I’ve been having an email discussion with someone on what exactly the above means. I’m on the side of “nothing”.

If I have an ordered list of courses of action (which is given by your statement that “a given course of action is better than another in every possible case”), what is it about the fact that some of those courses result in infinite rewards/punishment that keeps me from simply performing a max() on the list?

• Define an effective model of the field of the rationals with less than and (perhaps unnecessarily) a predicate for the integers, i.e. a computable model of (Q; 0, 1, +, x, -, < , N)
• We can then define a computable function on that model that I’ll call d(t), for decay, as follows. Define d(n) = 1/2^{n then extend to the rationals via the enumeration of Q and (dyadic) interpolation. For example, if d(2/3) = 2/3, d(3/4) = 1/5 and the next rational in the enumeration of Q is 7/10, then set d(7/10) = (1/2) [2/3 + 1/5] = 13/30. Note that d(t) is a strictly decreasing computable function on Q that is O(2}-n).
• In a similar way, we can define payoff functions f and g, with the appropriate limiting behaviors; for example, we could set f = 1/d and g = f – 1. These are, again, computable functions.
• The next step is a little tricky, but still manageable. We’d like to define the integral of these functions over Q in a finitistic fashion. There are two ways of approaching this: Method 1: Since f, g and d are all strictly monotonic, they have unique continuous extensions f*, g*, d* to the reals (either by Dedekind cuts or a Urysohn-style argument). Since these extensions are continuous, they are locally integrable—and note, in particular, that d* is integrable over the entire real line (being O(2^-n)) and that f*d* and g*d* have infinite integrals over R. Method 2: We define the “integral” of a function h over the interval [a, b], a & b rational, as follows. Let {x_1, …, x_n} be a finite—and thus computable—sequence of rationals between a and b. Let L(h, {x_1, … x_n}) be the usual lower Riemann sum associated with that partition; by the usual theorems on lower Riemann sums there is a well-defined limit alpha over R. Since we’re working over Q, however, alpha won’t necessarily exist in our model. We can, however, define a predicate “e-integral” E(z, e) as follows: E(z, e) iff every sequence {x_1,…,x_n} has a (definable) refinement {y_1,…,y_n} with | L(h, {y_1,…,y_n}) – z | < e. This is a definable predicate over PA, although no longer a computable one.
• Via a similar limiting trick (i.e. up to a fudge factor of e) we can define integrals over all of Q. That is, a function h is integrable over all of Q if there is a N such that given any sequence xbar, L(h, xbar) < = N; and, in that case, we can define the e-integral over all of Q as above.
• Finally, with all that under our belt, we can set up the appropriate paradoxes as before. Since fd and gd have infinite integrals over Q, we can (with finitely many parameters, if need be) prove the claim that, given any actual value of f relative to the distribution d, the expectation of g relative to the distribution d strictly exceeds that value—and is, indeed, infinite.