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Zeitsehr. f. math. Logik und Grundlagen d. Math. Bd. 8, S. 57-70 (1962) CONSTRUCTIVE METHODS OF NUMERATION by ARTHUR H.KRUSE in University Park, New Mexico (U.S.A.) 1. Indroduction The content of this paper may be developed in a set theory of the kind axio- matized by A. MOSTOWSKI [5]. This is the kind of set theory obtained by modifying that of K. GODEL [3] or that of P. BERNAYS [l] in such a way that the existence of elements which are not sets (Urelemente, or atoms) is not ruled out. Moreover, neither the axiom of choice nor the Fundierungsaxiom will be assumed. The use of some terminology and notation will be indicated briefly. Elements x and y determine the pair [x , y] = {{ x} , {x, y}} . For all classes a2 and 9Y, d x 9 = = {[z, y] I x E d; y E a}. The universe is @, the class of all elements. A relation is any subclass of % x %, and a function is any relation satisfying the usual condi- tions. For each relation 9,W-l = { [x, y] 1 [y , x] E a} is the inverse of 9, dom (9) = = {XI [x, y] E W for some y} is the domain of 92, and im(9) = dom(9-l) is the image of 9%'. A function from, or on, a2 has domain d. A function to, or into, 37 has image c 37. A function onto 37 has image 37. If p and y are functions, y o ip = = { [x, Z] I x = y (y) and y = p (x) for some y}. If p, y , 6 are functions, E o y o ip = =to (yo p) = (60 y)o p. For each function p and set X, pix= pn(Xx@). Each ordinal number consists of its predecessors; 0 is the void set and the first ordinal number. o is the first infinite ordinal number. 0 is the class of all ordinal numbers. For each set X, an X-tuple is a function on X. For each class d and each set X, dX) is the set of all X-tuples over (i.e., into) d, and a finite sequence over d is any object which is an n-tuple over d for some n E For each set X, I XI is the power of X, i.e., the class of all sets equi-potent with X. For each class d, 9(d) is the class of all subsets of d, Bw(d) is the class of all well-orderable subsets of d, and W (d) is the class of all one-one functions into d with domain E 0. For all classes d and 9Y, d \ 37 = {xI x E d; x B 37}. Other standard notations will be used without introduction when no confusion is likely to result. The purpose of the paper will now be indicated. Suppose 9 is a function from a set of sets to a set of sets. Suppose that for each X E dom(6), each member of %(X) arises from some finite subset of X in some finitary way. In terms of such functions e", this paper undertakes a somewhat systematic study of two problems. The first problem is to prove that for X infinite and well-orderable, IE(X)I 5 I XI. Special attention is paid to the uniformity of such inequalities. Consider a class d of pairs [E, [] such that E satisfies conditions of the sort vaguely suggested in the first paragraph, [ is a one-one function on an ordinal number, and im ([) E dom (3).

Constructive Methods of Numeration

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Zeitsehr. f. math. Logik und Grundlagen d . Math. Bd. 8, S . 57-70 (1962)

CONSTRUCTIVE METHODS O F NUMERATION

by ARTHUR H.KRUSE in University Park, New Mexico (U.S.A.)

1. Indroduction

The content of this paper may be developed in a set theory of the kind axio- matized by A. MOSTOWSKI [5 ] . This is the kind of set theory obtained by modifying that of K. GODEL [3] or that of P. BERNAYS [ l ] in such a way that the existence of elements which are not sets (Urelemente, or atoms) is not ruled out. Moreover, neither the axiom of choice nor the Fundierungsaxiom will be assumed.

The use of some terminology and notation will be indicated briefly. Elements x and y determine the pair [x , y] = {{ x} , {x, y}} . For all classes a2 and 9Y, d x 9 = = {[z, y] I x E d; y E a}. The universe is @, the class of all elements. A relation is any subclass of % x %, and a function is any relation satisfying the usual condi- tions. For each relation 9 , W - l = { [x, y] 1 [y , x] E a} is the inverse of 9, dom (9) =

= { X I [x, y] E W for some y} is the domain of 92, and i m ( 9 ) = d o m ( 9 - l ) is the image of 9%'. A function from, or on, a2 has domain d. A function to, or into, 37 has image c 37. A function onto 37 has image 37. If p and y are functions, y o ip = = { [x, Z] I x = y (y) and y = p (x) for some y}. If p, y , 6 are functions, E o y o ip = = t o ( y o p) = (60 y ) o p . For each function p and set X , p i x = p n ( X x @ ) . Each ordinal number consists of its predecessors; 0 is the void set and the first ordinal number. o is the first infinite ordinal number. 0 is the class of all ordinal numbers. For each set X , an X-tuple is a function on X . For each class d and each set X , dX) is the set of all X-tuples over (i.e., into) d, and a finite sequence over d is any object which is an n-tuple over d for some n E For each set X , I XI is the power of X , i.e., the class of all sets equi-potent with X . For each class d, 9(d) is the class of all subsets of d, Bw(d) is the class of all well-orderable subsets of d, and W (d) is the class of all one-one functions into d with domain E 0 . For all classes d and 9Y, d \ 37 = {xI x E d; x B 37}. Other standard notations will be used without introduction when no confusion is likely to result.

The purpose of the paper will now be indicated.

Suppose 9 is a function from a set of sets to a set of sets. Suppose that for each X E d o m ( 6 ) , each member of % ( X ) arises from some finite subset of X in some finitary way. In terms of such functions e", this paper undertakes a somewhat systematic study of two problems.

The first problem is to prove that for X infinite and well-orderable, IE(X)I 5 I X I . Special attention is paid to the uniformity of such inequalities. Consider a class d of pairs [E, [] such that E satisfies conditions of the sort vaguely suggested in the first paragraph, [ is a one-one function on an ordinal number, and im ([) E dom (3).

