Def(omega) is countable while P(omega) is not

Encode pairs of naturals by $\pi (a,b)=(a+b)(a+b+1)\mathrm{/}2+b$. For a given s, the values with $a+b=s$ are the consecutive integers from $s(s+1)\mathrm{/}2$ to $s(s+1)\mathrm{/}2+s$, so these disjoint intervals cover omega and pi is a bijection. Number the membership-language symbols by naturals (the variables are $v_i$, and the remaining logical and punctuation symbols form a fixed finite list); encode a finite symbol string by pairing its length with its nested symbol code ending in 0. Restricting this partial decoding to well-formed formulas and pairing each with its finite allowed arity gives a natural-number formula/arity code. Encode a finite tuple of naturals by the same length-and-nested-pair construction, then pair it with the formula/arity code. For each natural k decode a valid formula/tuple pair if possible and output its defined subset; otherwise output empty. Replacement gives a sequence $E:\omega \to \mathrm{Def}(\omega )$, and F1 makes it surjective.

Given any sequence $S:\omega \to \mathcal{P}(\omega )$, Separation forms $D=\{n\in \omega :n\mathrm{\notin }S(n)\}$. For each k, membership of k in D is the opposite of its membership in S(k), so $D\mathrm{\ne }S(k)$. Thus S is not surjective. Applied to E from step 1.1, D is not in Def(omega), although it is a subset of omega. This uses a defining formula in V, not necessarily a definition over the structure $(\omega ,\in )$, so there is no contradiction with the definition of Def.