Paradoxes and infinity
Following up my three-way classification of paradoxes, I want to argue that paradoxes involving infinity are always of type-3, that is, the result of ill-posed problems or inappropriate ways of taking limits. (Much the same position is defended in the comments thread by Bill Carone). In fact, I’d argue for the following general principle, applicable to all models relevant to human decisionmakers.
Whenever a result, true for all finite n, is strictly reversed for the infinite case, the problem in question has been posed incorrectly
To defend this, I rely on the premise that we are finite creatures in a finite universe. If a mathematical representation of a decision problem involves an infinite set, such as the integers or the real line, it is only because this is more convenient than employing finite, but very large bounds, such as those derived from the number of particles in the universe. Any property that depends inherently on infinite sets and limits, such as the continuity of a function, can never be verified or falsified by empirical data. Since we are finite, any result that is true for all finite n is true for us.
It’s not surprising then that paradoxes frequently introduce beings who can be assumed infinite – God, angels, demons and so on. There is, of course, a long theological tradition of asking what God can and cannot do – for example, make a logically contradictory statement true. It certainly appears from the various paradoxes that, if God is capable of handing out infinite rewards and punishments (which is, I think, generally supposed by believers), it’s not valid to say that, if a given course of action is better than another in every possible case (for some partition of the possible cases), then it is definitely the best choice.
It’s hard to imagine decision theory without this premise, so it’s reasonable to conclude that good theists should not be good decisionmakers, and vice versa paxil birth defects cases – a conclusion supported by consideration of Pascal’s wager. Similarly, to suggest that the Almighty can’t generate a probability distribution giving equal weight to every positive integer seems tantamount to denying Omnipotence altogether, but allowing for this possibility creates all sorts of problems and paradoxes, leading once again to the acceptance of strictly dominated choices.
This leads me on to this paper by Arntzenius, Elga, and Hawthorne, pointed out by Brian. As in the Peter Wakker piece I cited earlier, Arntzenius, Elga, and Hawthorne observe the crucial role of (the absence of) countable additivity in generating a number of paradoxes. But rather than adopting a type-3 solution like Wakker’s (in such circumstances always use a sigma-algebra and not just an algebra), Arntzenius, Elga, and Hawthorne seem to want to derive a type-1 solution, in which this result is supposed to have practical implications, such as that “when infinitely many decisions are involved, the difference between making the decisions simultaneously and making them sequentially can be the difference between riches and ruin”. Except in relation to dealings with supernatural beings, I think this conclusion is profoundly mistaken.
fn1. Insert obligatory joke about the world being divided into three sorts of people
fn2. To clarify the relevance of “strictly”, consider a sequence xn approaching y from below. It’s true for all finite n, but not in the limit, that xn is less than y. A strict reversal would arise if in the case where for infinite n, we had x strictly greater than y.