Gödel's 1931 paper about Incompleteness contained two results. Its Second Incompleteness Theorem stated that sufficiently strong axiom systems are unable to internally verify their own consistency. During the last 10 years, we have published several papers about this topic. They have identified Boundary Cases where adequately weak axiom systems can recognize their own internal consistency, and we have also developed new generalizations of the Second Incompleteness Theorem that more precisely identify the exact threshold where the Second Incompleteness Effect takes place. This talk will summarize the contents of our four papers [1,2,3,4] plus discuss further results that are scheduled to be published in the near future.

Our talk will include an explanation about why we believe the study of the Second Incompleteness Theorem and its Boundary-Case Exceptions will become increasingly relevant to Computer Science as the 21-st Century progresses. We will also examine this topic from Epistemological perspectives of Mathematics and Philosophy. The initial part of our talk will have a sufficiently simple and introductory nature so that the thrust of this talk should be comprehensible to an audience that is not fully familiar with Gödel's Second Incompleteness Theorem.