A note on Tsirelson type ideals

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