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Decius comments on On the importance of taking limits: Infinite Spheres of Utility - Less Wrong
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Does it matter if the number of people is countably infinite, or uncountably infinite?
If each person corresponds on a 1-1 basis with the real numbers, there are an infinite number people who will not be selected to change spheres on any of the integer-numbered days. Those people will never change spheres.
That can also happen if there are countably infinite people. Suppose that for each n, on the n-th day the 2n-th person is moved from hell to heaven. All the odd-numbered people will stay in hell forever.
Conclusions that involve infinity cannot generally be generalized to any finite solution; this seems like a 'each monkey now has two bananas' moment.
The finite analogue come from the fact that infinity minus infinity is undefined, and can be anything from "still infinity" (like if you had infinity people and then only sent away the even numbered ones) to any number you choose, to negative infinity. In a finite problem, the answer becomes well-defined, but there are multiple possible answers.
Yeah, that's the point.
Consider the converse: Is there a way to arrange the days and people such that it is better to start in hell? Trivially, it seems like the simple solution is that way, since each person leaves hell after a finite number of days and then spends infinity days in heaven, but I lack the concept which allows me to find the amount of time the average person spends in hell.
By the way, are you talking about this meme, or is there another problem with monkeys and bananas?
I was talking about the math that spawned that meme, yes.
Just compare the cardinality of the number of days to the cardinality of the number of people. If |days| < |people| then start them in the heaven sphere. If |days| = |people| then it doesn't matter (by symmetry the first are last, and the last first, so to speak). If |days| > |people| then start them in the hell sphere.
My first impression was the same as yours, but then I realized there was no guarantee about any of the cardinality, even for the set of days. The post assumes the reals, but comparing the cardinality should work for any sets (although if they're bigger than the reals can we really compare "utility" at all?)
I think that's a better statement of what I tried to say.
In the above example, the number of people and the number of days they live were uncountable, if I'm not mistaken. The take-home message is that you do not get an answer if you just evaluate the problem for sets like that, but you might if you take a limit.
Conclusions that involve infinity don't map uniquely on to finite solutions because they don't supply enough information. Above, "infinite immortal people" refers to a concept that encapsulates three different answers. We had to invent a new parameter, alpha, which was not supplied in the original problem, to come up with a well defined result. In essence, we didn't actually answer the question. We made up our own problem that was similar to the original one.
Provided you can assign a unique rational number to each day each person lives, they are countable.
I will note that the expected time for a given person to remain in the sphere in which they started is infinite, provided they don't know in what order they will be removed. The summation for each day becomes (total of an infinite number of people)+(total of a finite number of people); if we assume that a person-day in bliss is positive and a person-day in agony is negative, then the answer is trivial. An infinite summation of terms of positive infinity is greater than an infinite sum of terms of negative infinity- the cardinalities are irrelevant.
Thanks for clearing up the countability. It's clear that there are some cases where taking limits will fail (like when the utility is discontinuous at infinity), but I don't have an intuition about how that issue is related to countability.
You said 'discontinuous at infinity'. Did you mean 'the infinite limit diverges or otherwise does not exist'?
No, I mean a function whose limit doesn't equal its defined value at infinity. As a trivial example, I could define a utility function to be 1 for all real numbers in [-inf,+inf) and 0 for +inf. The function could never actually be evaluated at infinity, so I'm not sure what it would mean, but I couldn't claim that the limit was giving me the "correct" answer.
If you accept the Axiom of Infinity, there's no problem at evaluating a function at infinity. The problem is rather that omega is a regular limit cardinal, so there's no way to define the value at infinity from the value at the successor, unless you include in the definition an explicit step for limit cardinals.
You can very well define a function that has 1 as value on 0 and on every successor cardinal, but 0 on every limit cardinal. The function will indeed be discontinuous, but its value at omega will be perfectly defined (I just did).
The problem with saying a function is not continuous at infinity is that the definitions of 'continuous' requires the standard definition of 'limit' (sigma-epsilon), while the definition of limits at infinity uses the same nomenclature and similar notation, but expresses something different.
Consider the case where F(X-epsilon) is 1, F(X) is 0, and F(X+epsilon) is either 0 or undefined. The common thought there is that the limit at X does not exist; why is that any different just because X is infinite, without sacrificing the concept which allows us to talk about continuity in terms of limits?
Well... I guess you can see it that way if you want, but in set theory (again, everything I say is under the axiom of infinity) both notions are unified under the notion of limit in the order topology.
In this way, you can define a continuous function for every transfinite ordinal.
Yes, I understand that the concept of limit in calculus and set theory means something a little different. Possibly this is just arguing over definitions: in calculus, it is said that a limit doesn't exist when the function has a different value w.r.t. the value calculated using the topology of its domain, but in set theory a limit is defined in a different way, using only the order topology. In this sense, a function can be defined at omega, at omega+1, omega+2, etc. After all, that's the raison d'etre of the entire concept. Under this assumption, you just say that if the function is defined at omega, and if it has a different value at omega than the one defined from its order topology, you just say that it's discontinuous.
Let me clarify with an example: the function y = 2x, defined in set theory, would have at omega the limit omega (demonstration below). If you just define a similar function but that has at omega the value 0, then you have a discontinuous function, because the (topological) limit is different from the defined value.
Of course, if you want, you can just say that the (simple) limit doesn't exists when this situation arise, I was on the other side pointing to the fact that a function can be perfectly defined and continuous or discontinuous at infinity.
Thesis
Omega is the limit of y=2x, defined on the natural number.
Demonstration
The intersection of the range of the function with omega is still the range of the function.
The range is unbounded but every one of its member is finite: since omega is regular limit, it is never reached by the sequence. So, any ordinal greater then or equal to omega is an upper bound. But omega is also an initial ordinal, so is the least upper bound.
That is, the least upper bound of the intersection of the range of the function with omega is still omega, so by definition omega is the limit of the range of the function. QED.
TIL that mathematicians commonly strictly define terms in one context, then extend them into other contexts in ways that are not strictly compatible with the original context.
Now I understand that two people with significantly different levels of math education and understanding lack the common vocabulary required to trivially communicate basic math-related concepts.
It's still going to be hard to convince me that the sum of an infinite number of days, each of which has infinite positive utility and finite negative utility, will ever be lower than zero.
Um... -inf and +inf are not real numbers. (Noting that your function as described is undefined at -inf.)
In addition, the definition of continuous restricts it to points which exist on an open interval; if the limit from below and limit from above are equal to the value at X, then the function is continuous on an open interval containing X. How do you determine the limit as X approaches +inf from above?
MrMind explains in better language below.