I am curious about the history of the Whitehead's Conjecture, as this was the first natural mathematical statement, in the sense that mathematicians were actually interested in the answer, that was shown to be independent of ZFC. That is, in Gödel's language, it is undecidable. Before then, all statements shown to be undecidable were artificial, in the sense that no mathematician was actually interested in the statement itself. They were merely examples showing that such statements actually existed.
John Constatine Henry Whitehead, to give him his full name, and who also happens to be the nephew of Alfred North Whitehead who collaborated with Bertrand Russell on their Principia Mathematica, was lead to this question by his investigation of the Cousain problems in complex analysis/geometry. The linked article is clear enough for those with some mathematical maturity. It's expressed in the language of homological algebra but it's explained what this means in terms in a more prosaic manner. To rephrase:
An abelian group G, is called a Whitehead group when every surjective morphism, f, to it from any other abelian group splits. That is, it has a right inverse.
Now, it turns out that every free group is a Whitehead group. J. C. H. Whitehead asked whether the converse is true. That is:
Is every Whitehead group also a free group?
Ekloff, in an article in The American Mathematical Monthly (vol.83, no.10, Dec.'76) points out that this has a natural interpretation in terms of topological groups by taking advantage of Pontryagin duality: that is, is every path connected abelian group, a power of the circle group, that is a product of a number of copies of the circle group.
Stein actually showed that Whiteheads conjecture follows when the group is countable. But otherwise progress was slow until Saharon Shelah in '74, completely unexpectedly showed it was undecidable from ZFC. In fact more, he showed it was undecidable if one also assumes GCH, the Generalised Continuum Hypothesis.
Equally interestingly, he also showed that if one assumes that V=L, that the von Neumann universe is equivalent to Gödel's constructible universe then it does happen to be true!
Q. Have any mathematicians postulated that V=L is a natural axiom to add to ZFC as it resolves Whiteheads question in a natural way? More, is it also evidence that one ought to consider a plurality of set theories in the same way that we have a plurality of number systems (of many kinds: groups, modules etc).