I thought you said that strange things are odd, and that we should therefore expect reality to be a very unstrange thing, namely a perfectly good being (Vikram Valame)!
How are you counting persons? If two instances of a matter-configuration have identical experiences, are they one or two people? Is the observation they have weighted by two in likelihood?
If you have a mind and cut all its circuit wires lengthwise, is it now two minds?
I’m inclined to say no, these are the same mind and so they get no extra weight for there being two instances.
2)
The goal of hypothesis formation is to predict your observations. If you can specify your observations with fewer bits, you have a better hypothesis. If your hypothesis needs to pick you out of a population of size omega (unique elements), you need omega bits. That means that the hypothesis is the worst hypothesis possible. If the elements aren’t unique, and there are eg 2^100 unique beings (copied as many or as few times as you want) then you need 100 bits to pick yourself out. So increasing the number of people doesn’t strengthen the hypothesis.
One good reply to Pascal's mugging arguments is that the larger the mugger's threat, the less likely they are to actually fulfill it. Shouldn't SIA be counterbalanced by similar considerations too?
Well, "there are as many people as there are sets" seems like an incredible claim, and I think it's reasonable to put a lower prior on incredible claims. I don't know if you should put an infinitely lower/zero prior on it.
If a mugger threatens you with an infinite punishment, I don't know if you should put a zero prior on it either. But surely you shouldn't agree to Pascal's mugging. So the infinite considerations must end up balanced out somehow, so maybe they get balanced out in SIA's case too.
In general, probability theory is not well equipped to deal with big infinities.
Student of set theory and logic here. Cantor defined Ω as the set of all ordinals. But the Burali–Forti Paradox shows that such a set cannot exist. That is, if we assume it exists, we can derive a contradiction.
Another remark: it is perhaps misleading to say that "there are Ω people," because Ω is an ordinal (at least it would be if it existed), not a cardinal. An ordinal signifies the order of a number. For example, 1, 2, 3 are ordinals in the sense of 2 succeeding 1 and coming before 3. This is quite different from 1, 2, or 3 conceived as cardinals, which in this case signify the number of elements, in other words, what you attempted to capture. The study of "very big numbers" as quantities rather than orders manifests in set theory as the study of large cardinals. ZFC famously cannot prove that they exist, so postulating the existence of one or more of these creates a strictly stronger set theory. I also suspect that "There is no largest cardinal," is true in ZFC. I will write a proof in the comments if I come across it.
Okay, maybe I was using it in a slightly different way from Cantor. In any ways, I claim it can't be a set, for the same reason there can't be a set of all truths.
In order to establish the claim that Ω is not a set, one must first define it. So, what is your definition of Ω, and in which logic? The most standard way to go about it would be a set-theoretic definition in the language of sets in FOL, but lesser-known systems also exist.
In set theory, this is called a proper class. There are many different classes. For example, there is a class of all singletons, {{x} : x = x}, which does not exist in ZFC. There is also the class of all sets {x : x = x}, which also does not exist. The very point of set theory is to lift our guesses in natural language to the rigor of formal proof. In ZFC, there are formal proofs as to why none of these classes are sets. Namely, they both use Russell's Paradox in conjunction with the axiom schema of specification.
For a formal treatment of classes, see the NBG axiomatization of set theory.
Suppose there is the largest cardinal C. Let this mean that for all cardinals B, C ≮ B. But consider now the power set of C, P(C). Cantor proved that C ≺ P(C). Therefore, it holds that C < |P(C)|, a contradiction. The same proof can be run with the stronger and more intuitive premise, that for all cardinals B, B < C, because it implies C ≮ B.
For a detailed explanation of the reasoning used, consult The Foundations of Mathematics by Kenneth Kunen.
If SIA is true, should I assume that everyone is having my exact string of experiences, no matter what they’re doing. It’s surely infinitely more likely that I’d experience these experiences in that case.
I find anthropics super interesting thanks for all the work you’ve done on it. It is one of those fields where every answer sounds crazy in some sort of way.
//If SIA is true, should I assume that everyone is having my exact string of experiences, no matter what they’re doing. It’s surely infinitely more likely that I’d experience these experiences in that case.//
That actually has a prior of zero, because there are infinite possible experiences, and yours aren't special. Also, it's not so clear if this is right--for if there are infinite people having them in both cases, it's unclear how to do comparisons.
If my observations/experiences also have a prior of zero, which it sounds like is what you're suggesting, then conditionalizing on them can boost hypotheses' probabilities from zero to positive.
It's almost like you became a Guenonian, he called out the Infinite as the thing that unifies every religion. Maybe you would enjoy just jumping off the deep end and reading The Multiple States of the Being, which is about the Infinite, and a little over 90 pages.
Based on the anthropic data, wouldn't a theory that posits that the Earth has an average level of suffering be infinitely more probable than a theory that posits that ~100% of conscious existence is suffering-less and perfect? (theism)
Good poetry, but this post seems to be good evidence that your analysis has gone deeply off the rails somewhere.
Reality is often strange.
I thought you said that strange things are odd, and that we should therefore expect reality to be a very unstrange thing, namely a perfectly good being (Vikram Valame)!
