**Authors:** Colin James III

© Copyright 2017 by Colin James III All rights reserved. The Gödel incompleteness theorem states in effect that sequences of logic symbols can be assigned to strings of natural numbers, but because it is assumed there are more natural number than sequences of logic symbols, the symbols are incomplete as a self-referencing mechanism to describe repeatedly themselves as yet more numbers. (The Gödel completeness theorem states in effect that the sequences of logic symbols may be consistent to form a logic system that is sufficiently complete enough to prove theorems as tautology.) The arguments ultimately turn on the mapping of sequences of symbols into strings of natural numbers. The arguments also assume a function to map numbers as a domain into symbols as an image in a co-domain, also collectively named a range. The incompleteness theorem is that such a function exists and operates where all of its domain is larger than the smaller image existing within all of the co-domain. The completeness theorem is that all of the image, existing solely for its purposes, is self-sufficient unto the existence of itself. We are interested in mapping the image in the co-domain as sequences of logic symbols back into the originating domain which is named a preimage. Attempting such an inverse function is not allowed by the one-way definition of a function to an image as dictated by the incompleteness theorem. We show this invertive approach or reverse process is not tautologous by evaluating the misuse of the application of quantified operators in the one-way functional mapping in the first place. Our experimental results in mapping are the definitive evidence that constructionist logic and subsequently constructivistic logics are not complete (deny the completeness theorem) and hence deny the incompleteness theorem. The most important evidence is that those logics can exist only by ignoring the law of excluded middle (LEM) that "p or not p is a tautology", (p+~p)=(p=p). We argue that any system denying the LEM is not tautologous and hence unworkable.

**Comments:** 3 Pages. © Copyright 2017 by Colin James III All rights reserved.

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[v1] 2017-11-11 07:04:37

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