Given the convergence of several forms of non-demonstrative evidence for the conjecture that \(\textbf{P} \neq \textbf{NP}\), it is also reasonable to ask why this statement has proven so difficult to resolve in practice. For note that although this statement originates in theoretical computer science, it may be easily formulated as statements about natural numbers. In particular, \(\textbf{P} \neq \textbf{NP}\) is equivalent to the statement that for all indices \(e\) and exponents \(k\), there exists a propositional formula \(\phi\) such that the deterministic Turing machine \(T_e\) does not correctly decide \(\phi\)’s membership in \(\sc{SAT}\) in \(\lvert \phi\rvert^k\) steps. Using familiar techniques from the arithmetization of syntax, it is not difficult to see that this statement can be formalized in the language of first-order arithmetic as a \(\Pi^0_2\)-statement – i.e. a statement \(\Theta\) of the form \(\forall x \exists y \psi(x,y)\) where \(\psi(x,y)\) contains only bounded numerical quantifiers.