I'm sorry, I've read this a few times but I don't see the paradox? Conditioning on measure zero events *is* ill-defined, and that's why you can get different solutions to P(A|B) given different approaches. Is there something I'm missing?
Sure, the conditional probability is undefined since you’d have to divide by 0. But that’s just a problem with the math. This could totally happen, and if it did the angel would have to figure out what their credence should be.
Well then I would say that saying P(H|F’) = P(H|F) is also just a problem with the math and would see no qualms when the two conditioning events are presented as processes, taking the conditional probabilities to be 0 and 0.5
What I'm getting at here, is that the argument for P(H|F') is presented somewhat deceivingly, namely it seems less like we're computing a conditional probability in the natural sense, and more like
1. We already draw some sequence
2. We're told it has a finite amount of tails aside from 100
3. What should we take P(H) to be?
This seems uncontroversially 0.5 to me, but there's no conditionalizing happening here! This is because we draw the sequence of Heads, Tails first!
Whereas with P(H|F) the argument is different, namely saying that if we're drawing from a sequence with finite amt. of tails, what's the likelihood that the 100th flip is tails. I am less certain this number is 0, in fact I'd have to hear a bit more how this drawing occurs.
This is the correct response to this argument, imo. A contradiction was snuck in and so we’ve basically principle of explosion-ed to any conclusion. (I’m not saying this is the exact reasoning used, but I think this is fundamentally what’s going on, and your example with measuring an unmeasurable set is illustrative here.)
But could this “totally happen”? Empirically I was never able to complete an infinite series of coin tosses as I was always running out of time and space on my notebook to write down the outcomes.
I’m under the impression that those are being not uniquely determined is evidence or being poorly defined, and the primary reason why things are poorly defined, is this not the case?
I read "ill-defined" as meaning something like "not meaningful" rather than "not unique". On a reread, I see what you wrote is compatible with the latter.
Yeah on first read this argument sounds like that trick where you divide by zero at some point in a sequence of seemingly innocuous algebraic manipulations and you get to prove that 1=0.
Would causal finitists be okay with God throwing an infinitely precise dart? The same puzzle comes up if instead of coin flips you take the binary expansion of the y coordinate of where the dart landed (though this would rule out “eventually always 1” due to non-uniqueness of binary representations, but I think we could get around that). Or is the problem somewhere other than the act of flipping infinitely many coins?
Let’s say God has a square dart board, 1 unit wide. He throws a dart board and notes how high on the board it landed. Say it landed 9/32 units high. In binary, that’s 0.01001, which represents HTHHT and then infinitely many Hs. (Realistically, with probability 1, the number will be some weird irrational number with unpredictable digits). Then everything goes the same, replacing “there are finitely many tails” with “there are finitely many 1s” and “the 100th toss was tails” with “the 100th digit is 1”.
And here's maybe a motivation for that. It seems weird that there could be an infinite past causal sequence of coinflips, but there couldn't simply be a mind that was aware of each of the flips in the sequence.
This wouldn’t deal with the case mentioned in the final footnote where no physical quantities are involved, just God picking a number in His head though.
How about understanding omniscience as “God knows all that is knowable” so if there is no such a thing as the 10^100 digit in the decimal expansion of the dart’s position because that simply does not physically exist then this does not detract from omniscience? Should God have to pass an exam on unicorn anatomy to be declared omniscient?
> If you reply, “perhaps the angel isn’t certain that they’re really being told by God that there were finitely many tails,” you must still admit that the agent assigns nonzero probability to God having told the truth and them having understood God correctly and such, and so they assign nonzero probability to F = “there are only finitely many heads,” and the paradox re-arises with respect to the conditional probability distribution P(— | F).
Not necessarily. If the angel assigns a prior probability less than 1 to the proposition that God is really the one telling them this, and he is telling the truth (prior to hearing what exactly God tells them), then hearing that there are only finitely many tails should produce a Bayesian update such that the angel now puts a probability zero on the claim that God is really the one telling them this and is telling the truth.
