ω-consistent theory  From Wikipedia, the free encyclopedia
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In mathematical logic, an ω-consistent (or omega-consistent,
also called numerically segregative[1]) theory is a theory
(collection of sentences) that is not only (syntactically)
consistent (that is, does not prove a contradiction), but also
avoids proving certain infinite combinations of sentences that
are intuitively contradictory. The name is due to Kurt Gödel,
who introduced the concept in the course of proving the
incompleteness theorem.[2]   A theory T is said to interpret the
language of arithmetic if there is a translation of formulas of
arithmetic into the language of T so that T is able to prove the
basic axioms of the natural numbers under this translation.  A T
that interprets arithmetic is ω-inconsistent if, for some
property P of natural numbers (defined by a formula in the
language of T), T proves P(0), P(1), P(2), and so on (that is,
for every standard natural number n,