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JGWeissman comments on VNM expected utility theory: uses, abuses, and interpretation - Less Wrong
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Perhaps you mean that first-order Peano arithmetic cannot prove that the set of natural numbers is infinite.
Well, first-order Peano arithmetic doesn't have any notion of "set", much less "infinite"...
Right, that is my point. If an axiomatic system has no concept of "set" or "infinite", it will not notice if the thing it is describing is in fact an infinite set. But we can use a more complete system, and notice that it is an infinite set and make deductions from it. Which is why first-order Peano arithmetic does not invalidate my claim that Academian's characterization of immortality relies on a hidden reference to an infinite set.
The standard model of Peano arithmetic is 0={} and S(X)=Xunion{X}; its objects are all finite sets.
I fail to see any meaningful sense in which saying "There does not exist a time T after the present such that I am not alive" has a hidden reference to infinity which is somehow avoided by saying "For any duration N and probability epsilon, there exists a longer duration M such that..."
The set of all objects in this standard model is infinite, even though Peano arithmetic does not refer to this set.
The difference is that in my version, utility is assigned to states of a finite universe. To the extent that there is a reference to infinity, the infinity describes an abstract mathematical object. In your version, utility is assigned to a state of an infinite universe. The infinity describes physical time, it says that there is an infinite amount of time you will be alive.
That's true. You prefer to assign infinitely many utilities to infinitely many states of possible finite universes, and I allow assigning one utility to one state of a possibly infinite universe.
I think the reasons for favoring each case are probably clear to us both, so now I vote for not generating further infinities of comments in the recent comments feed :)
No, I mean that there are no infinite sets in first-order Peano arithmetic. The class "natural number" is not an object in Peano arithmetic.
First-order Peano arithmetic does not explicitly refer to sets, infinite sets, or natural numbers, but it describes them.
Correct. Nor does "There is no time of death" refer to a set. You can model it as quantifying over a "set of times" if you like, just as Peano arithmetic can be modeled as quantifying over a "set of numbers", but this does not mean the theory refers to sets, or infinity.