58 ARTHUR H. KRUSE

The uniformity of I S ( X ) l 5 1 XI with respect to d lies in the existence of a func- tion I' on d such that for each [c", 51 E d, I'([Z, 51) is a one-one function from E( im(5)) into im(5) . This problem is studied in $ 4 and $ 5.

The second problem is to prove under the hypothesis that X E dom (E) is suffi- ciently large the statement that I B ( X ) I [ S ( X ) 1 and similar statements in which 9 ( X ) is replaced by "Iw ( X ) , W ( X ) , or another set depending on X . One such result is due to E. SPECKER [7], who proved that [ P ( X ) I J [ X I a if [ XI 2 151 . In $ 6 SPECKER'S method is used to prove theorems much more general for the case [ XI 2 1 w [ . The developments in $ 6 use also the developments in $ 4 and $5.

An example of a function c" of the kind indicated above is provided by the finitary closure operator 9, which is studied 9 2 . For each set X , S I X ) is finitarily very comprehensive. The operator 9 turns out to be rather messy to handle for reasons indicated a t the end of 3 2.

The following lemnia follows from [6, p. 320 (Theorem 1) and p. 279 (Theorem l)]. It may also be proved directly from elementary principles of ordinal arithmetic.

1.1. Lemma. Suppose p , ci E 0 and p < ma. Then p + wa = ma.

'

2. The operator 9 For each class d, let B.(d) be the class of all finite subsets of d. Consider a set X . We define gn ( X ) for each n E w by induction so that go ( X ) = X

and for each n E w \ { 0 } ,

S n ( X ) = X w @ , ( U { F j ( X ) [ j E n } ) .

Trivially, Sm ( X ) c 9% ( X ) if m E n E w . We define

$ ( X ) = U { S t n ( X ) [ n E w } .

It is easy to prove that S ( X ) is the smallest set Y such that X c Y and each finite subset of Y is a member of Y . $ ( X ) will be called the f in i tary closure of X . It is easy to prove that w c S ( X ) , [x, y ] E S ( X ) for all x, y E S ( X ) , and each finite sequence over 9 ( X ) is a member of S ( X ) . X will be called F-ini t ia l if and only if no member of X is a finite set.

Consider a set X and a function f on X . We define fin] for each n E w by induction so that fin] is a function on Fn ( X ) for each n E w . Let f [ O ] = f . For each n E o, let

We define /["I = U { f [ " ] [ n E w } ; f [ " ] is a function on S ( X ) . Now im( f [ " l ) c Fm(im(f)) for each n E w , and im( f [ " ] ) c S ( i m ( f ) ) .

For each set X and each x E g ( X ) we define rank F(x; X ) to be the first n E w

for which x E S7$ ( X ) . Let a* = (z 1 x E F ( X ) for some S-initial set X } . For each x E %* we define rank .(x) to be the smallest n E w for which x E S n ( X ) for some $-initial set X .

The following two lemmas may be proved by induction. The details will be omitted.

f["+lI = p1 uw,{f[ml(d I z E Z H I Z ~ S 9 ? + 1 ( X ) \ $ " ( X ) } .

CONSTRUCTIVE METHODS OF NUMERATION 59

2.1. Lemma. Suppose X is an %-initial set and x E F ( X ) . Then rank F(x) ==

= rank g ( x ; X ) . In fact, for each F-init ial set Y and each n E w ,

S % ( X n Y ) = % % ( X ) n S % ( Y ) = S % ( X ) n % ( Y ) = a ( X ) n S n ( Y ) . 2.2. Lemma. Suppose X and Y are sets with X %-initial, and suppose f is a

junction from X onto Y . Then for each x E F ( X ) , rank F(f["](x); Y ) 5 rank g ( x ) .

2.3. Lemma. Suppose f i s a function with dom(f) an 9- ini t ial set. Then irn(fLn]) = 9 % ( i m ( f ) ) for each nE w , and im(f["]) = 9 ( i m ( f ) ) .

Proof. It suffices to prove that im(f[%]) = 9 * ( i m ( f ) ) for each n E w . Suppose it is false. Let k be the first n E w for which inequality holds. Then k > 0. Also, im(flk]) C F k ( i m ( f ) ) . WehavesomeZESk(im(f)) \ irn( f[kl) . ThenZ cSk-l ( i m ( f ) ) ,

9 k - l ( i m ( f ) ) = im(flk-lI). Now the axiom of choice for finite sets of sets is valid in the set theory assumed in this paper. Hence there is a function g from Z into dom(f["-'I) for which f [ k - l ] ( g ( z ) ) = z for each z E 2. Let W = i r n ( g ) . Then W c dom(f[k-l]) = 9 k - l ( d o m ( f ) ) is finite, and hence W E Sk(dom(f ) ) = dom(f[k]). Now Z 6 S k W l ( i m ( f ) ) by choice of k and Z. Hence there is c E Z such that c E gk-l(irn(f)) and c 6 S i ( i m ( f ) ) for each j E k - 1 . Then rank F(c; int(f)) =

= k - 1 . Sincec = f L k - l ] ( g ( c ) ) , rank F(g(c)) 2 k - 1 by 2.2. Thus W Q 9',(dom(f)) for each j E k - 1 . Hence W B Fk-l(dom(f)) , and W E 9k(dom(f)) \ Sk-l(dom(f)) . Hence f['](W) = {f["l](w) I 1u E W } = Z, and 2 E im(fLk]), contradiction. Q . e. d.

*

2.4. Example . We define a function f on F ( 0 ) so that for each zE 9 ( 0 ) ,

f(.) = ( ( w 2 ) \ w ) w z ,

Then f is one-one, and im(f) i s S-init ial . Moreover, S ( S ( 0 ) ) = % ( O ) , f [ " ] = f , and a ( i m ( f ) ) =+ im( f ) . Hence im( f [ " ] ) =I %(im(f)) .

Consider a set X. We define a function gX from S ( X ) into Pg,(X) by induction on rankg,( ; X ) is follows. If x E S ( X ) and rankg(%; X ) = 0, let W,(x) = {x}. If x E 9 ( X ) and rank F(x; X ) > 0, let g x ( x ) = U{gX(y) I y E x}. The following lemma is routine.

