Based on its logical form alone, we cannot currently exclude the possibility that \(\Theta\) is independent not just of first-order Peano arithmetic [\(\mathsf{PA}\)], but of even stronger axiom systems such as \(\textsf{ZFC}\). But although early results suggested \(\textbf{P} \neq \textbf{NP}\) may be independent of some very weak axiomatic theories (e.g. DeMillo and Lipton 1980), it is now believed that this statement is unlikely to be independent of stronger theories like \(\mathsf{PA}\) which approximate the mathematical axioms we employ in practice. In particular, Ben-David and Halevi (1992) showed that if it were possible to use current proof theoretic techniques to show the independence of \(\Theta\) from \(\textsf{PA}\), then this demonstration itself would also show that \(\textbf{NP}\) is very close to being in \(\textbf{P}\) in a sense which can be made precise using techniques from circuit complexity.[40] As this is thought to be implausible, Aaronson (2003) suggests that we currently possess no reason to suspect that \(\textbf{P} \neq \textbf{NP}\) is more likely to be independent of \(\textsf{PA}\) (let alone \(\mathsf{ZFC}\)) than other currently open number theoretic statements.