 Arrow’s point is that utilities must be bounded to avoid the St. Petersburg paradox and that traditional axiomatic accounts of the expected utility principle guarantee this to be the case. The well-known axiomatizations proposed by Ramsey (1926), von Neumann and Morgenstern (1947), and Savage (1954) do, for instance, all entail that the decision maker’s utility function is bounded. (See  section 2.3 in the entry on decision theory  for an overview of von Neumann and Morgenstern’s axiomatization.)