Kerodon

Warning 4.7.1.25 (The Burali-Forti Paradox). One can informally summarize Corollary 4.7.1.24 by saying that the collection $\mathrm{Ord}$ of all ordinals is well-ordered (with respect to the order relation of Definition 4.7.1.19). Beware that one must treat this statement with some care to avoid paradoxes. The proof of Proposition 4.7.1.22 shows that the order type of $\mathrm{Ord}$ is strictly larger than $\alpha $, for each ordinal $\alpha \in \mathrm{Ord}$. This paradox has a standard remedy: we regard the collection $\mathrm{Ord}$ as “too large” to form a set (so that its order type is not regarded as an ordinal).