It is said that P!=NP is an arithmetical statement, also that it is of order two, that is we can state is as $\forall_p \exists_i \text{(polynomial-time turing machine number $p$ fails to solve 3SAT instance number $i$})$. Can we write that sentence using only arithmetical and logical operators? How to do it? If not, what's the minimum number of quantifiers possible?
I currently can't state it using any number of quantifiers, and I heard it's possible, so I want to know how, but I wasn't able to find a reference. Additionally, I was just curious about the minimum number of quantifiers.
I don't need a detailed construction for this specific question, I could figure it out based on an explanation of encoding computation in arithmetic.