We have problems proved to be in classes P, NP, co-NP, NP-Complete, and NP-Hard. These problems have their time complexities deduced too. I am wondering if I am missing any key information on the topic.
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One of the standard tricks for solving this sort of problem is what's called using an "oracle". Suppose we could ask the oracle a question about a specific class of problems, and in one step it would answer "yes" or "no". It becomes reasonable to ask "What sort of problems are in P for a machine with this oracle"? "What sort of problems are in NP for a machine with this Oracle.
It has been shown that there are oracles for which P=NP. It has been shown that there are oracles for which P≠NP. Most of the techniques we know for proving hardness should give the same result, no. matter what oracle is used.