  A new computational method that uses polynomial equations and dynamical
systems to evaluate logical propositions is introduced and applied to Goedel's
incompleteness theorems. The truth value of a logical formula subject to a set
of axioms is computed from the solution to the corresponding system of
polynomial equations. A reference by a formula to its own provability is shown
to be a recurrence relation, which can be either interpreted as such to
generate a discrete dynamical system, or interpreted in a static way to create
an additional simultaneous equation. In this framework the truth values of
logical formulas and other polynomial objectives have complex data structures:
sets of elementary values, or dynamical systems that generate sets of infinite
sequences of such solution-value sets. Besides the routine result that a
formula has a definite elementary value, these data structures encode several
exceptions: formulas that are ambiguous, unsatisfiable, unsteady, or
contingent. These exceptions represent several semantically different types of
undecidability; none causes any fundamental problem for mathematics. It is
simple to calculate that Goedel's formula, which asserts that it cannot be
proven, is exceptional in specific ways: interpreted statically, the formula
defines an inconsistent system of equations (thus it is called unsatisfiable);
interpreted dynamically, it defines a dynamical system that has a periodic
orbit and no fixed point (thus it is called unsteady). These exceptions are not
catastrophic failures of logic; they are accurate mathematical descriptions of
Goedel's self-referential construction. Goedel's analysis does not reveal any
essential incompleteness in formal reasoning systems, nor any barrier to
proving the consistency of such systems by ordinary mathematical means.