Two issues:
1)
How are you counting persons? If two instances of a matter-configuration have identical experiences, are they one or two people? Is the observation they have weighted by two in likelihood?
If you have a mind and cut all its circuit wires lengthwise, is it now two minds?
I’m inclined to say no, these are the same mind and so they get no extra weight for there being two instances.
2)
The goal of hypothesis formation is to predict your observations. If you can specify your observations with fewer bits, you have a better hypothesis. If your hypothesis needs to pick you out of a population of size omega (unique elements), you need omega bits. That means that the hypothesis is the worst hypothesis possible. If the elements aren’t unique, and there are eg 2^100 unique beings (copied as many or as few times as you want) then you need 100 bits to pick yourself out. So increasing the number of people doesn’t strengthen the hypothesis.
One good reply to Pascal's mugging arguments is that the larger the mugger's threat, the less likely they are to actually fulfill it. Shouldn't SIA be counterbalanced by similar considerations too?
No, why think that infinite people has an infinitely lower prior?
Well, "there are as many people as there are sets" seems like an incredible claim, and I think it's reasonable to put a lower prior on incredible claims. I don't know if you should put an infinitely lower/zero prior on it.
If a mugger threatens you with an infinite punishment, I don't know if you should put a zero prior on it either. But surely you shouldn't agree to Pascal's mugging. So the infinite considerations must end up balanced out somehow, so maybe they get balanced out in SIA's case too.
In general, probability theory is not well equipped to deal with big infinities.
Student of set theory and logic here. Cantor defined Ω as the set of all ordinals. But the Burali–Forti Paradox shows that such a set cannot exist. That is, if we assume it exists, we can derive a contradiction.
Another remark: it is perhaps misleading to say that "there are Ω people," because Ω is an ordinal (at least it would be if it existed), not a cardinal. An ordinal signifies the order of a number. For example, 1, 2, 3 are ordinals in the sense of 2 succeeding 1 and coming before 3. This is quite different from 1, 2, or 3 conceived as cardinals, which in this case signify the number of elements, in other words, what you attempted to capture. The study of "very big numbers" as quantities rather than orders manifests in set theory as the study of large cardinals. ZFC famously cannot prove that they exist, so postulating the existence of one or more of these creates a strictly stronger set theory. I also suspect that "There is no largest cardinal," is true in ZFC. I will write a proof in the comments if I come across it.
Okay, maybe I was using it in a slightly different way from Cantor. In any ways, I claim it can't be a set, for the same reason there can't be a set of all truths.
In order to establish the claim that Ω is not a set, one must first define it. So, what is your definition of Ω, and in which logic? The most standard way to go about it would be a set-theoretic definition in the language of sets in FOL, but lesser-known systems also exist.
I don't know if I have a super precise definition. Maybe it's the amount of stuff too large to be a set.
In set theory, this is called a proper class. There are many different classes. For example, there is a class of all singletons, {{x} : x = x}, which does not exist in ZFC. There is also the class of all sets {x : x = x}, which also does not exist. The very point of set theory is to lift our guesses in natural language to the rigor of formal proof. In ZFC, there are formal proofs as to why none of these classes are sets. Namely, they both use Russell's Paradox in conjunction with the axiom schema of specification.
For a formal treatment of classes, see the NBG axiomatization of set theory.
It's too be to be a class too.
Here is the promised proof.
Suppose there is the largest cardinal C. Let this mean that for all cardinals B, C ≮ B. But consider now the power set of C, P(C). Cantor proved that C ≺ P(C). Therefore, it holds that C < |P(C)|, a contradiction. The same proof can be run with the stronger and more intuitive premise, that for all cardinals B, B < C, because it implies C ≮ B.
For a detailed explanation of the reasoning used, consult The Foundations of Mathematics by Kenneth Kunen.
If SIA is true, should I assume that everyone is having my exact string of experiences, no matter what they’re doing. It’s surely infinitely more likely that I’d experience these experiences in that case.
I find anthropics super interesting thanks for all the work you’ve done on it. It is one of those fields where every answer sounds crazy in some sort of way.
//If SIA is true, should I assume that everyone is having my exact string of experiences, no matter what they’re doing. It’s surely infinitely more likely that I’d experience these experiences in that case.//
That actually has a prior of zero, because there are infinite possible experiences, and yours aren't special. Also, it's not so clear if this is right--for if there are infinite people having them in both cases, it's unclear how to do comparisons.
If my observations/experiences also have a prior of zero, which it sounds like is what you're suggesting, then conditionalizing on them can boost hypotheses' probabilities from zero to positive.
It's almost like you became a Guenonian, he called out the Infinite as the thing that unifies every religion. Maybe you would enjoy just jumping off the deep end and reading The Multiple States of the Being, which is about the Infinite, and a little over 90 pages.
Based on the anthropic data, wouldn't a theory that posits that the Earth has an average level of suffering be infinitely more probable than a theory that posits that ~100% of conscious existence is suffering-less and perfect? (theism)
I address that here https://benthams.substack.com/p/three-niche-philosophical-objections