2.6. Lemma. Suppose X is a set. Then g X is a function from % ( X ) onto Pg,(X). For each x E S ( X ) , x E F ( 9 X ( x ) ) and rank $ ( x ; X ) = r a n k F ( x ; g , ( x ) ) .

For each class d, we define

S(J4 = U{S(X) I x E PW)}, and d will be called 9-ini t ial if and only if no member of d is a finite set.

We conclude this section with some remarks which we shall leave rather vague since we do not care to pursue the matter further. The class of all sets and the class of all functions whose domains are sets give rise to a category in the sense of S. EILENBER~ and S. MACLANE [2]. The class of all 9-initial sets gives rise to a subcategory. There seems to be no natural way in which 9 gives rise to a covariant functor on the first of these categories (cf. 2.3 and 2.4). However, 9 and the assign- ment f -+ f [ " ] (for each map f in the subcategory) provide a covariant functor from the subcategory to the first category.

GO ARTHUR EL KRUSE

3. The functions A , , . . . , A,, The functions A , , . . . , A14 will be defined and their needed properties exhibited

in lemmas. Since these lemmas are straightforward, their proofs will be omitted. One of the standard constructive proofs of the SCKRODER-BERNSTEIN equivalence

theorem (e.g., that in [l, pp. 114-1151) yields a function A, such that 3.1 follow- ing holds.

3.1. Lemma. A , is a function with domain { [ f , g ] I f and g are elements and one-one functions; im(f) c dom(g); im(g) c d o m ( f ) } . For each [ f , g ] E dom(A,), A , ( [ f , g ] ) i s a one-one function from dom(f) onto dom(g) .

There are unique functions f and g on w \ { O } such that for each nE w \ { O } , f ( n ) is the identity function on w (thus f (n) ( i ) = i for each i E w ) , g(n) is a one- one function from d' into w , and

g(n) ( .Z jG% wn-'-jh (7.)) = 1 7 i E n P ( i ) h ( j )

for each h E d), where p is the unique strictly increasing function from w onto the set of all prime finite ordinal numbers. We define a function A, on w '\ {0} so that A,(n) = A,([f (n ) , g(n)]) for each n E w \ (0).

3.2. Lemma. A , is a function on w \ (0). For each n E w \ { O } , A , (n ) is a one- one function from w onto wn.

We define a function A, on 0 x 0 so that for all p , v E 8, A, ( [ p , v ] ) is the unique function f on p v such that f ( p x + A ) = v il + x for each x E v and each A E p .

3.3. Lemma. A, is a function on 0 x 07. For all p , v E U , A 3 ( [ p , v ] ) is a one-one function from p v onto v p .

We define a function A , on U \ ( O } so that A,(n) =A,(n + l ) - l o & ( ? a ) for each n E w \ (0) and A , ( S ) = A 3 ( [ w , w']) for each S E 0 \ w .

3.4. Lemma. A, i s a function on 0 \ { O } . For each 6 E Q \ { 0 } , A4(S) i s n one- one function from w' onto

We define a function A, on 0 \ w as follows. Consider y E U\,w. Let ,9 = U {CX E 0 I w" 5 y } . Now w@ 5 y < up+'. Let f be the identity function on wp. Let g =A,(S)- l I y ; g is a one-one function from y into wb by 3.4. Let &(Y) = 4 ( [ f , sl) .

3.5. Lemma. A , is a function on U \ w . For each y E 0 \ w , A,(y) is a one-one function from wp onto y where p E 8 is such that wfl 5 y <

We define a function A, on U \ w as follows. Consider y E U \ w . Let p = U {CX E U [ wa 5 y } . Now wb 5 y <

3.6. Lemma. i s a function on 6' \ w . For each y E 0 \ o, A,(y) i s a one-one function from y onto wB+l where wP

We define a function A , on 0 x 0 so that for all p , v E 0 , A,([ ,u, v ] ) is the unique function f on p + v such that f ( x ) = v + x for each x E p and f ( p + A) = 2 for each L E Y.

Let A6(y) =A,(?) o (A , (B) - l ) .

y < w@+1.

CONSTRUCTIVE METHODS OF NUMERATION 61

3.7. Lemma. A, is a function on U x 8. For all p , v E 0, A,([p , v ] ) is a one-one function from p + v onto v + p .

There is a unique one-one function f on w such that for each n E w , f (n) is a 1-tuple and f (n) (0) = n. There is a unique one-one function g on the set S of all finite sequences over w such that for each s E S ,

g b ) = 17nEdom(e)P(n)8(n)+1 E w > p being the same here as before 2.2. We define A, = A, ( [ f , g ] ) .

3.8. Lemma. A, is a one-one function from w onto the set of all finite sequences

There is a unique one-one function f from w into P g ( w ) such that f (n ) = {n} for each n E w . There is a unique function h on P F ( w ) such that for each X E P g ( w ) , h ( X ) is the unique one-one increasing finite sequence whose image is X . By 3.8, A,-lo h is a one-one function from P s ( w ) into w . Let A, = A 1 ( [ f , A 8 - l o h]) .

ouer w .

3.9. Lemma. A, is a one-one function from w onto g g ( w ) .

Define the function Alo on @(*) so that for each f E Alo(f) is the function g on w such that g(n) = { f ( z ) I z E A, (n)} for each n E w . The proof of the following lemma uses the axiom of choice for finite sets of sets.

3.10. Lemma. A,, is a function on %(*). For each f E @("), A, , ( / ) is a function from w onto g s ( i m ( f ) ) . For each one-one f E @(O), A,,(f) is one-one.