3. I read the footnote now and it's not super convincing
Of course, I've never really been into the more abstract mathematics, so I don't know that my intuition as to the shakiness of sampling infinity is truly well-founded.
Not convincing? Maybe you don't like a series of events extending infinitely far back into the past, but at the very least with respect to a sequence ordered like the negative integers (..., -3, -2, -1), talk of the 100th latest term is coherent
I probably won't get into the weeds further on this but with these specific infinite sequences, there is a defined end and an undefined end
Taking the hundredth term from the defined end of the sequence is easy, but taking the hundredth term from the undefined side is not. You can't point to the negative numbers as a sequence where the latest 100th term is defined because you dont start counting the negative numbers from -∞ and end at -1, you start at -1 and end up at -∞
I think Flo was clear that we know the end of Bob's infinite series of coin-flips, so "100th most recent" is fine; we just count backwards from the end.
My objection to this is that the existence of God is not a statement about abstract math, it's a statement about the real world, and you can't actually have an infinite series of coin-flips in the real world. And infinity is central to the argument. So regardless of its mathematical validity or lack thereof, the argument is ultimately irrelevant.
Two quick comments: (i) David Builes presents basically this puzzle in his 2020 paper 'A Paradox of Evidential Equivalence', to which Cian Dorr, John Hawthorne and Yoaav Isaacs reply - convincingly, to my mind - in 'Solving a Paradox of Evidential Equivalence' (Mind 2021). [The solution they give is to deny that P(H|F') = 0.5.] (ii) Conditionalising on credence zero propositions is fine if you take conditional probability as primitive work with Popper functions (as I believe Builes points out in his paper, but it's been a while since I've looked at it).
Conditional expectations admit different versions. Kolmogorov's theorem shows existence, not uniqueness. If you define this formally you're going to be conditioning on S := the limsup of the count of tails in the sequence after the n'th outcome. By Kolmogorov's 0-1 law we know that the event S=0 has probability 1 or 0, and it's easy to verify that S>0 has positive probability, so S=0 has probability 0. So, when we go to define the regular conditional probability, the integral of the joint density of X_{100} and S over {X_{100}=T&S=0} will be 0. So the regular conditional probability can take an arbitrary value. Similarly for S' defined like S but excluding the 100'th toss.
I'm not seeing that F and F' are NOT relevantly different. I'd need to think about it more, but it seems like the information in F tells me something important relevant to my assessment of the probability.
I've never trusted infinity since I realized that the decimal expansion of 1/3 is 0.333..., you can add two of them and you get a decimal expansion of 2/3 as 0.666... but then add a third and you get 1 rather than 0.999.... Clearly no good can come of thinking too much about infinity it can only drive one mad, take Cantor, please!
Feel like there is a Bertrand's type paradox lurking here. Someone comes up to you and says I have a machine that generates a non-uniform sequence of digits that will be some integer, but I'm not going to tell you which one, but let me assure you whatever resources you think I have, I have more. What is my credence going to be about which number the machine is going to generate (sample from the integers), it feels like I have equal credence for every number. Not because I believe the machine can generate any number, but because I think for any number I can think of the machine might be the one that generates that number from its biased sampling. This doesn't seem to require infinite information to bring about. I suppose the lesson is that my credence would actually be spread about across a non-uniform distribution also, but it would be as close to uniform as my Baysian cognitive powers would allow given the resources available to it?
The traditional proof of atheism goes (I'm giving a parallel argument):
Nothing is better than a whole loaf of bread when you are hungry.
Half a loaf o bread is better than nothing.
Therefore half a loaf of bread is better than a whole loaf of bread when you are hungry.
(It is left to the reader to substitute "God" for "whole loaf of bread when you are hungry" and consider the proof one might so derive)
This is a good argument for causal finitism. But surely if causal finitism is true, then God exists. So this isn't a proof that God doesn't exist, but in fact a strong argument for His existence!
I believe Knuth thinks that God, though very large, could be finite. Seems like this gets you out of a lot of theological trouble if the universe is also finite.
On the other hand, it might get you into some theological trouble, like with the Inquisition.
