Gödel’s Second Incompleteness Theorem states axiom systems of sufficient strength are unable to verify their own consistency. We will show that axiomatizations for a computer’s floating point arithmetic can recognize their cut-free consistency in a stronger respect than is feasible under integer arithmetics. This paper will include both new generalizations of the Second Incompleteness Theorem and techniques for evading it.
Dan E. Willard. "On the available partial respects in which an axiomatization for real valued arithmetic can recognize its consistency." J. Symbolic Logic 71 (4) 1189 - 1199, December 2006. https://doi.org/10.2178/jsl/1164060451