Inconsistency Robustness in Foundations: Mathematics self proves its own
Consistency and Other Matters
release_ggiauep5qrbnjkhmbis26mctxm
by
Carl Hewitt
2011
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.
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