# Sniffnoy comments on VNM expected utility theory: uses, abuses, and interpretation - Less Wrong

20 17 April 2010 08:23PM

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Comment author: 18 April 2010 12:02:28AM 2 points [-]

Well, first-order Peano arithmetic doesn't have any notion of "set", much less "infinite"...

Comment author: 18 April 2010 12:10:54AM 0 points [-]

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.

Comment author: 18 April 2010 12:31:58AM *  0 points [-]

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..."

Comment author: 18 April 2010 12:52:05AM 0 points [-]

The standard model of Peano arithmetic is 0={} and S(X)=Xunion{X}; its objects are all finite sets.

The set of all objects in this standard model is infinite, even though Peano arithmetic does not refer to this set.

I fail to see any meaningful sense in which ""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 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.

Comment author: 18 April 2010 01:05:31AM *  0 points [-]

The difference is that in my version, utility is assigned to states of a finite universe.

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 :)