Consider a function f on w . We define a function h on w by induction as follows so that for each n E w , h (n) is a function from w onto gm ( im (1)). Let h (0) = f . Suppose n E w and h(n) has been defined. Let h(n + 1) be that function k on w such that lc(2i) = h(n) (i) and k ( 2 i + 1) = rllo(h(n)) (i) for each i E w . We define a function h, on w2 so that h,(w n + i) = h(n) (i) for all n , i E w ; h, is a function from w2 onto g ( i m ( f ) ) . Let h, = hoo (A, (Z)) . By 3.2, h, is a function fro'm w onto F ( i m ( f ) ) . Let g, be the identity function on w . Then g, is one-one, and im(g,) = w c F ( i m ( f ) ) . Let g, be the unique function g on F ( i m ( f ) ) such that for each 5 E g ( i m ( f ) ) , g(x) is the first n E w for which h,(n) = 2. g, is one-one, and im (g2) c w .

We define a function A,, on a(*) so that for each f E a(*), A,,(/) = A,( [g , , g 2 ] ) where g, and g, are defined in terms of f as in the preceding paragraph.

3.11. Lemma. All is a function on a(*). For each f E @(*), A,, ( f ) is a one-one funcfion from w onto F ( im ( f ) ) .

We define a function A,, on the class d of all functions f such that dom(f) is a subset of U so that for each f E d, A12( f ) is a function on i m ( f ) , and for each xE i m ( f ) , A,,(/) (x) is the first iy E dona(/) for which f ( a ) = z.

3.12. Lemma. A,, is a function on the class of all functions f such that dom(f) is a subset of 0. For each f E dom(A,,), Alz( f ) is a one-one function from im( f ) into donz ( f ) , and f o (A,, ( f ) ) is the identity function on im ( f ) .

62 ARTHUR H. KRUSE

We define the function A13 on 8 ( U ) so $hat 3.13 below holds.

3.13. Lemma.A,, is a function on .!?(U). ForeachXE 9 ( 0 ) , R 1 3 ( X ) is the un iqw one-one increasing function from a n ordinal number onto X .

We define a function A,, on the set of all pairs [ f , g] such that f and g are functions and dom (f) and dom ( 9 ) are ordinal numbers. For such f and g , AI4 ( [ f , g ] ) is defined to be that function h on dom(f) + dom(g) such that h ( x ) = f ( x ) for each 3c E dom(f) and h(donz ( f ) + A) = g ( A ) for each iZ E dom(g) .

3.14. Lemma. A,, is a function on the set of all pairs [ f , g ] such that f and g are functions whose domains E 0 . For such f and g , dom (Al4 ([f , 91)) = dom ( f ) + dom ( 9 ) , im(nl,crf, 91)) = i m ( f ) u i m ( 9 ) , a n d A , , ( [ f , sl) I &om(/) = f .

4. The classes 3 and 2

Suppose E is a function from a class of sets to a class of sets. Then 5 will be called inclusion-preserving if and only if for all Y , 2 E dom(E) with Y c 2 , E ( Y ) c t (2 ) .

Let 3 be the class of all 5-tuples (a, ,!I, [, t, e) such that (i)-(v) below hold.

(ii) 6 is an inclusion-preserving function on ( B + 1) \ a . (iii) For each non-zero limit ordinal number ,u with a < ,u 5 #?,

(iv) t is a function from a onto E (a). (v) e is a function on { f I f is a one-one function from an ordinal number onto

an ordinal number; a 2 dom(f) < im ( f ) 5 p } ; for each f E dom(e) , e ( f ) is a func- tion from E (dom(f)) onto 5 (im (f)) .

Consider g = (a, 8, l , t, e) E 9'. There are ordinal numbers y and S such that 0.9' a < my+1. We shall define va for y < I 5 6 by transfinite recursion so that y 1 is, a function from w' onto [(a') and qa I a = t for y < A 5 6 and so that y 1 = p l P I w 1 f o r y < A < p i d . I f y + 1 5 6 , let

(i) a , p E B\w; a 2,8.

E(p) = u { E ( & I a < A < P I .

V y f l = -4 , ( [Z> e(A,(a)) O t o (&b)-I)l) (cf. (iv), (v), 3.6, and 3.14); vy+l meets the requirements. For each ordinal number A s u c h t h a t y ( A < 3 , + 1 ~ 6 , l e t

(cf. (v), 3.4, and 3.14); pll+I meets the requirements. For each limit ordinal num- ber p such that y < ,u 5 6 , let yrL = U { f n I y < A < ,u} ; cpP meets the requirements (cf. (iii)). The definition of vn for y < 3, 5 6 has been completed. We now define @ ( g ) as follows. If a =,!I, let @(g) = Z . If a < B < dfl, let

VA+I = A I ~ ( [ V A , @(A4(3,)) O P r o (As(A)-')])

@(d = e(A5(B, O ( A 5 b - l ) ) O (A5b)) O ( 4 ( B ) - 1 )

@ ( 9 ) = e(A5(B)) O (A5(P)F1)

(cf. 3.5 and (v)). If p = d, let @ ( g ) = ~ 8 . If cob < B , let

(cf. 3.5 and (v)). The definition of @(g) has been completed.

CONSTRUCTIVE METHODS OF NUMERATION 63

We now have a function @ = ([g, @(g)] 1 g E 9). The following lemma may be

4.1. Lemma. @ i s a function on 9. For each g = (a,B, l ,z, e ) E 9, @ ( g ) i s

Let 9' ={(a, j3, 5 , z, e ) E 9 I j3 = w8 for some 6 E 0). Let < c9' x 9' be defined

verified.

a function from ,!? onto 5 (j3).

so that for all (MI, B1>51, t l 9 el) 7 (4 3 j 3 2 , t 2 9 z2, ez) E 9 ' 7

( f i l , B 1 , E l , t l , e l ) < ( a 2 , 8 2 , t 2 , % , @ * )

if and only if a, = a,, Bl c &, l1 c 5,, z1 c t2, and el c Q ,

< B,, t1 = 6, I dom(t,), tl = t,, and el = e, I dom (el)). It may

4.2. Lemma. For all g,, g2 E 9' with g, < g,, @(g,) c @(g,) (i.e., @(gl) =

Let H be the class of all couples ((a, j3, 5 , z, e ) , ~ l ) such that (vi)-(viii) below hold.

