# Are there infinite possibilities to the outcome of the P vs. NP question?

The P vs. NP poll provides 3 possibilities: equal, not equal, and independent. This is reasonable, because despite the law of the excluded middle you must supply a proof for your answer, which itself is insufficient. But then the statement "P vs. NP is independent" itself can be independent, with a proof. And this process can go on indefinitely. One of them must be provable, or did I miss something? Anyway the number of possibilities is infinite, instead of 3. Is this correct?

• you already asked this question 8 years ago did you not? – Giacomo Alzetta Aug 24 '18 at 14:20
• Either you can prove that it's true or you can prove that it's false or you can prove neither. Whether or not you consider further refinements of the "you can prove neither" case as different "possibilities" is purely a matter of opinion. – David Richerby Aug 24 '18 at 14:48
• @GiacomoAlzetta No, that's a different question -- essentially, "How could it possibly be independent?" – David Richerby Aug 24 '18 at 14:48

This isn't really a question about P vs. NP - any other statement would do in its place. The point is that there are a couple different "philosophical contexts" being mixed up here, and depending on exactly how you ask the question each of $2$, $3$, or infinity are reasonable answers (I'm not claiming that these are exclusive, but I think they frame the question well):