In his book The Philosophy of Nature, Ellis presents "the new essentialism" as resting on the notions of a property, an intrinsic property, an essential property, natural necessity and possibility, a natural kind, a fixed natural kind, and a natural law. The present paper argues that (1) the central notions in this group are susceptible of a logical analysis, (2) Ellis's notion of natural possibility has a historical precedent in the work of Abéelard, (3) the notion of natural possibility contains both de re and de dicto elements, and (4) Ellis's essentialist claims, when joined to any plausible definition of natural possibility, lead to inconsistency.

We show that the theory of 𝒟(≤ g), where g is a 2-generic or a 1-generic degree below 0ʹ, interprets true first-order arithmetic. To this end we show that 1-genericity is sufficient to find the parameters needed to code a set of degrees using Slaman and Woodin's method of coding in Turing degrees. We also prove that any recursive lattice can be embedded below a 1-generic degree preserving top and bottom.

We say that a subset A of M is implicitly definable in M if there exists a sentence $\varphi(P)$ in the language $\mathcal{L}(M) \cup \{P\}$ such that A is the unique set with $(M,A) \models \varphi(P)$. We consider implicit definability of subfields of a given field. Among others, we prove the following: $\overline{\mathbb{Q}}$ is not implicitly $\emptyset$-definable in any of its (proper) elementary extension $K \succ \overline{\mathbb{Q}}$. $\mathbb{Q}$ is implicitly $\emptyset$-definable in any field K (of characteristic 0) with tr.deg $_{\mathbb{Q}}K < \omega$. In a field extension $\mathbb{Q} < K$ with K algebraically closed, $\mathbb{Q}$ is implicitly definable in K if and only if tr.deg $_{\mathbb{Q}}(K)$ is finite.