  This article describes a Turing machine which can solve for $\beta^{'}$ which
is RE-complete. RE-complete problems are proven to be undecidable by Turing's
accepted proof on the Entscheidungsproblem. Thus, constructing a machine which
decides over $\beta^{'}$ implies inconsistency in ZFC. We then discover that
unrestricted use of the axiom of substitution can lead to hidden assumptions in
a certain class of proofs by contradiction. These hidden assumptions create an
implied axiom of incompleteness for ZFC. Later, we offer a restriction on the
axiom of substitution by introducing a new axiom which prevents impredicative
tautologies from producing theorems. Our discovery in regards to these
foundational arguments, disproves the SPACE hierarchy theorem which allows us
to solve the P vs NP problem using a TIME-SPACE equivalence oracle.
