Take a reasoner who can make pre-commitments (or a UDT/TDT type). This reasoner, in effect, only has to make a single decision for all time.
Let A, B, C... be pure outcomes, a, b, c,... be lotteries. Then define the following pseudo-utility function f:
f(a) = 1 if the outcome A appears with non-zero probability in a, f(a) = 0 otherwise. The decision maker will use f to rank options.
This clearly satisfies completeness and transitivity (because it uses a numerical scale). And then... It gets tricky. I've seen independence written both in a < form and a <= form (see http://en.wikipedia.org/wiki/Expected_utility_hypothesis vs http://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem ). I have a strong hunch that the two versions are equivalent, given the other axioms.
Anyway, the above decision process satisfies <= independence (but not < independence).
To see that the decision process satisfies <= independence, note that f(pa+(1-p)b)=max(f(a),f(b)). So if f(a) <= f(b), then f(pa+(1-p)c)=max(f(a),f(c)) <= max(f(b),f(c)) = f(pb+(1-p)c)).
Yes, the two versions of independence are equivalent given the other axioms. If you don't have continuity to make them equivalent, I think the natural thing to do is to ask for both types of independence.
(The intuition is: independence feels like it should demand all of these things. Normally it's not stated like that because it's clunky to add extra statements when one is enough.)
In a previous post, I left a somewhat cryptic comment on the continuity/Archimedean axiom of vNM expected utility.
Here I'll explain briefly what I mean by it. Let's drop that axiom, and see what could happen. First of all, we could have a utility function with non-standard real value. This allows some things to be infinitely more important than others. A simple illustration is lexicographical ordering; eg my utility function consists of the amount of euros I end up owning, with the amount of sex I get serving as a tie-breaker.
There is nothing wrong with such a function! First, because in practice it functions as a standard utility function (I'm unlikely to be able to indulge in sex in a way that has absolutely no costs or opportunity costs, so the amount of euros will always predominate). Secondly because, even if it does make a difference... it's still expected utility maximisation, just a non-standard version.
But worse things can happen if you drop the axiom. Consider this decision criteria: I will act so that, at some point, there will have been a chance of me becoming heavy-weight champion of the world. This is compatible with all the other vNM axioms, but is obviously not what we want as a decision criteria. In the real world, such decision criteria is vacuous (there is a non-zero chance of me becoming heavyweight champion of the world right now), but it certainly could apply in many toy models.
That's why I said that the continuity axiom is protecting us from "I could have been a contender (and that's all that matters)" type reasoning, not so much from "some things are infinitely important (compared to others)".
Also notice that the quantum many-worlds version of the above decision criteria - "I will act so that the measure of type X universe is non-zero" - does not sound quite as stupid, especially if you bring in anthropics.