Bulletin The of Symbolic Logic Volume 12, Number 1, March 2006
REVIEWS
The Association for Symbolic Logic publishes analytical reviews of selected books and articles in the eld of symbolic logic. The reviews were published in The Journal of Symbolic Logic from the founding of the Journal in 1936 until the end of 1999. The Association moved the reviews to this Bulletin, beginning in 2000. The Reviews Section is edited by Alasdair Urquhart (Managing Editor), Steve Awodey, John Baldwin, Lev Beklemishev, Mirna D amonja, David Evans, Erich Gr del, Denis z a Hirschfeldt, Roger Maddux, Luke Ong, Grigori Mints, Volker Peckhaus, and Sawomir Solecki. Authors and publishers are requested to send, for review, copies of books to ASL, Box 742, Vassar College, 124 Raymond Avenue, Poughkeepsie, NY 12604, USA. In a review, a reference JSL XLIII 148, for example, refers either to the publication reviewed on page 148 of volume 43 of the Journal, or to the review itself (which contains full bibliographical information for the reviewed publication). Analogously, a reference BSL VII 376 refers to the review beginning on page 376 in volume 7 of this Bulletin, or to the publication there reviewed. JSL LV 347 refers to one of the reviews or one of the publications reviewed or listed on page 347 of volume 55 of the Journal, with reliance on the context to show which one is meant. The reference JSL LIII 318(3) is to the third item on page 318 of volume 53 of the Journal, that is, to van Heijenoorts Frege and vagueness, and JSL LX 684(8) refers to the eighth item on page 684 of volume 60 of the Journal, that is, to Tarskis Truth and proof. References such as 495 or 2801 are to entries so numbered in A bibliography of symbolic logic (the Journal, vol. 1, pp. 121218).
Ilijas Farah. Analytic quotients. Memoirs of the American Mathematical Society vol. 148 no. 702, American Mathematical Society, Providence, R.I., 2000, xvi + 177 pp. The Boolean algebra PN of subsets of the natural numbers has some very well-known ideals, starting with the ideal [N]< of nite sets and the ideal Z of sets of asymptotic density zero; going a little farther we have the ideal Zlog of sets A N such that
n
lim
1X1 = 0. ln n i An i + 1
For any such ideal we can consider the corresponding quotient Boolean algebra PN/I. The algebra PN/[N]< has long been recognised as one of the fundamental objects of set-theoretic analysis, topology and combinatorics. The others have not been studied so systematically, but show on the briefest of acquaintanceships the potential for generating fascinating questions. In this extraordinary monograph we are given some tools for tackling these questions which are surely going to be part of the essential kit for anyone working in the area. The most striking results concern the representation of Boolean homomorphisms between quotient algebras in terms of functions from PN to itself or between conite subsets of N. The rst steps are already far from being obvious; we need denitions which will lead to a useful classication scheme. One is well known. A P-ideal is an ideal I such that for every sequence In nN in I there is an I I such that In \ I is nite for every n. The next is natural enough to the lateral thinker: an ideal I is analytic if it is an analytic
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subset of PN when PN is given its usual compact metric topology, that is, is identied with the Cantor set in [0, 1]. Returning to the combinatorial aspect, I is ccc over [N]< if for any uncountable family A PN \ I there are distinct a, b A such that a b is innite. Concerning homomorphisms between quotient spaces, let us say that a function F : PN PN represents : PN/I PN/J if (a ) = (F (a)) for every a N. We shall be looking for representations F which are continuous, or have the Baire property, or are asymptotically additive, that is, for which there are disjoint sequences Ki i N , Li i N S in [N]< such that F (a) = i N F (a Ki ) Li for every a N. A striking facteasy to prove when you know howis that a Boolean homomorphism between quotients has a continuous representation i it has a representation with the Baire property. Our starting point is the following remarkable result due to S. Solecki (Annals of Pure and Applied Logic, vol. 99 (1999) pp. 5172): a proper ideal of PN containing all nite sets is an analytic P-ideal i it is of the form Exh( ) for some lower semi-continuous submeasure on PN. I see I have to give some more denitions. A submeasure on PN is a functional : PN [0, ] such that () = 0 and (a) (a b) (a) + (b) for all a, b N. It is lower semi-continuous if it is lower semi-continuous for the topology of PN, that is, if (a) = limn (a n) for every a N. Now Exh( ) is the ideal {a : limn (a \n) = 0}. 1 If (a) = 1 for every non-empty set a, Exh( ) = [N]< ; if (a) = supn1 n #(a n), Exh( ) = Z. One more denition. A submeasure on PN is entirely non-pathological if whenever a N and > 0 there is an additive functional : PN [0, [ such that and a + a; it is easy to check that naturally arising submeasures generally have this property. Now I can state one of the new theorems from this memoir. If is an entirely non-pathological lower semi-continuous submeasure on PN, I is an ideal including [N]< , and : PN/I PN/ Exh( ) is a Boolean homomorphism, then has a continuous representation i there is an h : N N such that a = h 1 [a] for every a N. The proof uses some remarkable calculations on nite sets. The requirement that a homomorphism should have a continuous representation is visibly a strong one, and even after discovering that our favourite ideals are derivable from good submeasures most of us would expect the last theorem to have limited applicability. And indeed there are simple examples of homomorphisms, either injective or surjective, which do not have continuous representations. But we come now to the heart of the monograph. a S. Shelah and J. Stepr ns (Proceedings of the American Mathematical Society, vol. 104 (1988) pp. 12201225) showed that if the Proper Forcing Axiom is true then every automorphism of PN/[N]< is trivial, that is, there are conite sets I , J N and a bijection h : J I such that a = h 1 [a] for every a N. Here we nd that if PFA is true, I and J are analytic P-ideals, and : PN/I PN/J is a Boolean isomorphism, then has a continuous representation; if J is of the form Exh( ) for some entirely non-pathological submeasure , then is trivial. For these results we have to work hard. The key seems to be to look at the ideal K of those a N such that the restriction of to the principal ideal generated by a can be represented by a continuous function, and show that K is ccc over [N]< , that a single continuous function can represent on the ideal {a : a K}, that thi...
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