The second proof gives a stronger result (Theorem 2) which implies both Theorem ..... By Corollary 11 (c), Lemma 3 (b), and Konig's theorem. COROLLARY 19.
Inequalities for cardinal powers By FRED GALVIN* and ANDRAS HAJNAL Introduction
Silver [7] recently proved that, if GCH holds below ~"' 1 , then it holds at (See Corollary 12 for a general statement of Silver's result.) Silver's proof used metamathematical ideas . Direct proofs of Silver's theorem were found independently by Baumgartner, Jensen, and Prikry (see [1]). Working independently of each other, and independently of Magidor and Solovay (who obtained a similar result assuming the existence of a Ramsey cardinal), but using Prikry's ideas, we proved that, if ~"'i is a strong limit cardinal, then 2N"'1 < ~< 2 N1>+· (See Theorem 1 for a general statement of this result.) We have two different proofs for our result. The first one gives a proof of Theorem 1. The second proof gives a stronger result (Theorem 2) which implies both Theorem 1 and Silver's theorem. Though both proofs use the same ideas, we think it worthwhile to give them in detail. The work of the second author was motivated by an unpublished result of Erdos, Hajnal, and Milner stated in [2]. This result says that, if 2N1 = ~ 2 , then p--+ [w~ 1 ]k.No for some p < W3 • (Here w~ 1 is an ordinal power; in the rest of this paper, only cardinal exponentiation is used.) We thank Prikry and Silver for communicating their results to us. ~"' 1 •
I. Statement of Theorem I. Proof of the corollaries
THEOREM 1. Let K and /.,, be uncountable regular cardinals such that for all o < /., (o" fr if and only
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INEQUALITIES FOR CARDINAL POWERS
if {a: cp(a) ~ -ifr(a)} ~Stat tr, where Stat tr is the class of all stationary subsets of tr. For cp E •ON we define the rank of cp, denoted II cp II, as the least ordinal µsuch that 11 Y' II )
=
pJ.
~A +11 \" ll
•
Proof. For (a), put A"= Xfi m, let m,
Proof. Note that A. is a strong limit cardinal. Let f!'Aa) = otl't(a). Note that {a: cpt(a);;;; cp(a)} = {a: cp(a) = O} i;t Stat IC by Lemma 3 (a). Hence cpt < cp and II cpt II < v for all fEF. That is, F= LJµf!'Aa) for a< IC, it follows that I H(f) I ~ T(of!'t)· But '>f!'Aa) = otl"t(a) and II cpt II = µ
