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RobinZ comments on Rationality Quotes September 2012 - Less Wrong
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If life were of infinite value, trading a life for two new lives would be a meaningless operation - infinity times two is equal to infinity. Not unless by "life has infinite value" you actually mean "everything else is worthless".
Not quite so! We could presume that value isn't restricted to the reals + infinity, but say that something's value is a value among the ordinals. Then, you could totally say that life has infinite value, but two lives have twice that value.
But this gives non-commutativity of value. Saving a life and then getting $100 is better than getting $100 and saving a life, which I admit seems really screwy. This also violates the Von Neumann-Morgenstern axioms.
In fact, if we claim that a slice of bread is of finite value, and, say, a human life is of infinite value in any definition, then we violate the continuity axiom... which is probably a stronger counterargument, and tightly related to the point DanielLC makes above.
If we want to assign infinite value to lives compared to slices of bread, we don't need exotic ideas like transfinite ordinals. We can just define value as an ordered pair (# of lives, # of slices of bread). When comparing values we first compare # of lives, and only use # of slices of bread as a tiebreaker. This conforms to the intuition of "life has infinite value" and still lets you care about bread without any weird order-dependence.
This still violates the continuity axiom, but that, of itself, is not an argument against a set of preferences. As I read it, claiming "life has infinite value" is an explicit rejection of the continuity axiom.
Of course, Kaj Sotala's point in the original comment was that in practice people demonstrate by their actions that they do accept the continuity axiom; that is, they are willing to trade a small risk of death in exchange for mundane benefits.
You could use hyperreal numbers. They behave pretty similarly to reals, and have reals as a subset. Also, if you multiply any hyperreal number besides zero by a real number, you get something isomorphic to the reals, so you can multiply by infinity and it still will work the same.
I'm not a big fan of the continuity axiom. Also, if you allow for hyperreal probabilities, you can still get it to work.
At which point why not just re-normalize everything so that you're only dealing with reals?
You could have something have infinite value and something else have finite value. Since this has an infinitesimal chance of actually mattering, it's a silly thing to do. I was just pointing out that you could assign something infinite utility and have it make sense.
True
Only if you have a way to describe infinity in terms of a real number.
You just pick some infinite hyper real number and multiply all the real numbers by that. What's the problem?
Oh, you're saying assign a hyperreal infinite numbers to the value of individual lives. That works, but be very careful how you value life. Contradictions and absurdities are trivial to develop when one aspect is permitted to override every other one.
Nitpick, I think you mean non-commutativity, the ordinals are associative. The rest of your post agrees with this interpretation.
Oops, yes. Edited in original; thanks!