If it's infinite, then you could potentially argue that God enforces induction in order to benevolently prevent people from being confused, but you're still going to have to solve measure theory in order to say anything else about universes.

My claims are three-fold:

1) There is no elegant way to compare infinites of the same cardinality (or, if they’re too big to have cardinality, whatever represents their size) except in very, very rare cases.

2) Even if there’s no way to compare infinites of the same cardinality and there are two infinite sets of the same cardinality (I’m going to talk about cardinality, but the same points apply if it’s too big to have a cardinality) you shouldn’t think you’re equally likely to be a member of two different infinite sets as long as the subjective probability of you being in one of the sets is higher than the other. For example, if you roll a die in an infinite hotel, the same number of people will get 1-5s as 6s, but you should expect to probably get a 1-5, because you’re doing something that mostly (in terms of subjective probability—or even the objective probability as you keep flipping it) mostly comes up 1-5.

3) This means that as long as we have reason to think that for any particular person, probably they’re placed in an inductive world, induction is safe! Yay! Theism has that—God is good, and for any particular person, it’s likely that their flourishing involves putting them in an inductive universe—while Tegmark’s world has no single process that’s done to everyone and in most cases produces an inductive world. Thus, Tegmark can’t solve the problem—you need something to be true of everyone that in most cases places them in an inductive world, rather than simply having a lot of worlds, some of which are inductive, others of which are not.

(I also have a vague memory that if you weight universes by something like Kolmogorov complexity of their starting conditions you might be able to rescue induction, but I can't even give you a link, let alone a proof)

That view, unfortunately, has the problem of being extremely false, perhaps at levels that no one has ever seen before :) joecarlsmith.com/2021/1…

My concern is, when I actually talk to you about all this, your position sounds reasonable. You admit that everything around infinity creates paradoxes, that there's no good solution, and that you have a slight preference for the horrible paradoxical solutions of SIA or theism compared to the horrible paradoxical solutions of SSA or Tegmark or something. I kind of disagree, but whatever.

When I read your blog, it sounds like you think you have totally proven SIA and theism, that you're delighted by them, that you are comfortable applying anthropic reasoning to prove things about other unrelated fields without restraint, and that it's impossible for thoughtful people to disagree.

Let me be very clear: I give maybe 2:1 odds to theism being true, and am nowhere near certain in it. I am extremely confident in SIA being true—at about 90% odds, and even that is mostly because smart people disagree. I think it’s very difficult and tricky to figure out how to solve various infinite paradoxes (compare: I’m pretty confident in utilitarianism or something like it, even though infinite ethics paradoxes are horrendous. I’m also very confident that SIA raises the probability of theism because, while some of the infinite anthropics stuff gets tricky, it’s only when you’re comparing infinite sets of the same cardinality—here you’re not doing that, but comparing infinite sets to either finite ones or ones of smaller cardinality.

Why am I so confident in SIA despite horrendous paradoxes? Mostly because they are not paradoxes of SIA. They are paradoxes of any kind of anthropic reasoning, and whatever solution anyone else has, an SIAer can adopt. So the infinite ethics stuff is, in my view, not a good way to reject SIA, particularly because, while it’s a mess, it’s not totally obvious that there’s no elegant way to do it, and infinity ruins everything.

The other arguments against SIA—that don’t involve infinite paradoxes—are, in my view, extremely weak. Here’s how pretty much the entire literature happens:

STEP 1: person introduces SIA and explains that it solves the doomsday argument.

STEP 2: they present the presumptuous philosopher result (that SIA leads you to be confident that the universe is big or infinite, depending on the setup).

Then, they reject it. But the presumptuous philosopher objection is not very good—you can show that anyone who avoids the result must accept lots of things that are much worse. link.springer.com/artic…

Then you see the arguments for SIA, and they are about as good as any argument in philosophy. There are about two dozen facts favoring SIA—some of which are quite knockdown arguments that would be enough to move pretty much everyone in any other area of philosophy (I gave 4 in my article arguing for God—but there are plenty of others; see here, for instance arxiv.org/pdf/gr-qc/000…). I think, therefore, the case for SIA is quite overwhelming and the objections are just meh.

You worry that SIA leads to thinking that there are infinite states. Well, remember SIA leads to thinking there are more people that you might currently be—so if the people in the other states know they’re in other states, it won’t help. But I accept there are infinite states—in the multiverse! Now, you can posit that there are more states in any particular region (e.g. 1,000,0000,000,000 underground people who are hallucinating being Californians) but once you conclude that every possible person exists, stuffing universes with more people won’t actually raise the number of people with any property (if our universe is stuffed with hallucinating Californians, so are the others, and so it won’t affect the relative frequencies). It’s also at least unclear how to do this once there are infinite people, so you should at least not be confident, even if you don’t think all possible people exist, that there are infinite states.

Now, this may seem like an absurd consequence. But we have independent arguments for it that are convincing. We can show that anyone who doesn’t think you should be infinitely confident that there are infinite copies of you think that you can get psychic powers just by learning you’re early—that’s much worse!

Here's another one: shouldn't SIA believe that the moral law requires you to maximize your number of children,

Well, once you think that all possible people exist, SIA doesn’t lead you to person max in the same way. Fortunately, that’s what SIA leads you to think.

Even if you don’t think that, I don’t think this is true because:

1) It’s not clear how many more copies of me there will be if it’s objectively moral to maximize people—maybe like twice as many or something if there’s an elegant way of comparing it, but you won’t get extreme presumptuousness.

2) Given that very different collections of people would exist, I’m very skeptical that there’s any elegant way of doing these sorts of comparisons.

The only escape I can find from this is that maybe you can't use anthropics to prove necessary truths (like the moral law) because it can only distinguish between possible worlds. But then I don't think you could prove God either (since if He exists, He exists necessarily).

This isn’t the route I’d like to go, but it’s a route you could go. First of all, you could deny God is necessary. I’m pretty agnostic on whether God is necessary, and some very good philosophers (e.g. Richard Swinburne) don’t think he’s necessary). But the view probably wouldn’t be about necessity strictly, but instead something similar (I don’t have a precise account of the view, but there could be something like it). For instance, you might think it would be suspicious to conclude that you can have infinite credence in propositions—more than 100%—based on the fact that it more strongly predicts your specific observations (if I apply Bayes theorem to probabilities more than 100, then I get more confirmation). Maybe you’d think that something is going awry there in the same way that it’s going awry here.

(Sorry, this comment has grown very long).

I agree that the best objection to SIA is that it makes it’s not super clear what it says about the real world. It tells you the world is infinite and then makes it unclear how to reason about infinites. Now, I don’t think this is totally unworkable—SIA probably once you conclude that the world is infinite mostly works out to being normal—but it’s a puzzle.

Worryingly, its rivals are much worse. SSA implies a 110 billion to 1 update in favor of solipsism (jc.gatspress.com/pdf/SI…), and compartmentalized conditionalization implies even more confidence in skepticism core.ac.uk/download/pdf…). The problem is that there are plausible anthropic principles that conflict, so it’s not possible to be a moderate—and the rivals to SIA imply near certainty in extremely unlikely scenarios.

Related to my confusion about how strongly you believe these things, can I ask whether these considerations impel you to practical theism (eg do you go to church, confess your sins, live a Biblically-inspired lifestyle, etc)? If not, why not?

I do not do these things because I’m not a Christian. However, I’ve looked into Christianity pretty carefully, as well as various other religions, I pray regularly, and I now act differently on account of my theism (if the purpose of our world is to build connections that last forever, then that lowers the value of various pleasures—and I’ve acted accordingly).

(Sorry, that ended up being much longer than I expected).