You can't flip for infinitely long and then "come to a stop." Mathematically you can't talk about the *end* of an infinite sequence or what's 100 places back from it.
I'm not a theist myself, so I have no objection, intellectually or emotionally, to disproving the existence of God, but I don't think this argument does it.
You have made a basic mathematical error here, because division by infinity is undefined in standard mathematics. For any n > 0, if n divided by infinity is 0, then infinity multiplied by 0 is n, and we know that isn't true, because anything multiplied by zero is zero. So your claim that the probability of any specific infinite series occurring is zero is wrong. It's not zero, it's undefined.
The omnipotence of God does not necessarily extend to logical contradictions. God doesn't have to be able to create a rock so heavy that he can't lift it, and he doesn't have to be able do anything that depends on a mathematical absurdity.
There are other problems here, but I think that's enough to undermine the argument. I suspect you're just trolling with this post, anyway.
The part where I argue the probability can’t be 0 also works for any number other than ½ (the ½ possibility being addressed after that), so my using loose speaking to argue for 0 is inconsequential
That doesn't really help, because the whole argument is based on problems related to infinity; you've devised a purely mathematical absurdity, since in the real world you can't actually have an infinite series of coin-flips. If you could make the same argument with a finite series, I suspect you would have (note that the link you labeled "there is no uniform distribution on a countable set" actually only applies to countable infinite sets; I assume that's just a typo on your part). So we're back at my statement (though it's hardly original with me) that God's omnipotence need not extend to absurdities.
This is very interesting... I have a few issues with the argument, but I'm not sure any definitively defeat it.
1) Why not use the hyperreal numbers and say that the probability of any infinite sequence is infinitesimal? I'm not sure if this would work, but it makes more sense to me than saying that it's 0 and yet could happen. Not sure if this makes too much difference to the argument.
2) You say that it's a fair coin, but you've also given incredibly strong evidence that it's a very unfair coin. What does it actually mean to say that the odds of getting a heads or tails are half, considering the monumental evidence against this? We need to establish what probability really means, which is not a trivial task.
3) I think (2) offers the clue. P(H|F') is (roughly) treated as the prior probability, sticking with the assumption that the coin is fair. P(H|F) is (roughly) treated as the posterior probability, implicitly taking into account the actual results of the coin toss that offer essentially infinite evidence that the coin is very strongly rigged. If we just try to accommodate the evidence that the coin is rigged, the two should come out as equal.
4) So long as it is possible to be perfectly certain of something, and for that something to be incorrect, it is possible to gain infinite information. That is because the information we gain from an event is -log(p), where p is the probability of the event. If we are certain p will never happen, p is 0, in which case the information gained from the event would be infinite.
And actually, if the probability of any infinite sequence of coin tosses is 0, then we find out that an infinite sequence of coin tosses has occurred, even if we don't know the sequence, then we have gained infinite information. The fact that we don't know which sequence it was doesn't matter, we know that an event we assigned p=0 to has occurred, and so our information gained is -log(0)=infinity (well, not really... but it tends to infinity as p tends to 0).
5) This is a boring one but we can't really prove anything from pushing an area of maths beyond its limits and finding that it breaks down.
6) Angels can solve this easily using angel maths :)
Nonstandard (hyperreal) probabilities can't be used to solve this problem, and in general typically don't work when applied to standard structures rather than nonstandard ones.
In any case, I think we'd need to either try something similar, or just say the probability of infinite coin tosses is not defined (but that's boring).
Hyperreal summation is really weird/particular in terms of the allowed sets or allowed sequences you can sum over. Only sets with a particular property called "being internal" will work. The natural numbers aren't internal, so you can't assign an infinitesimal probability to (for example) each natural and have the sum over every such number even be defined, much less add up to 1. Things would be even more ill-defined in the case of the space of all infinite coin flips, so you couldn't talk about "the infinitesimal probability of getting a finite number of heads in your infinite sequence of flips;" the set of all outcomes with finitely many heads is not internal.