(vi) (a, p , E , 7, e) E 9. (vii) There are y , 6 E 0 such that a = wy, w8

(in which case be verified that < partially orders 9'. The following lemma may be verified.

= @ ( 9 2 ) I dom(@(g,))).

/? < d + l , y is a one-one func- tion on cod, im (y IwY) c [ ( w Y ) , and im(y1 ( w n + l \ d ) ) c 5 ( w A + l ) \ [ (wA) for each 1 E 6 \ y .

Consider h = ((a, j3, 5 , t, e ) , y) € H. Let y and 6 be given by (vii). Let

Ql' =A1([Y I w Y , 4 2 ( 7 ) 1 )

(cf. 3.12 and 3.1); 9' is a one-one function from my onto E(wY). Let yA for y < 1 5 6 be given as in the definition of @ ( g ) for g E 9. Then py+l I w8 = t if y + 1 5 6; moreover, im (t) = [ ( m y ) , and im (qA) = 6 (wy) for y < 1 5 6 . Hence

t (w"1) \E(wA) c irn(QIA+ll (wA+1 \ w")

f r l = 4 , ( Q l A + , I (w1+l \ W Y ) ) I (E(w"+l) 1 E(w")

and the function

is a one-one function from [(aA+') \ l ( w A ) into w'+l\ w' for y 5 1 < 6 . For y 5 1 < 6 the function

. 94 = 4" I (w'+l 4 I f a ] )

Ql = Ql'U(U {Qli I y 51< 6))

(cf. 3.1) is one-one from wA+l \ w' onto E(w'+'l) \ ((mi). The function

is one-one from w8 onto E(w8) . We now define Y ( h ) as follows. If Y ( h ) = ~ l . If w 8 < B , let

= w8, let

y(h) =',([yo ( ~ 5 ( f f l ) - 1 ) , n 1 2 ( @ ( r 1 5 ( 8 ) ) 0 Yl) (cf. (v), 3.5, 3.12, and 3.1).

64 ARTHUR H. RRUSE

We now have a function Y = { [ h , Y ( h ) ] I h E A"}. The following lemma may

4.3. Lemma. Y is a function on A"- For each h = ( (a, ,f3, E , t, e ) , y ) E A", Y ( h )

Let X' = ((9, y ) E A" I g E 9'). Let < * c X' x 2' be defined so that for all

which case y1 = y , I dom(y,)) . It may be verified that < * partially orders 9. The following lemma may be verified.

4.4. Lemma. For all h,, h, f X' with h, < * h,, 'Y(h,) c Y(h,) (i. e., F(h,) =

be verified.

is a one-one function from B onto t(j3).

(g l?Yl ) , (gZ?Y2)E* ' , (g, ,yl)<*(g,,y,) if andonly if g 1 i g 2 and y1cy2 (in

= Y(h2) I d o m ( ~ ( h 1 ) ) ) .

6. The. functions rl, . . ., r, Consider n E w \ (0). We may specify unique functions E n , z,, en, yn on 0 \ w

(a) En (p ) is a function on (,5 + 1) \ w . For each A E (/? + 1) \ w , ln (B) (12) = A@). (b) t,(/3) is a function from w onto dn), zn(/3) (0) is the (only) member of { O } ( " ) ;

for each i E w \ { 0 } , zn(/?) ( i ) is the n-tuple of exponents of the first n primes in the prime factorization of i.

(c) en(B) is afunction on {f I f is a one-one function from an ordinal number onto an ordinal number; w 5 dom(f) < i m ( f ) 5 ,?I}; for each f E dom(e,(B)), en(/?) ( f ) is a function from (dom(f))(") onto (im(f))(*), and en(,!?) ( f ) ( t ) = f o t for each t E (dom(f))(*).

< w B c l ; for each 12 E w', yn(@) (A) is the (only) member of {A}(*).

so that (a)-(d) below hold for each ,8 E 0 \ w .

(d) yn(B) is a function on w' where w' 5

We define a function y n on U\w so that

9% (B) = ((w.9 B 3 En ( B ) > zn (B) > en ( P I ) 9 Yn ( P I ) for each p E U \ w . Now yn(j?)E&' for each B E U \ w , p,&(wY)Ei%" for each yEO\{O) , and ~ n ( w Y ) < * p m ( w d ) for a l l y , d E U with O < y < d .

We define a function rl on (w \ (0)) x (U \ w ) so that for all n E w and j3 E 0 with 0 < n and w 5 B , T', ( [ n , B ] ) = Y ( y n ( B ) ) . The following theorem follows from 4.3 and 4.4.

5.1. Theorem. r, is a function on (w\{O))x (U\w). For each nEw\{O) and each /? E U \, w , r, ( [n , /I]) i s a one-one function from ,!3 onto @("). For each n E w and alZ y , 6 E U with O < y < 6, I',([n, my]) cTl([n, w ' ] ) , i .e . , F,([n, wy] ) = = F l ( [ n , w'] ) I wy.

For each n E w \ {0 } we define

A n = U { r l ( [ n , ~ " I ) I y E U \ { O ) } . 5.2. Corollary. Suppose n E w \ (0). A , is a one-one function from U onto U(,).

FOT each y E 0 \ ( O } , im(d, I wY) =

CONSTRUCTNE METHODS OF NWERATION 65

The existence of functions satisfying the conditions on I‘, and A , in 5.1 and 5.2 is rather generally known. E. SPECKER [7] develops a result such as 5.1 for the case n = 2 .

We may specify unique functions Sand on 0 \ w so that (e) and (f) below hold for each B E U \ w .

(e) i (p ) is a function on (/I + 1) \, w. For each 1 E (B + 1) \ w, g((8) (1) is the set of all finite sequences over 1.

