A note on Tsirelson type ideals

Using Tsirelson's well-known example of a Banach space which does not contain a copy of c(0) or l(p), for p greater than or equal to 1, we construct a simple Borel ideal I-T such that the Borel cardinalities of the quotient spaces P(N)/I-T and P(N)/I-0 are incomparable, where I-0 is the summable ideal of all sets A subset of or equal to N such that Sigma(n is an element of A) 1/(n + 1) < infinity. This disproves a "trichotomy" conjecture for Borel ideals proposed by Kechris and Mazur.