) G34 n
1 e v f
if
( x
=
y
)
i
f
f
<
I
f
(
x
)
,
I
f
(
y
)
>
e
I
f
+
(
=
)
0 E y i
e
lf
(
X
=
y
)
i
f
f
<
I
f
(
x
)
,
I
f
(
y
)
>
E
I
f

(
=

)
I E l l it
(
A )
i f f

0
e
v f
/f
(
A
)
0 e i f
f f
(
A )
i f
f

1
e
v
f
I
f
(
A
)
l e v f
/ f
( A
v
B
)
i
f
f
I
E
V
f
i
f
(
A
)
o
r
1
E
V
f
i
f
(
B
)
0 E vf if (A v B) if f 0 e vf
if ( A ) a n d
0 e
v f
i f
( B )
1 e vf
/ f
( a v
A
( x )
)
i
f
f
t
h
e
r
e
i
s
a
n
I
f
'
t
h
a
t
d
i
f
f
e
r
s
f
r
o
m
I
f
a
t
most in the value of I f (x) such that I e v f
/ f
, ( A ( x ) ) .
0 e vf if (axA (x)) if f for all If ' that diffe r from I f at mo st in
38 JEAN PAUL VAN BENDEGEM
the value of 1f(x), 0 E 1 1
4

,
( A ( X ) ) .
A formula A is rfvalid l f A ) i f f 1 e i l
i f
( A ) f o r a l l
m o d e l s
M f .
Note that in the d efinitio n above, there is no need to re fer to classica l
mathematics a t all. Hen ce, i t is p e rfectly possible to formulate stric t
finitist mathematics on its o wn (a t le a st f o r ele mentary a rithme t ic).
But, it is not hard to see that the fo llowing theorem can be proved:
Theorem 4. r f A if f r * A
Proof L et Nf and N* be the domains o f resp. M f and M* . De f ine the
function N f —› N* , such th a t J (i ) = [i ] , f o r i < L , a nd I ( L ) =
[L,L 0 1,...]. Note th at J is a bijection . I t is n o w a rou tine ma tter t o
show tha t ( h p correspon ds e xa ctly t o (
J i * ) a n d v i c e
v e r s a .
T h e
conditions on the valuation function are identical in both cases, s o w e
need not bother about this. A s an example, I will take (13*):
/*(x + y)= [1(x )C) 1(y)1 (wh ere I i s th e classica l in t e rp retatio n
function wherefrom / * is derived)
Now, e ither 1f(x) and I f(y) are defined or not. In the f irst case, it fol
lows that both / (x ) = l f (x ) and 1 (y) = If (y) (or, i f necessary, a p e rmu 
tation can be found). I t then follo ws that:
1*(x + y)= [1 (x) 0 1(y )]= J (1(x)C) 1 (y)) =
I (I f (x ) C) 1 f(y)) = .1(1f(x + y)).
In the second case, one o r both o f /(x ) and 1(y) are not defined. I t then
follows that / (x ) > L and 1(y) > L and, certainly, / ( x + y) L. Thus:
/ * ( x + y) = [L, 1,01, „ J . J(L )=. 1 (If (x + y)).
Summarizing, we find that for all x and y:
I l f ( x + y) 1 * ( x + y)
Finally, note the necessity in (15f) to include <L, L > in the negative e x
tension o f T h i s is necessary to obtain the translation. QED
Corollary: c l A then t f A
One might st ill object that, on the synta ctical le vel, the strict f init ist is
STRICT FINITISM AS AN ALTERNATIVE FOUNDATION 3 9
still using the complete vocabulary o f classical mathematics. Th is is in 
deed the case, but so lely f or the purpose o f setting up a translation. I f
the strict finit ist is, in any reasonable sense o f the word, strict, then he
or she will allow only a finite number of names for numbers. I f we accept
LF1 or L.L as names, then, ob viously, one cannot avo id a (pote n tially)
infinite number o f names. Thus, an additional restriction must be made,
to the effect that, e.g., on ly terms o f a limite d co mp lexity are allowed,
where the limitations refer to the number o f operations (additions a nd
multiplications) mentioned in the t e rm. I t therefore f o llo ws tha t, syn 
tactically, the language o f the theory o ve r Nf is le ss exp re ssive than
the language o f P A, hence it cannot be an extension. Wh a t remains,
however, is that it is still the case that, i f A i s a statement that is f i n i
tistica lly expressible, then i f A holds cla ssically, it holds fin itistica lly.
Thus, we remain as close as possible to classical mathematics.
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