> What does “Undecidable” mean, anyway A big and relatively recent example is to take (1) the axioms of Zermelo-Fraenkel set theory and (2) the axiom of choice

> What does “Undecidable” mean, anyway A big and relatively recent example is to take (1) the axioms of Zermelo-Fraenkel set theory and (2) the axiom of choice and ask if (1) can be used to prove or disprove (2). The surprising answer is "No", i.e., (1) cannot be used either to prove or disprove (2). So, can work with (1) and, whenever convenient, assume (2), continue on, and never encounter a problem. So, given (1), (2) is undecidable. The work was by Paul J. Cohen as at https://en.wikipedia.org/wiki/Paul_Cohen https://en.wikipedia.org/wiki/Paul_Cohen The field of research about what is undecidable also goes back to Kurt Gödel as at https://en.wikipedia.org/wiki/Kurt_G%C3%B6del https://en.wikipedia.org/wiki/Kurt_G%C3%B6del Another example, starting with (1), of an undecidable statement is the continuum hypothesis: There is no set whose cardinality is strictly between that of the integers and the real numbers. In simple terms, "cardinality" means that there is no set X that is too large to be put into 1-1 correspondence with the set of integers and too small to be put into 1-1 correspondence with the set of real numbers.