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Hindman's Theorem, Ultrafilters, and Reverse Mathematics (To Appear in Jsl)

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Hindman's Theorem, Ultrafilters, and Reverse Mathematics (To Appear in Jsl) The Journal of Symbolic Logic Volume 00, Number 0, XXX 0000 HINDMAN’S THEOREM, ULTRAFILTERS, AND REVERSE MATHEMATICS (TO APPEAR IN JSL) JEFFRY L. HIRST APPALACHIAN STATE UNIVERSITY Abstract. Assuming CH, Hindman [2] showed that the existence of certain ultralters on the power set of the natural numbers is equivalent to Hindman’s Theorem. Adapting this work to a countable setting formalized in RCA0, this article proves the equivalence of the existence of certain ultralters on countable Boolean algebras and an iterated form of Hindman’s Theorem, which is closely related to Milliken’s Theorem. A computable restriction of Hindman’s Theorem follows as a corollary. Throughout this paper, proofs are carried out in the formal system RCA0.For a full exposition of RCA0 and reverse mathematics, Simpson’s book [6] is the best source. For the following discussion, the salient features of RCA0 are that it is a 0 subsystem of second order arithmetic with induction restricted to 1 formulas and comprehension restricted to relatively computable sets. The language of second order arithmetic is remarkably expressive, and the following concepts are easily formalized. A countable eld of sets is a countable sequence of subsets of N which is closed under intersection, union, and relative complementation. We use Xc to denote {x ∈ N | x/∈ X}, the complement of X relative to N. Given a set X N and an integer n ∈ N, we use X n to denote the set {x n | x ∈ X ∧ x n}, the translation of X by n.Adownward translation algebra is a countable eld of sets which is closed under translation. Given a countable collection {Gi | i ∈ N} of subsets of N, RCA0 suces to prove the existence of the downward translation algebra generated by {Gi | i ∈ N}, which is denoted by h{Gi | i ∈ N}i and consists of all nite unions of nite intersections of translations of elements and complements of elements of {Gi | i ∈ N}. RCA0 can also prove that h{Gi | i ∈ N}i is a downward translation algebra. Note that in RCA0, we encode h{Gi | i ∈ N}i as a countable sequence of countable sets, and that each set in h{Gi | i ∈ N}i may be repeated many times in the sequence. By allowing this repetition, we can organize the encoding sequence so that given the indices of any elements of the algebra we may deterministically compute indices for their unions, intersections, complements, and translations. 1991 Mathematics Subject Classication. 03B30, 03F35, 03D80, 05D10. Key words and phrases. Hindman’s Theorem, Milliken’s Theorem, reverse mathematics, computability. c 0000, Association for Symbolic Logic 0022-4812/00/0000-0000/$00.00 1 2 JEFFRY L. HIRST APPALACHIAN STATE UNIVERSITY Suppose that F = {Xi | i ∈ N} is a countable eld of sets. In RCA0, we dene an ultralter on F as a set U N satisfying the following four properties: If Xi = ∅ then i/∈ U. If i, j ∈ U and Xk = Xi ∩ Xj, then k ∈ U. If i ∈ U and Xj Xi, then j ∈ U. c ∈ ∈ If Xj = Xi , then i U or j U. Where no confusion will arise, we will abuse notation by writing U for {Xi | i ∈ U}, so that we may write statements using the standard notation for ultralters. For example, consider the following denition and lemma. An ultralter U is an almost downward translation invariant ultralter if ∀X ∈ U ∃x ∈ X (x =6 0 ∧ X x ∈ U). While the denition only requires one translating x for each set, the existence of many is shown by the following lemma. Lemma 1. (RCA0) If U is an almost downward translation invariant ultralter and X is an element of U, then the set {x ∈ X | X x ∈ U} is unbounded. Proof. Suppose by way of contradiction that x is the largest number in X such that X x ∈ U. Since X x ∈ U, there is a y ∈ X x such that y =06 and (X x) y ∈ U. Since y ∈ X x, we have x + y ∈ X. Additionally, (X x)y = X (x+y), so we have x+y ∈ X, x+y>x, and X (x+y) ∈ U, contradicting our choice of x. a If X N, then the notation FS(X) denotes the set of all nonrepeating sums of nonempty nite subsets