Math. Log. Quart. 39 (1993) 353 - 356

Mathematical Logc Quarterly

@ Joham Ambrosius Barth 1993

Variations of Rado’s Lemma

Paul Howard

Department of Mathematics, Eastern Michigan University, Ypsilanti, MI 48197, U.S.A.

Abstract

The deductive strengths of three variations of Rado’s selection 1 mma are studied in set theory without the axiom of choice. Two are shown to be equivalent to Rado’s lemma and the third to the Boolean prime ideal theorem. MSC: 03B25, 04A25, 06E05. Keywords: Rado’s lemma, Axiom of choice, Boolean prime ideal theorem.

Many theorems which were originally proved using the axiom of choice (AC) have been strengthened by replacing appeals to the axiom of choice with appeals to some essentially weaker principle. In this note we will consider one of those principles due to R. RADO [6] known as Rado’s selection lemma. (RL) L e f T be a family ofjiniie sets such ihai fhere is a funciion (p wifh fhe following

properiy: For each finiie S T , cp(S) = cps is a funcfion w i fh domain S such fhai cps(K) E K for all K E S. Then ihere is a function f wiih domain T and wiih ihe following properly: For all finiie S T such fhaf S

Rado’s lemma has been used in combinatorics (see [4]), and variations of i t have been used by Y. RAV [5] in the study of commutative algebra and ordered algebraic structures. In the hierarchy of weak versions of the axiom of choice the position of (RL) can be seen in the following diagram:

T , ihere is a finiie S’ S‘ and f and cpsl agree on S .

/ (RL) - (WUF) (AC) - (BPI)

(ACF)

where (BPI) is the Boolean prime ideal theorem (every Boolean algebra has a prime ideal), (ACF) is the axiom of choice for families of finite sets and (WUF) is the weak ultra-filter principle ( fhere is a Ron principle ulfrafilter on w ) .

354 Paul Howard

None of the reverse implications hold in Zermelo-Fraenkel set theory with the possible exception of (RL) + (BPI) which is unknown in Zermelo-Fraenkel set theory but is known not to hold in set theory weakened to permit the existence of atoms (see [l], [2], [3] and [5]). However (RL) and (ACF) jointly imply (BPI) (see IS]).

In [5] Y. RAV showed that Rado's lemma for families of pairs has many of the same consequences as (RL) itself, for example this "weakening" of (RL) and (ACF) jointly imply (BPI). However RAV did not settle the question of whether or not the two versions of Rado's lemma are equivalent. We will show that the following consequence of Rado's lemma for families of pairs implies (RL). (WRL) Lei T be a set such ihai there is a function 'p with ihe following property:

For each finiie S C TI p(S) = ps is a funciion from S inio {0,1}. Then there is a function f with domain T such that for all finiie S C TI i h e n is a finiie S' T such ihai S E St and f and 'ps~ agree on S.

(WRL) is due to E. SCHECHTER [7].

We will also consider two versions of (RL) which are obtained from (RL) by weakening the hypotheses and therefore appear to be stronger than (RL). These are (SRLl) and (SRL2) given below: (SRLl) Lei T be a family of finite sets and lei 3 be a colleciion of finite subseis of

T saiisfying (1) (VK E T)(3S E 3 ) ( K E S), (2) (VSi,Sz E 3)(3S3 E F)(Si USz E S3). Assume fhere is a function 'p with domain 3 such that for each S E TI 'p(S) = 'ps is a funciion wiih domain S saiisfying (VK E S)( 'ps(K) E K ) . Then fhere is a funciion f wiih domain T and wiih ihe following property: For all S E 3 there is an S' E 3 such that S E S' and f, 'psi agree on S.

(Condition (1) could be shortened to T = U 3 and condition (2) could be written T is a directed set when ordered by inclusion.) (SRLP) is identical to (SLRl) except that (2) is replaced by (2') (vsi , sz E 3)(si u sz E 3).