(f) @ ( p ) is a function on dom(e,(p)) (cf. (c)); for each f E dom(e,(B)), $ ( B ) ( f ) is a function from T ( p ) (dom(f)) onto j ( B ) ( im(f)) , and ?(@) ( f ) ( t ) = f o t for each

1 E m (dam ( f ) ) * We define a function ij3 on 0 \ w so that

- d B ) = ( ( w , B 7 C((B), 4 3 , i j ( B H 7 Y,((B))

for each /3 E 0 \ w (cf. 3.8 and (d)). Now @ ( p ) E Z for each ,!? E 0 \ w , ~1 (my) E 2’ for each y E 0 \ { 0 } , and ?(d) .< * pi(w8) for all y 7 6 E U with 0 < y < 6 .

We define a function I’, on 0 \ w so that F2(/3) = Y(F(p)) for each p E Q \ w . The following theorem follows from 4.3 and 4.4.

5.3. Theorem. I‘, is a fu,nction on 0 \ w . For each /3 E 0 \ w , I‘,(/?) is a one- me function from ,3 onto the set of all finite sequences over p . Por all y , 6 € 0 with o < y < s , ~ , ( w ~ ) c ~ , ( w ~ ) , i.e., r,(d) =rZ(wd)Iwd.

We define r3 = u{r2 (wy) I y E 0 \ (0)). 5.4. Corollary. r, is a one-one function from 0 onto the c h s of all finite sequences

over U. For each y E 0 \ { 0 } , im(r31 wy) is the class of all finite sequences over my.

The existence of a function r, for which 5.4 holds is mentioned and used without proof in [4, bottom of p. 5441, in which “Yma’’ should be replaced by “TWa” in accordance with [a, top of p. 5371.

We now define a function I’, on the class of all one-one functions 5 such that dom(5) E 0 \ w and im(5) \ im(CIw) is 5-initial (cf. $ 2). Consider a such C. Let f j = dom(5). Let 5 be the function on (/3 + 1) \ a defined so that E ( A ) = F((im(5I A ) ) for each 1 E ( p + 1) \a. Let t = All([lw) (cf. 3.11). Let e be the function on { f I f is a one-one function from an ordinal number onto an ordinal number; w 5 dom(f ) < im(f) 5 8) such that for each f E dom(e ) , e(f) = (5 o f o (-l)[m]

(cf. 0 2). Let y = clws where ad 5 p < Let a = w . Now (i)-(vii) of $ 4 may be verified (use 2.3 for (v)). Let h = ((a, p , ( , t, e) , y ) . Then h E 2. Now let r4(o = wh) .

The following theorem may be verified.

5.5. Theorem. r, is a fulcction on {el[ is a one-one function; dam([) E O \ w ; im(C) \ irn(c1 w ) is 9-initiaZ}. For each 5 E dom(T,), I‘4([) is a one-one function from dom(5) onto g ( i m ( < ) ) . For all y , 6 E 0 \ (0) and el, c2 E dom(r,) with dom(cl) = wY and dam((,) = d, i f y I 6 and 5, = 121wy (i.e., if 5, cc,), then r,cc,) = r,cc,)iwy i.e., r4(c1) c r 4 c c 2 ) ) .

5 Ztschr. f . math. Logik

66 ARTIIUR H. KRUSE

5.6. Corollary. I f 5 is a one-one function on 8 such that im(5)'\im([1w) is 9-initial, the function

5' = u { u C l w Y ) l Y E O\{O}}

is a one-one function from U onto F ( i m ( [ ) ) , und im(('1WY) = -F(im(SIw")) for each y E U \ { O } .

Let r, be the function [' of 5.6 with 5 the identity function on 0 (for 0 \, w is 9-initial) .

5.7. Corollary. r, is a one-one function on 8, and im(T5) = S ( U ) . For each y E U \ {O} , im(T,lwY) = F(wY).

We define a function I'6 on the set of all functions 5 with dom(5) E U \ w as follows. Consider such a function 5 . Define a function f on dom(5) so that f ( I ) = = w + A for each 3, E dom(5). Then im(f) is $-initial, and f is one-one. Let

r6(0 = ( 5 ( f - ' ) ) ' " ] ( '4(f))

cf. 9 2).

The following theorem follows from 5.5, 2.3, and the definition of (50 (f-*))[" '].

8.8. Theorem. r6 is a function on {[I[ is a function; dom(5) E U \ w ) . For each 5 E dom(r6 ) , r6(Q is a function from dom(5) onto F( im(5 ) ) . For all y , S E U \ (0) and ll, CZ E dom(r6) with dom(5,) = wY and dom(5,) = w', if c CZ (i.e., if y 5 6 and Cl = Sz IwY) , then r6(<1) cr6(52) (i.e., r 6 ( 5 1 ) = r6 (5Z)IwY).

6.9. Corollary. If [ is a function on 9, the function

5' = u { r 6 ( 5 1 w ~ ) i Y ~ w w isa function f romUontoS( im(13) ,and im(r /~Y) = S(irn(5lwY)) foreachyE O ' , { O ) .

We define a function r, on the set of all one-one functions 5 such that dom(t) E 0 \ w as follows. Consider such a function 5 . By 5.8 and 3.12, AlZ(F6([)) is a one-one function from S ( i m ( 5 ) ) into dom(5) . Let r7([) = A1([~,AlZ(r6(~))]) (cf. 3.1).

The following theorem may be verified.

5.10. Theorem. r, is a function on (51 5 is a one-one function; dam(() E 8 \ L O } .

For each 5 E d o m ( r 7 ) , r7(() is a one-one function from dom(5) onto F ( i m ( 5 ) ) . The following theorem may be proved by essentially the method of proof of 5.10

(use 5.9 to get a function 5'' from 5, use the method of defining A12 to get a one-one function 5'' from S ( i m ( 5 ) ) into 9, and apply the ~CHRODER-BERNSTEIN theorem (cf. [ l , pp. 114-1151) to [ and 5" to get 5").

