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Introduction

Logic discovered its own limitations in Gödel's Incompleteness Theorems. Before Gödel's proofs, mathematicians sought a logical mechanistic process that could, with enough time and resources, decide any mathematical question. Gödel proved this was impossible. Any consistent sufficiently powerful set of mathematical assumptions or axioms can be expanded in simple obvious ways and complex ones. The standard example is to add the axiom that the system itself is consistent. Gödel proved this cannot be derived within any consistent system strong enough to define an ideal computer. An ideal computer can run forever error free and has access to unlimited storage.

Any single path approach to expanding mathematics will encounter a Gödelian limit.1Progress can be made forever, but all the results, over an infinite time, are subsumed in a single axiom that will never be discovered. Only a divergent process that explores an ever increasing number of alternatives, providing increasing resources to each viable path, can escape a Gödelian limit. There is no way to select between alternative expansions of mathematics beyond eliminating those that are inconsistent or proved wrong for more complex reasons. For example, a contradiction exists only if two consequences are inconsistent, but there may still be an infinite sequence of consequences that are mutually inconsistent.

The proof that all mathematical truth can be explored in this way is trivial. If resources are available, all alternatives can be considered, each with ever greater effort. There is evidence that this is relevant to the evolution of the mathematically capable human mind, to the future expansion of mathematics and to cultural creativity in general.


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