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Using Tsirelson’s well-known example of a Banach space which does not contain a copy of $c_0$ or $l_p$, for p ≥ 1, we construct a simple Borel ideal $I_T$ such that the Borel cardinalities of the quotient spaces $P\left(\mathbb{N}\right)/I_T$ and $P\left(\mathbb{N}\right)/I_0$ are incomparable, where $I_0$ is the summable ideal of all sets A ⊆ ℕ such that ${\sum }_{n\in A}1/(n+1)<\infty$. This disproves a “trichotomy” conjecture for Borel ideals proposed by Kechris and Mazur.