5.11. Theorem. Suppose 5 is a one-one function on 0. Then there is a one-one function [* from 0 onto F( im(5 ) ) .

CONSTRUCTIVE METHODS OF NUMERATION 67

6. A generalization of a lemma of E. SPECKER E. SPECKER [7] proved that for each set X with a t least 5 members, I P ( X ) I $ I XI2.

For all sets M and X , M will be called X-ancestral if and only if M c P ( X ) and

6.1. Theorem. Suppose X , M , 8, T, and A are sets such that (a)-(d) below hold. (a) M i s X-ancestral. (b) E i s an inclusion-preserving function from M onto a class of sets; for each set

C c M which i s c-well-ordered, %(U C ) = U {%(2)[ 2 E C } . There i s a set Y E M such that I&'( Y)I 5 I YI and I Y I i s an aleph.

(c) r i s a function with domain { [ r , f ] I f is a one-one function; dom(f) c im(f) c X ; dom(f) E M ; r i s a one-one function from an infinite ordinal number onto dom(f)}; for each [ r , f ] E d o m ( r ) , I'([r, f]) i s a function from E(dom( f ) ) onto S ( im( f ) ) .

(d) d i s a function on S ( X ) . For each u E S ( X ) , A (u) i s a function, dom(A (u) ) E 0 , i m ( A (u)) E M , and u E B ( i m ( d (u ) ) ) .

Suppose R i s an equivalence relation on W ( X ) such that for all f , g E W ( X ) , if f R g , then im(f) = im(g). Then I W ( X ) / B l $ l E ( X ) l . Hence I W ( X ) l Z& I % ( X ) [ ,

Proof. Suppose I W ( X ) / R [ 2 I % ( X ) l . There is a one-one function F from W ( X ) / R into 8 ( X ) .

There is a function B from W ( X ) / R onto P w ( X ) such that for each W E E W ( X ) / R and each f € W , B ( X ) = i m ( f ) .

By the last part of (b) there are a , A , tl, and t such that a E U \ w , A E M , z1 is a one-one function from a onto A , and z is a function from a onto E ( A ) .

We shall define recursively a one-one function cp from N ( X ) into X , N ( X ) being the f i r s tpEU such that I p I $ l X I .

Let p ( A ) = tl (A) for each A € w . Consider ,$ E N ( X ) \, o. Suppose p(A) has been determined for each A E B . We

wish to define p (B) . Now be the function on (B + 1) '\ a such that for each A € (B + l)\a,Lf(A) ==B(im(plA)) (observe that i r n ( p 1 I ) E M by (a) since A c im(p1 A ) c X ) . Let e be the function on { f I f is a one-one function from an ordinal number onto an ordinal number; a 5 dom(f) < im(f) 5 p } such that for each f E dom (e ) ,

This result will now be generalized for the case in which I XI 2 I w I . for all sets T and Y with T E M and T c Y cX, Y E M .

p w m 1 55 Ip(xx)I 3 and ] P ( X ) I 2& p(q.

is one-one. Let

e(f) = W f , (v ldom( f ) )O f o ( v Idom( f ) ) - l l ) .

Now (i)-(v) of $ 4 hold. Let g = ( c x , p , L f , z , e ) . Now g E 9 . Let k = @ ( g ) . By 4.1, k is a function from B onto ( ( B ) . Let

S = { I E p l f o r some W E W ( X ) . / R , k ( A ) = F ( W ) and y ( A ) $ B ( W ) } .

Let k, = A13(S) (cf. 3.13); k, is a one-one function from an ordinal number onto S. Let k, = (p,/B)o k,. Then k , E W ( X ) , and k,E W , for some W , E W ( X ) / R .

6 .

68 ARTHUR H. KRUSE

Then also B( W,) = i m ( k l ) = im(y1S). If F ( W,) = k (R) for some R E p , then, for such R , since pllB and F are one-one,

p,R) E B(W,) e=+ 3, E @ y ( 4 6 B(W,)>

a contradiction. Hence P(W,) $: k (R) for each A € f l , and F(W, ) $ t(fl). If

(d) and (b), contradiction. Hence im(A(F(W, ) ) ) aim(plB). Let a be the first, REdom(d(P(W,))) for which d ( P ( W , ) ) (A)Bim(vJB). Let y(B) = A ( F ( W , ) ) ( a ) .

The definition of pl is complete.

Via pl, 1 N(X)[ 5 ( X I contrary to fact. Thus the supposition that 1 W ( X ) /R( 5 5 IE(X)I is incorrect.

To get ( W ( X ) I & ( S ( X ) / , let R be the identity relation on W ( X ) . To get 19w((x)I g l E ( X ) ( , let R c W ( X ) x W ( X ) be such that for all f , g E W ( X ) , f Rg if and only if i m ( f ) = i m ( g ) ; then B is a one-one function from W ( X ) ,' R onto P p w ( X ) . Since also S,(X) c 9 ( X ) , I Y ( X ) I

i m ( A ( P f W 1 ) ) ) c i m ( y I B ) , then F ( W , ) E 9 ( i m l d ( F ( W , ) f ) ) cE( im(y /B ) ) = t ( B ) by

I E ( X ) I . Q.e.d.

6.2. Corollary. Suppose that X is a set and IwI 5 1x1. Suppose R is as described

( W ( X ) / RI S I U {X(")I n E w} l

and hence also I W ( X ) / R I $ 1 X I '' for each n E w . The inequalities obtained from these by replacing W ( X ) / R by any one of W ( X ) , g W ( X ) , and P ( X ) hold also.