In fact, to start getting actually interesting internal sets, we have to imagine not flipping a coin an "ordinary" infinite number of times (i.e., a sequence of flips indexed by the naturals) but rather a *specific hyperreal infinite* number of times (i.e., a sequence of flips indexed by all hypernatural numbers from 0 up to N, where N is some specific infinite hypernatural, of which there's a huge variety to pick from, often uncountably many). And that's completely different from the blog post's setup.
Basically, exotically small probabilities only work on exotically large domains, outside of trivial cases where there's only a finite number of things to begin with, in which case hyperreals don't help with much anyway. That said, they *are* useful, but only as a kind of dispensable intermediary where at the very last step of your argument you go back to standard (real) probabilities. For instance, this is *sort of* what Loeb measure does, how the nonstandard approach to constructing Brownian motion works, how nonstandard Ito integrals work, etc.
I love (1)! Very creative idea to do probability with hyperreals -- I wonder what that would lead to.
Regarding (5), I do wonder if Flo's argument is just a dressed up version of the following:(premise) God, as an omnipotent being, should be able to divide by zero.
(premise) Dividing by zero is impossible.
(conclusion) God doesn't exist.
But mathematically Flo's argument is very interesting regardless of what it says about God. I'm wracking my brain to make sense of it. I think your ideas 1-4 are all promising. I also think maybe we need more care in understanding the idea that "probability zero events can happen". Like for any sequence of coin flips r I specify in advance, it is true I won't get r. It is only in retrospect that some sequence r is obtained. Maybe that is somehow relevant.
I'm sorry, I've read this a few times but I don't see the paradox? Conditioning on measure zero events *is* ill-defined, and that's why you can get different solutions to P(A|B) given different approaches. Is there something I'm missing?
Sure, the conditional probability is undefined since you’d have to divide by 0. But that’s just a problem with the math. This could totally happen, and if it did the angel would have to figure out what their credence should be.
Well then I would say that saying P(H|F’) = P(H|F) is also just a problem with the math and would see no qualms when the two conditioning events are presented as processes, taking the conditional probabilities to be 0 and 0.5
Could you explain this a bit more?
What I'm getting at here, is that the argument for P(H|F') is presented somewhat deceivingly, namely it seems less like we're computing a conditional probability in the natural sense, and more like
1. We already draw some sequence
2. We're told it has a finite amount of tails aside from 100
3. What should we take P(H) to be?
This seems uncontroversially 0.5 to me, but there's no conditionalizing happening here! This is because we draw the sequence of Heads, Tails first!
Whereas with P(H|F) the argument is different, namely saying that if we're drawing from a sequence with finite amt. of tails, what's the likelihood that the 100th flip is tails. I am less certain this number is 0, in fact I'd have to hear a bit more how this drawing occurs.
If we ignore issues that are "just problems with the math", then the angel / God should also be able to tell us the measure of non-measurable sets.
God could just construct such a nonmeasurable object and then ask how much water he needs to fill it. Boom, paradox, God does not exist.
(The actual answer is that it's just nonsensical to talk about ill-defined math concepts like credence in this case, but let's ignore that here.)
This is the correct response to this argument, imo. A contradiction was snuck in and so we’ve basically principle of explosion-ed to any conclusion. (I’m not saying this is the exact reasoning used, but I think this is fundamentally what’s going on, and your example with measuring an unmeasurable set is illustrative here.)
But could this “totally happen”? Empirically I was never able to complete an infinite series of coin tosses as I was always running out of time and space on my notebook to write down the outcomes.
Sorry to nitpick, but this is a bugbear of mine. Conditioning on measure zero events isn't ill-defined, it's just not uniquely determined.
I’m under the impression that those are being not uniquely determined is evidence or being poorly defined, and the primary reason why things are poorly defined, is this not the case?
I read "ill-defined" as meaning something like "not meaningful" rather than "not unique". On a reread, I see what you wrote is compatible with the latter.
I’m not sure how it could be we
Yeah on first read this argument sounds like that trick where you divide by zero at some point in a sequence of seemingly innocuous algebraic manipulations and you get to prove that 1=0.
Seems one easy solution is to be a casual finitist.
