Gödel's Theorem Quotes

“A consistent theory T cannot postulate its own consistency, although the consistency of T can be postulated in another consistent theory. (An exercise for the reader: Is it possible to have a pair of consistent theories S and T such that each postulates the consistency of the other? It follows from what has been said above that the answer is no.)”
― Torkel Franzén, Gödel's Theorem: An Incomplete Guide to Its Use and Abuse

“The incompleteness theorem is a mathematical theorem precisely because the relevant notions of truth and provability are mathematically definable. Nonmathematical “Gödel sentences” and Liar sentences give rise to prolonged (or endless) discussions of just what is meant by a proof, by a true statement, by sound reasoning, by showing something to be true, by convincing oneself of something, by believing something, by a meaningful statement, and so on.”
― Torkel Franzén, Gödel's Theorem: An Incomplete Guide to Its Use and Abuse

“In the quoted passage, the suggested conclusion is that a system of laws must be “inconsistent or incomplete.” Given the accompanying explanation of what “inconsistent” and “incomplete” mean here, it is an easy observation that all systems of laws, rules of living, and so on, are both inconsistent and incomplete and will remain so. In other words, in the case of legal systems, there will always be actions and procedures about which the law has nothing to say, and there will always be actions and procedures on which conflicting legal viewpoints can be brought to bear. Hence the need for courts and legal decisions. References to Gödel’s theorem can only add a rhetorical flourish to this simple observation.”
― Torkel Franzén, Gödel's Theorem: An Incomplete Guide to Its Use and Abuse