(SRLl) was introduced in [3] where i t was shown that (BPI) + (SRLl) and incorrectly claimed that (SRLl) is true in a model of set theory where (BPI) fails. The author probably intended to claim that (SRL2) was true. In any case, we will show that (SRL1) c-) (BPI) and that (RL) H (SRLP).

We begin with

T h e o r e m 1. (WRL) implies (RL).

P r o o f . Assume (WRL) and let T be a family of finite sets and 'p a function which satisfies the hypotheses of (RL). Let T = { ( K , z ) : h' E T and t E K}. For each finite subset U of T', let Su = { K : (3z)(K, t) E U } and define a function p, whose domain is the family of finite subsets of TI, by P ( U ) = pu : IJ - {0,1} is the function defined by

1 if 'psu(K) = t, 0 otherwise.

Variations of Rado's Lemma 355

T' and /3 satisfy the hypotheses of (WRL). Let g be the function defined on T' given by the conclusion of (WRL). Define f as follows: f = { (K, z) : g ( K , z) = 1 }. There are three parts to the argument that f satisfies the conclusion of (RL).

First we show that f is a function. Assume that ( K , 21) and (K, 22) are in f . Then g ( K , z l ) = g ( K , x z ) = 1. Let U = { ( K , z l ) , ( K , z ~ ) } , then there is a subset V of T' such that U V and Pv agrees with g on U. Therefore P v ( K , z l ) = 1 = Pv(K,z2) so that 'psV(K) = zl and 'psV(K) = 2 2 . Hence z1 = z2.

Next we show that dom(f) = T. Choose K E T and let U = { ( K , z) : z E K }. There is a finite subset V E TI such that U V and g and PV agree on U. Let x o = ' p s v ( K ) . Since (K, 20) E U s dom(Pv), Pv(K,zo) = 1 and therefore g ( K , z o ) = 1,

Finally we show that if S C T is finite then there is a finite S' C T such that S E S' and f agrees with cpsl on S. Assume S C T is finite. Let I/ = { (K, x) : K E S and z E K }. Since g satisfies the conclusion of (WRL) there is a finite V s TI such that U s V and Pv agrees with g on U. Let S' = Sv = { K : (3z ) ( (K , z ) E V}. Let K be any element of S and let 2 = ' p s ~ ( K ) . Then z E K so (K,z) E U and

so ( K , 20) E f.

Pv(K, x) = 1. I t follows that g ( K , z) = 1 and therefore that f ( K ) = z. 0 T h e o r e m 2. (SRL1) implies (BPI). P r o o f . In view of the equivalence of (BPI) and (RL) + (ACF) it suffices to show

that (SRL1) implies (ACF). Let Z be a family of finite sets and let T be the family of all non-empty,

finite subsets of FS(2) = { u : (3n E W ) ( U : n - UZ)}, that is, T consists of all non-empty, finite sets of sequences from UZ. For each element u E FS(Z) let S, = { Y E T : (Vr E Y) ( r is a subsequence of u)} and let T = { S, : u E FS(Z) }.

We first argue that T and T satisfy (1) and (2) in the hypotheses of (SRLl). For ( l ) , i f x = ( u 1 , ..., u , , } E T t h e n z E S , w h e r e u = u ~ b ~ ^ ~ ~ ~ ^ u ,,. (Here, i f r a n d q are sequences r-q denotes the concatenation of T and q. ) . For (2), S, U S, s S,-,.

We now want to define a function 'p so that TI 3 and 'p satisfy the hypotheses of (SRLl). We first note that if u : n - UZ and r : m - UZ, then r is a subsequence of u if and only if there is an order preserving (and hence one to one) function P : m - n such that (Vj E m ) ( r ( j ) = u(p( j ) ) ) , that is r = uop. Now we fix a linear ordering of the set { /3 : (3m E w)(P : m -+ w and P is order preserving) }. For any u E FS(Z) this will induce a linear ordering on the subsequences of u by ordering subsequences r of u according to the least P such that r = u o P. We denote this ordering by <,. We can now define the function cp as follows: Let S, E 5, then 'p(A,) = PA, is the function with domain A, defined by

cp~, (Y) = the <v least element of Y. By (SRLl) we conclude that there is a choice function f for T . The function f can be used to define a choice function g for 2. For example, for each w E Z we could let Y, = { u : u is a sequence of length 1 from w } and define g ( w ) to be the first (and only) term of f (Y , ) . 0

