Problem with enumerating formulas

This will come down to the following: what is the precise definition of "logical formula?" The point is that although the $Q$ you've described is definitely "meaningful," it is of greater complexity in a sense than any of the $P_i$s. When we pin down a precise notion of "logical formula" and an enumeration of such, e.g. "first-order formula in the language of arithmetic" ordered lexicographically via some reasonable ordering on the symbols involved, the corresponding $Q$ will turn out to not be expressible by a formula in that particular sense.

For a concrete example of how this plays out, see e.g. Tarski's undefinability theorem. And compare this to similar "paradoxes of expression:" Berry's paradox, Richard's paradox, and Grelling's paradox.

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Logic

Paradoxes

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