Would causal finitists be okay with God throwing an infinitely precise dart? The same puzzle comes up if instead of coin flips you take the binary expansion of the y coordinate of where the dart landed (though this would rule out “eventually always 1” due to non-uniqueness of binary representations, but I think we could get around that). Or is the problem somewhere other than the act of flipping infinitely many coins?
Sorry can you explain how it works there?
Let’s say God has a square dart board, 1 unit wide. He throws a dart board and notes how high on the board it landed. Say it landed 9/32 units high. In binary, that’s 0.01001, which represents HTHHT and then infinitely many Hs. (Realistically, with probability 1, the number will be some weird irrational number with unpredictable digits). Then everything goes the same, replacing “there are finitely many tails” with “there are finitely many 1s” and “the 100th toss was tails” with “the 100th digit is 1”.
I think maybe a casual finitist could just say you can’t have infinitely divisible space
Solved a billion times over with zeno's paradoxes, no theism or anti-theism required
And here's maybe a motivation for that. It seems weird that there could be an infinite past causal sequence of coinflips, but there couldn't simply be a mind that was aware of each of the flips in the sequence.
This wouldn’t deal with the case mentioned in the final footnote where no physical quantities are involved, just God picking a number in His head though.
How about understanding omniscience as “God knows all that is knowable” so if there is no such a thing as the 10^100 digit in the decimal expansion of the dart’s position because that simply does not physically exist then this does not detract from omniscience? Should God have to pass an exam on unicorn anatomy to be declared omniscient?
> If you reply, “perhaps the angel isn’t certain that they’re really being told by God that there were finitely many tails,” you must still admit that the agent assigns nonzero probability to God having told the truth and them having understood God correctly and such, and so they assign nonzero probability to F = “there are only finitely many heads,” and the paradox re-arises with respect to the conditional probability distribution P(— | F).
Not necessarily. If the angel assigns a prior probability less than 1 to the proposition that God is really the one telling them this, and he is telling the truth (prior to hearing what exactly God tells them), then hearing that there are only finitely many tails should produce a Bayesian update such that the angel now puts a probability zero on the claim that God is really the one telling them this and is telling the truth.
Correct & importantant
I'm not sure that "the 100th most recent coin flip" is a coherent concept when it comes to an infinite series.
Did u read the footnote
1. Footnotes are ontologically evil
2. No I didn't read the footnote
3. I read the footnote now and it's not super convincing
Of course, I've never really been into the more abstract mathematics, so I don't know that my intuition as to the shakiness of sampling infinity is truly well-founded.
Not convincing? Maybe you don't like a series of events extending infinitely far back into the past, but at the very least with respect to a sequence ordered like the negative integers (..., -3, -2, -1), talk of the 100th latest term is coherent
I probably won't get into the weeds further on this but with these specific infinite sequences, there is a defined end and an undefined end
Taking the hundredth term from the defined end of the sequence is easy, but taking the hundredth term from the undefined side is not. You can't point to the negative numbers as a sequence where the latest 100th term is defined because you dont start counting the negative numbers from -∞ and end at -1, you start at -1 and end up at -∞
I think Flo was clear that we know the end of Bob's infinite series of coin-flips, so "100th most recent" is fine; we just count backwards from the end.
My objection to this is that the existence of God is not a statement about abstract math, it's a statement about the real world, and you can't actually have an infinite series of coin-flips in the real world. And infinity is central to the argument. So regardless of its mathematical validity or lack thereof, the argument is ultimately irrelevant.
"The lord works in mysterious ways"
Two quick comments: (i) David Builes presents basically this puzzle in his 2020 paper 'A Paradox of Evidential Equivalence', to which Cian Dorr, John Hawthorne and Yoaav Isaacs reply - convincingly, to my mind - in 'Solving a Paradox of Evidential Equivalence' (Mind 2021). [The solution they give is to deny that P(H|F') = 0.5.] (ii) Conditionalising on credence zero propositions is fine if you take conditional probability as primitive work with Popper functions (as I believe Builes points out in his paper, but it's been a while since I've looked at it).
