> 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

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

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.