Inconsistency Robustness in Foundations: Mathematics self proves its own Consistency and Other Matters

TL;DR

This paper challenges G"odel's theorem by demonstrating that mathematical consistency can be proven within a typed grammar framework, emphasizing the importance of inconsistency robustness in mathematical foundations.

Contribution

It introduces a typed grammar approach to prove mathematical consistency, countering G"odel's incompleteness theorem and advancing the sociological understanding of mathematical foundations.

Findings

G"odel's proof relies on self-referential sentences that cannot be constructed in typed grammar.

Mathematical consistency can be proven without G"odel's limitations using typed grammar.

Inconsistency robustness influences the development of mathematical foundations for computer science.

Abstract

Inconsistency Robustness is performance of information systems with pervasively inconsistent information. Inconsistency Robustness of the community of professional mathematicians is their performance repeatedly repairing contradictions over the centuries. In the Inconsistency Robustness paradigm, deriving contradictions have been a progressive development and not "game stoppers." Contradictions can be helpful instead of being something to be "swept under the rug" by denying their existence, which has been repeatedly attempted by Establishment Philosophers (beginning with some Pythagoreans). Such denial has delayed mathematical development. This article reports how considerations of Inconsistency Robustness have recently influenced the foundations of mathematics for Computer Science continuing a tradition developing the sociological basis for foundations. The current common understanding is that G\"odel proved "Mathematics cannot prove its own consistency, if it is consistent." However, the consistency of mathematics is proved by a simple argument in this article. Consequently, the current common understanding that G\"odel proved "Mathematics cannot prove its own consistency, if it is consistent" is inaccurate. Wittgenstein long ago showed that contradiction in mathematics results from the kind of "self-referential" sentence that G\"odel used in his argument that mathematics cannot prove its own consistency. However, using a typed grammar for mathematical sentences, it can be proved that the kind "self-referential" sentence that G\"odel used in his argument cannot be constructed because required the fixed point that G\"odel used to the construct the "self-referential" sentence does not exist. In this way, consistency of mathematics is preserved without giving up power.