The Logical Necessity of Inconsistency

Carl Hewitt

MIT EECS (emeritus)

Kurt Gödel first formalized and proved that it is not possible to decide all mathematical questions by inference in his 1^st incompleteness theorem.

However, the incompleteness theorem (as generalized by Rosser) relies on the assumption of consistency. This talk proves a generalization of the Gödel/Rosser incompleteness theorem: a reflective paraconsistent theory in Direct Logic [Hewitt 2007] is incomplete (without relying on the assumption of consistency); that is, for each reflective paraconsistent theory, the theory proves that there is a Gödelian paradoxical sentence which can neither be proved or disproved in the theory.

In order to be useful for large software systems, paraconsistent theories in Direct Logic make use of a powerful form of reflection between abstract logical statements and reified manifestations in XML. Direct Logic makes use of a criterion of Admissibility that bars the Liar, Russell, and Curry paradoxes. However, the argument based on the Gödelian paradoxical statement is admissible and results in a logically necessary inconsistency!

So why did Gödel and the logicians who followed him not go in this direction? Solomon Feferman (who personally worked with Gödel) has remarked on “the shadow of Hilbert that loomed over Gödel from the beginning to the end of his career.” Also Feferman conjectured that “Gödel simply found it galling all through his life that he never received the recognition from Hilbert that he deserved.” Furthermore, Feferman maintained that “the challenge remained well into his last decade for Gödel to demonstrate decisively, if possible, why it is necessary to go beyond Hilbert’s finitism in order to prosecute the constructive consistency program.”

Also Gödel was a committed Platonist, which has an interesting bearing on the issue of the status of reflection. Gödel invented arithmetization to encode abstract mathematical statements as integers. Direct Logic provides a similar way to easily formalize and paraconsistently prove Gödel’s argument (and even an extension due to Löb). But it is not clear that Direct Logic is fully compatible with Platonism.

Consequently, with an argument just a step away from inconsistency, Gödel (with his abundance of caution) was not prepared to go in that direction.

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Carl Hewitt

Logical Necessity of Inconsistency