Proof. In 6.1 take M : E, I', and d so that M = P ( X ) , 5'( Y ) is the set of all finite sequences over Y for each Y E 9 ( X ) , I'( [r , f ] ) = { [ g , f o g] I g E %(dam ( f ) ) } for each [ r , f ] E dom( I ' ) , and d (9) = g for each g E c"(X) . Q.e.d.

in 6.1. Then

6.3. Remark . In 6.2, for the conclusion I W ( X ) / RI 1x1" for each nE w , one m y replace the hypothesis that I w I 5 I X I by the hypothesis that X i s infinite. If X i s infinite and I W ( X ) / RI 5 IXln, a rather obvious extension of a method of SPECKER [7] leads to the existence of a one-one function from w into X . The hypo- thesis IwI 5 ( X I i s thus established, and 6.2 may be applied. Similar inequalities for X finite but sufficiently large may be established by the usual finitary combinatorial methods.

6.4. Theorem. Suppose X , M , 8, and r are sets and (a), (b), and (c) of 6.1 hold. Suppose X is well-orderable. Then ) Z ( X ) I 2 1x1.

Proof. There are a , A , z l , and t as in the proof of 6.1. There are B E 0 \ a and a one-one function 9 from onto X such that y I a = z, . We may define g and k = @ ( g ) as in the proof of 6.1; k is a function from ,B onto E ( X ) . By 3.12, q - l o ( A 1 2 ( k ) ) is a one-one function from E ( X ) into X . Q.e.d.

The following theorem gives more information on the structure of E in 6.1 for the case in which X is well-orderable and M = 9 ( X ) .

CONSTRUCTIVE METIIODS OF NUMERATIOSN 69

6.5. Theorem. Xuppose X is a well-orderable set, and suppose E' is an inclusion- preserving function from P ( X ) into a class of sets. Then (a) and (b) below are equi- valent. (Moreover, (b)+ (a) does not depend on X being well-orderable.)

(a) For each c-well-ordered set C c P ( X ) , 3 ( U C ) = U {E(Z)I Z E C } . (b) For each Y E B ( X ) , 3 ( Y ) = U { E ( Z ) l Z E P p ( Y ) } . Proof. Trivially, (b) implies (a). Suppose (a) holds. Since X is well-orderable, to prove (b) it suffices to show that

for each p E 0, for each one-one function f from p into X ,

(*) E(im(f)) = U {E(Z) IZE gp( im( f ) ) } .

Suppose this fails for some ,8 E 0 . Consider the first such p. It is easy to show that ,9 = a + 1 for some iy E 0 \ w . There is a one-one function f from #I into X for which (*) fails. Define the function g on iy so that g(0) = /(a) and g ( l + A) = /(A) for each A E a . Then g is one-one, im(g) = im( f ) , and

E(im(f)) =S( im(g) ) = U { E ( Z ) [ BE g%(im(g))} = U {%(Z)IZE Y%( im( f ) ) } , contradiction. Thus (b) holds. Q.e.d.

6.6. Theorem. Xuppose X is a set and IwI 5 1x1. Let Z ( Y ) = { [ x , y](y is a finite sequence over Y , and xE P ( i m ( y ) ) } for each Y c X . Suppose R is as de- scribed in 6.1. Then the conclusion of 6.1 holds.

Proof . Let F = ( r E W ( X ) I d o m ( r ) E Lo\,}. We define a function G on Pas follows, using the fact that S ( Y ) c g ( Y ) for each Y cX. Consider r E F. Let

G, = {[A, {[o, r(~)i}il a E dom(r)}, G, = ~ - 1 p w ( r ) ) (cf. 5.10). Now G, is a one-one function from dom(r) into E(im(r)) , and G, is a one-one function from S(im(r)) into dom(r). Now let G ( r ) =A,([G,, G,]) (cf. 3.1); G ( r ) is a one-one function from dom(r) onto E( ih ( r ) ) .

In 6.1 take M , r, and d so that M = Y(X), T ( [ r , f]) = (G(f 0 r ) ) o (G( r ) - - I ) for each [ r , f ] E dom(I'), and il ([x, y ] ) = y for each [ x , y] E 5 ( X ) . Q.e.d.

6.7. Corollary. Suppose X is a set and I w l S 1x1. Suppose S i s a function on P s ( X ) such that S ( Y ) is a finite sequence with image Y for each YE gp(X). Then the conclusion of 6.1 holds with E ( X ) = g ( X ) . In fact, F ( X ) is equi-potent with the E ( X ) of 6.6.

Proof . In this proof E ( X ) will be given by 6.6. Define a function G on F ( X ) so that for each x E g ( X ) , G ( x ) = [ x , S (g , ( x ) ) ] (cf. 2.5). Then G is a one-one function from S ( X ) into E ' ( X ) . Let H be the identity function on E ( X ) . Then [G, HI E dom(A,) (cf. 3.1), and A,([G, h] ) is a one-one function from F ( X ) onto S ( X ) . Now apply 6.6. Q.e.d.

The writer does not know the answer to the following question. Is the conclusion of 6.1 for every set X with J w / 5 1x1 and with S ( X ) = F ( X ) provable without the axiom of choice? In fact, the writer does not know the answer even when F ( X ) is replaced by {{x, y}lx, y E X } . 6.6 and 6.7 illustrate what can be proved by the method of E. SPECKER [7].

TO ARTHUR H. KRUSE

References

[l] P. BERNAYS and A. A. FRAENKEL, Axiomatic Set Theory, Amsterdam, North-Holland, 1958. [2] S. EILENBERG and S. MACLANE, General theory of natural equivalences, Trans. Amer. Math.

[3] K. GODEL, The Consistency of the Axiom of Choice and of the Generalized Continuum- Hypothesis with the Axioms of Set Theory, Princeton, Princeton University Press, 1940.

[4] A. KRUSE, Some developments in the theory of numerations, Trans Amer. Math. Soc. 97

[5] A. MOSTOWSKI, uber die Unabhiingigkeit des Wohlordnungssatzes vom Ordnungsprinzip,

[6] W. SIERPI~SKI, Cardinal and Ordinal Numbers,’;Warsaw, Pahatwowe Wydawnictwo Naukowe,

[7] E. SPECKER, Verallgemeinerte Kontinuumshypothese und Auswahlaxiom, Arch. Math. 5

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(Eingegangen am 11. Dezember 1961)