Just some questions:

Does the halting problem <=> P = NP or P != NP? If not, what is the relation between the two problems? Does it exist?
Is there a class (set) of problems equivalent to the halting problem? –yes
What is the cardinality of this set?

If the cardinality of this set is uncountably (or even countably) infinite what does it mean for the implication arrow above? If it is uncountably infinite (moreover, if it is isomorphic to Aleph_alpha) does it follow the 2^n halting problem (the problem of halting an exponential number of turing machines) is of order 2^(Aleph_alpha)?

Allow m to be the cardinality of the set P and n to be the cardinality of the set NP. The halting problem is a subset of NP (classification grouping: MAX-3SAT problem).

Assume m < n or n > m. If we replace the value m with 2^m or n with 2^n (the problem of finding the solution to the power set of the problems in m/n) and m = n then we have constructed a problem such that m < n and/or n < m depending on one’s frame of reference.

Discuss?

-harbinger of night