of X. For example, FS({1, 2, 5})={1, 2, 3, 5, 6, 7, 8}. Using this notation, it is easy to state Hindman’s Theorem. Theorem 2 (Hindman’s Theorem). If f : N → k is a function mapping N into the natural numbers less than k, then there is an innite set X N such that f is constant on FS(X). Proof. The original non-formalized proof appears in [3]. For a proof of Hind- + a man’s theorem in the subsystem ACA0 , see [1]. The set X in the statement of Theorem 2 is called an innite homogeneous set for the partition. Given any set G N, we refer to the statement of Theorem 2 with f(x) as the characteristic function for G as Hindman’s Theorem for G. 1. Equivalence results. We begin with the central result of the section, linking an iterated form of Hindman’s Theorem to the existence of almost down- ward translation invariant ultralters. Theorem 3. (RCA0) The following are equivalent: (1) IHT (Iterated Hindman’s Theorem) If {Gi | i ∈ N} is a collection of subsets of N, then there is an increasing sequence hxiii∈N N such that for every j ∈ N, {xi | i>j} satises Hindman’s Theorem for Gj. (2) Every countable downward translation algebra has an almost downward translation invariant ultralter. Proof. To prove that (2) implies (1), suppose {Gi | i ∈ N} is a collection of subsets and let U be an almost downward translation invariant ultralter on h{Gi | i ∈ N}i, the downward translation algebra generated by {Gi | i ∈ N}. Dene a sequence of (indices for) nested sets hXiii∈N and an increasing sequence HINDMAN’S THEOREM, ULTRAFILTERS, AND REVERSE MATHEMATICS (TO APPEAR IN JSL) 3 h i c of integers xi i∈N as follows. Let X0 be whichever of G0 and G0 is in U. Let x0 = 0. Suppose that xn has been chosen and that Xn has been chosen so that ∈ c Xn U and either Xn Gn or Xn Gn. Since U is an almost downward translation invariant ultralter, by Lemma 1 we can nd a least xn+1 ∈ Xn such b that xn+1 >xn and Xn xn+1 ∈ U. Let Gn+1 denote whichever of Gn+1 and c ∩ ∩ b Gn+1 is in U, and dene Xn+1 by Xn+1 = Xn (Xn xn+1) Gn+1. h i We will show that FS( xn n>t) Xt. Let xi1 ,...,xik be a nite sequence of elements of hx i . Note that i > 0, so X exists, and x ∈ X . n n>t k ik 1 P ik ik 1 k Treating this as the base case in an induction, we have x ∈ X .For P m=k im ik 1 k the induction step, suppose that x ∈ X . Since i i 1, P m=j+1 im ij+1 1 j j+1 k x ∈ X . Since i 1, by the denition of X , X X ∩ m=j+1 im ij P j Pn ij ij 1 b k k ∩ ∈ ∈ (Xij 1 xij ) Gij ,so m=j+1 xim Xij 1 xij . That is, m=j xim Xij 1, completing theP induction step. By induction on quantier-free formulas, we have k shown that x ∈ X . Since t<i , we have t i 1, so X X P m=1 im i1 1 1 1 i1 1 t k ∈ and m=1 xim Xt. Since xi1 ,...,xik was an arbitrary non-repeating sequence of elements of hxnin>t, this suces to show that FS(hxnin>t) Xt. c To complete the argument, note that for each t, Xt Gt or Xt Gt .Thus h i h i c h i for each t, FS( xn n>t) Gt or FS( xn n>t) Gt , so the sequence xn n>0 satises the iterated version of Hindman’s Theorem presented in (1). To prove that (1) implies (2), suppose that {Gi | i ∈ N} is a countable down- ward translation algebra. Since {Gi | i ∈ N} is a countable collection of subsets, we may apply (1) to nd an increasing sequence of integers hxiii∈N such that h i c for all j, FS( xi i>j) Gj , where either Gj = Gj or Gj = Gj. We will show { | ∈ N} that U = Gi i is an almost downward translation invariant ultralter on {Gi | i ∈ N}. ∈{ | ∈ N} c If X Gi i , then for some j, Gj = X or Gj = X , so either ∈ c ∈ h i 6 ∅ ∅ ∈ X U or X U. For each j, Gj FS( xi i>j) = ,so / U. Suppose ∈ ∈ N X1,X2 U. Then for some k, j , X1 = Gk and X2 = Gj . Also, for ∩ ∩ c { } some m, Gm = X1 X2 or Gm =(X1 X2) . Let t = max j, k, m . then h i ∩ ∩ ∩ ∩ ∩ ∩ 6 ∅ FS( xi i>t) Gm Gk Gj = Gm X1 X2. Since Gm X1 X2 = , ∩ ∈ ∩ ∈ ∈ Gm = X1 X2. Thus, if X1,X2 U then X1 X2 U. Finally, if X U and Y is an element of the downward translation algebra satisfying X Y , { } then for some j and k, X = Gj and Y = Gk. Let t = max j, k and note that ∩ h i 6 ∅ ∈ X Gk FS( xi i>t) = ,soGk = Y and Y U.ThusU is an ultralter on the downward translation algebra.
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