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.
In the last century, Kurt Gödel's incompleteness theorems [15] sent shockwaves through the world of mathematical logic. The conventional wisdom is that Gödel's theorems and his interpretations thereof are correct; the prevalent discussion concerns what these results mean for logic, mathematics, computer science, and philosophy [26,20,16]. But as I shall demonstrate here, Gödel's theorems are profoundly misleading and his interpretations were incorrect: his analysis was corrupted by the simplistic and flawed notions of truth value and proof that have troubled logic since antiquity, compounded by his misapplication of a static definition of consistency to a dynamical system. Exposing these errors reveals that reports of logic's demise have been greatly exaggerated; we may yet realize the rationalist ideals of Leibniz and complete the logicist and formalist programs of Frege, Russell, and Hilbert.
There are two key principles here: first, that proof in formal reasoning systems is an exercise in solving systems of polynomial equations, yielding solutions that are sets of elementary truth values; and second, that certain selfreferential formula definitions are recurrence relations that define discrete dynamical systems (and in turn infinite sequences of basic solution sets). It is a corollary to these principles that all of the various syntactic results from such calculations make semantic sense as the truth values of formulas, including solution sets that are empty or have multiple members, and dynamic solutions that change with each iteration and depend on initial conditions.
These principles are familiar and uncontroversial in the contexts of elementary algebra and dynamical systems; they apply just as well when the basic mathematical objects are logical truth values instead of ordinary numbers. If you understand how to do arithmetic in different number systems, what it means to solve equations, and how to deal with recursive constructions like the Fibonacci sequence, then you can understand Gödel’s mistakes. You will find the powerful new paradigm of dynamic polynomial logic, which is a continuation of the pioneering 19th-century work of George Boole [4,5]. Dynamic polynomial logic is grounded in the intrinsic unity of logic and mathematics.
Relative to classical logic, dynamic polynomial logic is paraconsistent, paracomplete, and modal. In this algebraic framework the misguided principle of explosion is corrected: inconsistent axioms are shown to prove nothing instead of everything. Moreover the principle of the excluded middle is clarified; in classical logic this idea is applied incorrectly, reflecting confusion between arithmetical and algebraic systems. Dynamic polynomial logic computes precise solutions that can be interpreted as alethic and temporal modalities.
Gödel considered formal reasoning systems as described in Whitehead and Russell’s Principia Mathematica [31], which was an epic attempt to formalize the whole of mathematics. Gödel’s basic argument was that every formal reasoning system powerful enough to describe logical formulas, proof, and natural numbers (like PM) must allow the construction of a special formula that is semantically correct but syntactically undecidable: true by metalevel consideration of its content, but impossible to prove or disprove by mathematical calculation within the formal system itself. This special formula, denoted both [R(q); q] and 17 Gen r in Gödel’s paper, asserts that the formula itself cannot be proven within the system. Thus follows the apparent paradox, which Gödel described in this way (as translated in [30]):
From the remark that [R(q); q] says about itself that it is not provable it follows at once that [R(q); q] is true, for [R(q); q] is indeed unprovable (being undecidable). Thus, the proposition that is undecidable in the system PM still was decided by metamathematical considerations. The precise analysis of this curious situation leads to surprising results concerning consistency proofs for formal systems, results that will be discussed in more detail in Section 4 (Theorem XI).
Gödel’s Theorem VI states that there must exist (in a formal system like PM) a formula such as his special [R(q); q] which can neither be proven nor disproven within the formal system that contains it. His claim that his special formula [R(q); q] is semantically true is presented in the text of his paper but is not called out as a theorem. Gödel’s Theorem XI states that the existence of such an undecidable formula renders the consistency of the enclosing formal system itself an undecidable proposition.
The surprising result from my analysis is that Gödel’s special formula is neither semantically correct nor semantically incorrect; instead it is exceptional in a particular way, relative to the expectation that a formula should have a definite elementary value. Such exceptions, which appeared to Gödel as ‘undecidability,’ are features not bugs in formal reasoning system
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