SICK thanks for the reference
Conditional expectations admit different versions. Kolmogorov's theorem shows existence, not uniqueness. If you define this formally you're going to be conditioning on S := the limsup of the count of tails in the sequence after the n'th outcome. By Kolmogorov's 0-1 law we know that the event S=0 has probability 1 or 0, and it's easy to verify that S>0 has positive probability, so S=0 has probability 0. So, when we go to define the regular conditional probability, the integral of the joint density of X_{100} and S over {X_{100}=T&S=0} will be 0. So the regular conditional probability can take an arbitrary value. Similarly for S' defined like S but excluding the 100'th toss.
Wait a minute, this is just a retelling of the problem of evil! You almost got me!
Is it? How?
I'm not seeing that F and F' are NOT relevantly different. I'd need to think about it more, but it seems like the information in F tells me something important relevant to my assessment of the probability.
I've never trusted infinity since I realized that the decimal expansion of 1/3 is 0.333..., you can add two of them and you get a decimal expansion of 2/3 as 0.666... but then add a third and you get 1 rather than 0.999.... Clearly no good can come of thinking too much about infinity it can only drive one mad, take Cantor, please!
Feel like there is a Bertrand's type paradox lurking here. Someone comes up to you and says I have a machine that generates a non-uniform sequence of digits that will be some integer, but I'm not going to tell you which one, but let me assure you whatever resources you think I have, I have more. What is my credence going to be about which number the machine is going to generate (sample from the integers), it feels like I have equal credence for every number. Not because I believe the machine can generate any number, but because I think for any number I can think of the machine might be the one that generates that number from its biased sampling. This doesn't seem to require infinite information to bring about. I suppose the lesson is that my credence would actually be spread about across a non-uniform distribution also, but it would be as close to uniform as my Baysian cognitive powers would allow given the resources available to it?
The traditional proof of atheism goes (I'm giving a parallel argument):
Nothing is better than a whole loaf of bread when you are hungry.
Half a loaf o bread is better than nothing.
Therefore half a loaf of bread is better than a whole loaf of bread when you are hungry.
(It is left to the reader to substitute "God" for "whole loaf of bread when you are hungry" and consider the proof one might so derive)
Hah, I see what you did there. XD
This is a good argument for causal finitism. But surely if causal finitism is true, then God exists. So this isn't a proof that God doesn't exist, but in fact a strong argument for His existence!
I believe Knuth thinks that God, though very large, could be finite. Seems like this gets you out of a lot of theological trouble if the universe is also finite.
On the other hand, it might get you into some theological trouble, like with the Inquisition.
You can't flip for infinitely long and then "come to a stop." Mathematically you can't talk about the *end* of an infinite sequence or what's 100 places back from it.
That was addressed in the article - you don't need infinite time for infinitely many flips.
I'm not a theist myself, so I have no objection, intellectually or emotionally, to disproving the existence of God, but I don't think this argument does it.
You have made a basic mathematical error here, because division by infinity is undefined in standard mathematics. For any n > 0, if n divided by infinity is 0, then infinity multiplied by 0 is n, and we know that isn't true, because anything multiplied by zero is zero. So your claim that the probability of any specific infinite series occurring is zero is wrong. It's not zero, it's undefined.
The omnipotence of God does not necessarily extend to logical contradictions. God doesn't have to be able to create a rock so heavy that he can't lift it, and he doesn't have to be able do anything that depends on a mathematical absurdity.
There are other problems here, but I think that's enough to undermine the argument. I suspect you're just trolling with this post, anyway.
The part where I argue the probability can’t be 0 also works for any number other than ½ (the ½ possibility being addressed after that), so my using loose speaking to argue for 0 is inconsequential
That doesn't really help, because the whole argument is based on problems related to infinity; you've devised a purely mathematical absurdity, since in the real world you can't actually have an infinite series of coin-flips. If you could make the same argument with a finite series, I suspect you would have (note that the link you labeled "there is no uniform distribution on a countable set" actually only applies to countable infinite sets; I assume that's just a typo on your part). So we're back at my statement (though it's hardly original with me) that God's omnipotence need not extend to absurdities.
