Conceivability

Yesterday's LOGOS-seminar saw Manolo M. give a talk on conceivability (indeed, on ideal conceivability, but I don't want to focus on that here). As Roman pointed out, conceivability is a somewhat loose term. However, it seems that most LOGOS-members share a very similar conception of conceivability, differing only in the details (I also encountered signs of this fact in the RG on Fictionalism).

My question is simply, what is that common conception? (...) It is part of another project that this blog can be used for, that is making the common ground of the group explicit such that new members, like me, can position themselves relative to it (I take the essentialism-discussions earlier to be part of that project).

Maybe you don't think that you in fact share any view on conceivability in the group. To illustrate, let me briefly tell you what I think of conceivability, and I think most of you (except for the other newcomers) will have similar objections to make.

I can make sense of three ways to explicate conceivability:
1) The first is very close to imaginability, and in that sense an object that is green and red all over is inconceivable, but so is an object displaying a billion different colours, because that's just too much for my imagination.
2) The second includes, but is not exhausted by, anything that can be expressed by a sentence I can understand. In this sense, it's perfectly conceivable that 1=0.
3) And then there is conceivability relative to a set of ex- or implicitly stated assumptions. In that sense a proof for the continuum hypothesis from ZFC is inconceivable; it's also (in most contexts) inconceivable that Spain will invade Iran over the next few months.

So, straighten me out!