This can perhaps be considered more of a meta-conjecture than a conjecture: Hilbert's program, http://en.wikipedia.org/wiki/Hilbert's_program. The conjecture would be: that set theory (or some set of axioms suitable for doing math) can be proven consistent. Gödel's Incompleteness Theorem disproved this conjecture.

I don't have a reference, but I have the impression that, at the time, Hilbert's program seemed attainable, and Gödel's result came as a surprise.

This can perhaps be considered more of a meta-conjecture than a conjecture: Hilbert's program, https://en.wikipedia.org/wiki/Hilbert's_program. The conjecture would be: that set theory (or some set of axioms suitable for doing math) can be proven consistent. Gödel's Incompleteness Theorem disproved this conjecture.

I don't have a reference, but I have the impression that, at the time, Hilbert's program seemed attainable, and Gödel's result came as a surprise.