T h e o r e m 3. (RL) implies (SRL2). P r o o f . Assume that TI T and 'p satisfy the hypotheses of (SRL2). For each

A E F let K A = { f : f is a choice function for A } and let TI = { K A : A E 3}.

356 Paul Howard

Now we define a function (PI whose domain is the collection of finite subsets of T' as follows: If S C T' and S is finite, say S = { K A ~ , .. . , KA,} then cp'(S) = pi is the function with domain S such that for i = 1,. . . , n , ' p / s ( K ~ ~ ) is the element of K A ~ (that is, the choice function defined on Ai) defined by

pp's ( KA, )( w ) = 'PA u.. .UA I ( w )

for all w E Ai. By condition (1) in (SRLP) A1 U - - . U A,, is in 3 and therefore p ~ ~ u . . . u ~ , , is defined. T' and cp' satisfy the hypotheses of (RL). Let g be a function for which the conclusion is true. That is, (*) for allfinite S1 C T' fhere i s a finite S2 C T' such thaf cpi, and g agree on S1.

L e m m a . If w E T , A , B E F and w E A n B, fhen g ( K A ) ( W ) = g ( K B ) ( W ) .

P r o o f . Let S1 = { K A , K B } C T'. By (*) there is a finite S2 C T' such that S1 C_ Sz and pi, and g agree on Sl. Hence

But g ( K A ) = &,(KA) and g ( K B ) = 'P's,(KB)*

cp$ , (K~) (w) = ' P A ~ U - U A , ( ~ ) = & ( K B ) ( ~ ) (where S2 = { K A ~ , . . . , K A , } ) so g(KA) (w) = g(KB)(w) proving the lemma. 0

Define a function f on T by f ( w ) = g ( K A ) ( W ) , where A is any element of F such that w E A . By condition (1) in (SRLS), f is defined for every w . By the lemma f is well defined. We show that f satisfies the conclusion of (SRL2). Suppose A E 3. We need to produce a B E F such that A s B and ' p ~ and f agree on A. 5'1 = { K A } C T' so by (*) there is a finite Sz T' such that S1 2 S2 and g and pi, agree on S1. That is, for all w E A , (**) p's, (KA = d K A ) ( w ) . Assume S2 = { K A ~ , . . . , KA,,} and let B = A1 U . U A,,. By definition of f, the right hand side of (**) is f ( w ) and by the definition of p' the left hand side of (**) is

References

'PB (4' 0

[l] HALPERN, J. D., The independence of the axiom of choice from the Boolean prime ideal theorem. Fund. Math. 55 (1964), 57 - 66.

[2] HALPERN, J. D. and A. LBvY, The Boolean prime ideal theorem does not imply the axiom of choice. In: Axiomatic Set Theory, Proc. Symposia in Pure Math., Vol. 13, Part 1, American Mathematical Society, Providence 1971, pp. 83 - 134.

[3] HOWAFLD, P., Rado's selection lemma does not imply the Boolean prime ideal theorem. Zeitschrift Math. Log& Grundlagen Math. 30 (1984), 129 - 132.

[4] MIRSKY, L., Transversal Theory. Academic Press, New York 1971. [5] RAV, Y., Varients of Rado's selection lemma and their applications. Math. Nachr. 79

[6] RADO, R., Axiomatic treatment of rank in infinite sets. Canad. J. Math. 1 (1949),

[7] SCHECHTER, E., Choice, Completeness, Compactness: Foundations of Abstract Analysis.

(1977), 145 - 165.

337 - 343.

In preparation.

(Received: March 25, 1992)