Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2008 / B / 8 / ii / d

The same complete theory $T_{\infty }$ of an infinite pure set provides this property directly. For $n$ logical variables, each complete type specifies exactly which logical variables are equal. First-order consistency requires an equivalence relation on the finite index set, and infinitude ensures every such partition is realizable. Hencewhere $B_n$ is the $n$th Bell number, counting partitions of an $n$-element set. For example the first values are $B_1=1$, $B_2=2$ and $B_3=5$. The isolating conjunction lists logical equality within blocks and inequality between blocks. This explicitly verifies the hypothesis which Ryll-Nardzewski identifies with ${\aleph }_0$-categoricity.