Am I missing something? A quick read through seems to tell me that this attempts to solve the double-spending problem by... ignoring it?

I see some hand waving about a 24 hour waiting period and some verbiage about how different sub-currencies would handle it, but no actual solution.

Solving that problem is what the block chain is for, so the "without a ledger" bit seems to be missing the point.

Or maybe I didn't skim this carefully enough.

And how does this exactly prevent double spends?

I guess the emphasis is on trust and not so much on coin. Double-spends and coin creation are handled by trust. Everyone's sort of their own central bank. I guess that could be acceptable but it's concerning that anyone who's previously exchanged the coin you currently hold also needs to be trusted (since they can double spend and ruin the trustworthiness of your coin).

It might be a good communication protocol generally but a lot of the details aren't really addressed (and there's no code). I'd be interested to see this explained as a spec and with less focus on currency.