This is very interesting... I have a few issues with the argument, but I'm not sure any definitively defeat it.
1) Why not use the hyperreal numbers and say that the probability of any infinite sequence is infinitesimal? I'm not sure if this would work, but it makes more sense to me than saying that it's 0 and yet could happen. Not sure if this makes too much difference to the argument.
2) You say that it's a fair coin, but you've also given incredibly strong evidence that it's a very unfair coin. What does it actually mean to say that the odds of getting a heads or tails are half, considering the monumental evidence against this? We need to establish what probability really means, which is not a trivial task.
3) I think (2) offers the clue. P(H|F') is (roughly) treated as the prior probability, sticking with the assumption that the coin is fair. P(H|F) is (roughly) treated as the posterior probability, implicitly taking into account the actual results of the coin toss that offer essentially infinite evidence that the coin is very strongly rigged. If we just try to accommodate the evidence that the coin is rigged, the two should come out as equal.
4) So long as it is possible to be perfectly certain of something, and for that something to be incorrect, it is possible to gain infinite information. That is because the information we gain from an event is -log(p), where p is the probability of the event. If we are certain p will never happen, p is 0, in which case the information gained from the event would be infinite.
And actually, if the probability of any infinite sequence of coin tosses is 0, then we find out that an infinite sequence of coin tosses has occurred, even if we don't know the sequence, then we have gained infinite information. The fact that we don't know which sequence it was doesn't matter, we know that an event we assigned p=0 to has occurred, and so our information gained is -log(0)=infinity (well, not really... but it tends to infinity as p tends to 0).
5) This is a boring one but we can't really prove anything from pushing an area of maths beyond its limits and finding that it breaks down.
6) Angels can solve this easily using angel maths :)
Nonstandard (hyperreal) probabilities can't be used to solve this problem, and in general typically don't work when applied to standard structures rather than nonstandard ones.
Ahhh ok. Why is that?
In any case, I think we'd need to either try something similar, or just say the probability of infinite coin tosses is not defined (but that's boring).
Hyperreal summation is really weird/particular in terms of the allowed sets or allowed sequences you can sum over. Only sets with a particular property called "being internal" will work. The natural numbers aren't internal, so you can't assign an infinitesimal probability to (for example) each natural and have the sum over every such number even be defined, much less add up to 1. Things would be even more ill-defined in the case of the space of all infinite coin flips, so you couldn't talk about "the infinitesimal probability of getting a finite number of heads in your infinite sequence of flips;" the set of all outcomes with finitely many heads is not internal.
In fact, to start getting actually interesting internal sets, we have to imagine not flipping a coin an "ordinary" infinite number of times (i.e., a sequence of flips indexed by the naturals) but rather a *specific hyperreal infinite* number of times (i.e., a sequence of flips indexed by all hypernatural numbers from 0 up to N, where N is some specific infinite hypernatural, of which there's a huge variety to pick from, often uncountably many). And that's completely different from the blog post's setup.
Basically, exotically small probabilities only work on exotically large domains, outside of trivial cases where there's only a finite number of things to begin with, in which case hyperreals don't help with much anyway. That said, they *are* useful, but only as a kind of dispensable intermediary where at the very last step of your argument you go back to standard (real) probabilities. For instance, this is *sort of* what Loeb measure does, how the nonstandard approach to constructing Brownian motion works, how nonstandard Ito integrals work, etc.
I love (1)! Very creative idea to do probability with hyperreals -- I wonder what that would lead to.
Regarding (5), I do wonder if Flo's argument is just a dressed up version of the following:(premise) God, as an omnipotent being, should be able to divide by zero.
(premise) Dividing by zero is impossible.
(conclusion) God doesn't exist.
But mathematically Flo's argument is very interesting regardless of what it says about God. I'm wracking my brain to make sense of it. I think your ideas 1-4 are all promising. I also think maybe we need more care in understanding the idea that "probability zero events can happen". Like for any sequence of coin flips r I specify in advance, it is true I won't get r. It is only in retrospect that some sequence r is obtained. Maybe that is somehow relevant.