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Douglas_Knight comments on Conventions and Confusing Continuity Conundrums - Less Wrong
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Thanks. Though the continuity assumption is itself the thing I felt to be the problem area. Unless I'm misunderstanding your argument, you're assuming the very continuity property I want to derive.
(Incidentally, I may be wrong on this, but I think the closure property you're referring to would follow directly from the continuity. (alternately, one might need to show that closure property to show the continuity. Point being, I don't think those two properties are separable, except at A and B))
Continuity is merely preserving order: if A > B and p > q, then pA+(1-p)B > qA+(1-q)B. It is a not-being-stupid assumption. or an interpretation of probability.
Mendel seems to be working in an extremely abstract version of probability where p cannot be described as a size. But once you insist on p being a number, there are many possibilities. You might allow p only to take rational values, so that they can be finitely represented. Or you might allow p to take all real values, in which case some p exists solving the problem.
The issue of closure is about where the p's live, that is, what kinds of lotteries you can build. It isn't about preferences or states of the world (except in that lotteries are states of the world).
ETA: Actually...the axiom of independence gives you order preservation. The axiom of continuity does only one thing: it rules out lexicographic preferences. It says that if you lexicographically care more about X than about Y, you aren't allowed to use Y as a tie breaker, but must simply not care about about Y at all.