my forty years on his shoulders - Department of Mathematics

4theorem is demonstrably implied by the consistency statement - hence theconsistency statement is not provable. It was later established that the two are infact demonstrably equivalent.)Nevertheless, the consistency statement is obviously of a logical naturerather than of a mathematical nature. This is a distinction that is readily noticedby the general mathematical community, which naturally resists the notion thatthe incompleteness theorem will have practical consequences for their ownresearch.Genuinely mathematical examples of incompleteness from substantial settheoretic systems had to wait until the well known work on the axiom of choiceand the continuum hypothesis by Kurt Gödel and Paul Cohen. See (Gödel 1940),(Cohen 1963-64).Here, the statement being shown to be independent of ZFC - thecontinuum hypothesis - is of crucial importance for abstract set theory.However, mathematicians generally find it easy to recognize an essentialdifference between overtly set theoretic statements like the continuumhypothesis (CH) and “normal” mathematical statements. Again, this is aparticularly useful observation for the mathematicians.Specifically, the reference to unrestricted uncountable sets (of realnumbers) in CH readily distinguishes CH from “normal” mathematics, whichrelies, almost exclusively, on the “essentially countable”, (e.g., the continuous orpiecewise continuous).A more subtle example of an overtly set theoretic statement that requires asecond look to see its overtly set theoretic character, is Kaplansky’s Conjecture