P=NP - An impossible question
A proposed proof of undecidability by Nicholas Argall, 25 March, 2003.
There has been much debate surrounding answers to the question of P=NP.
The problem is that we cannot answer the question until we have successfully
asked the question. The question is impossible to ask, that it why it will
never be answered.
1) A provable answer to the question P=NP requires a complete and consistent
formal statement of the question.
Rationale: Hopefully, this is self-evident. It is certainly axiomatic that
a formally provable statement be expressed in formal terms. Completion and
consistency follow from the requirement to provide a proof that is not
subject to challenge.
2) A complete and consistent formal statement of the question must
incorporate a complete and consistent formal definition of the sets P and NP
Rationale: Hopefully, this is also self-evident. (I have left out the
requirement to define the equality operator, since it is defined for us by
set theory.)
3) A definition based on a potentially undetectable characteristic is
incomplete
Rationale: We cannot accept the definition of the set NP purely in terms of
its members having a property (a solution test in polynomial time) that we
have no reliable mechanism to detect. Therefore, a complete definition of
the set NP must be arrived at via some other means.
4) The only possibility for a complete definition of the set NP is a
language
Rationale: Once we rule out observation of characteristics, our only means
towards a definition of the set NP is to formulate a language, a procedure
for testing the formal expression of the candidate problem that will accept
the problem or reject it.
5) No formal language capable of expressing non-trivial mathematical
problems can be consistent and complete
Rationale: As proven by Godel.
6) Therefore, no consistent and complete definition of the set NP is possible
Rationale: If we accept that the set NP can only be rigorously defined via a
language, this conclusion follows from the premises above.
7) Therefore, no consistent and complete statement of the problem of P=NP is
possible
Comment: A conclusion which is not only proven in this paper, but supported
by the years of argument between mathematicians regarding the relevance of
proposed answers to the problem.
8) Therefore, P=NP is undecidable
Comment: Given our inability to ask, we are unable to answer.