What else does one blog from Siracusa?
(A.3) Archimedean Axiom: if p, q, r Î D (X) such that p >h q >h r, then there is an a , b Î (0, 1) such that a p + (1-a )r >h q and q >h b p + (1-b )r.
…The Archimedean Axiom (A.3) works like a continuity axiom on preferences. It effectively states that given any three lotteries strictly preferred to each other, p >h q >h r, we can combine the most and least preferred lottery (p and r) via an a Î (0, 1) such that the compound of p and r is strictly preferred to the middling lottery q and we can combine p and r via a b Î (0, 1) so that the middling lottery q is strictly preferred to the compound of p and r. Notice that one really needs D (X) to be a linear, convex structure to have (A.3).
The full treatment is here. In other words, the Archimedean Axiom means no lexicographic preferences for certainty. For some expected reward, you will accept a very small chance of a very bad outcome. Either extreme fear, or an extreme attachment to a symbolic value, or an extreme attachment to a “no stochastic trade-offs” principle can stop an Archimedean axiom from holding. There is further analysis here. Here is Wikipedia on Archimedes, a very impressive figure.