Some of the logic uses that because a chance is positive then it must be finite. In the infinite context infinidesimal chances might be relevant and can break this principle. For expected value calculations it helps that for any transfinite payoff there is a corresponding infinidesimal chance which would make that option ambivalent with a certain finite payout. And for example 4 times the lizards this threshold would be 4 times lower. Mere possiblity giving a finite (even if small) chance seems overgenerous althought I would expected the theory on what kind of evidentiary or reasoning processes can legitimately output infinidesimals seems hard and beyond me atm. But it leaves the door open for me to say “just because its possible doesn’t mean its possible enough”.
In general thinking with surreal numbers has really hammered down to me that “infinite” is an adjective like “finite” and “infinity—inifinity” is undefined for similar reasons that “finite—finite” is undefined. A lot this analysis would go differently if we kept the notion that we need to be asking “how much” even for the transfinite amounts.
(For hyperreals/surreals, I think “finite” means “smaller magnitude than some real,” and so where you say “finite” in your first paragraph I think you mean positive-standard-part, viz., positive and not infinitesimal.)
I’m sympathetic to this point, and I think it’s deep, but I’m pretty sure that it ultimately doesn’t make a difference. We should have positive-standard-part credence in the possibility of infinite utility. We should also have positive-standard-part credence in various subpossibilities. And so they can dominate our expected value calculations. It doesn’t all cancel out (unless you stipulate that it does, which seems to be an incredibly strong empirical assumption that I can’t imagine justifying). I strongly disagree with “for any transfinite payoff there is a corresponding infinidesimal chance which would make that option ambivalent”; this is an empirical claim that we have no reason to believe.
Depending on definitions, you could say that there’s an infinitesimal chance of any particular possible future, but you can’t say that there’s an infinitesimal chance of a class of possible futures such as (some precisification of) the possible futures with exactly one immortal happy lizard.
yes, I mean the primary archimedean field with “finite”. The fields go also “inside” in that 1/ω is in one that 1/ω2 is not in.
I mean the ambivalence point to be a mathematical statement rather than an empirical one about how expected value and infinites work. That is formulas of the form ω∗x=1 have a solution with x=1/ω and that w∗x<a,a=n∗r,n∈N,r∈R can be made true
The empirical part would involve a situation where the correct credence about it was truly infinidesimal. Currently we just designate the nearest real and call it a day. I have a suspicion that events that can happen but have probability 0 would benefit from going beyond real precision.
One can imagine dart competition. $20 for hitting the upper half of the board. $40 for hitting the upper and left half of the board. These two “challenges” or bets would be comparable in expected money if the dart lands uniformly on the board. How much money would be fair for landing squarely on the bullseye? For any small radius around the bullseye such a value is somewhat straighforward to get, yet any finite amount of money is not enough for the center itself. Yet presumably the center is not special and is as likely as any other. So if we are restricted to having a finite money payoff we can’t make an actually arbitrary target but must include infinitely many points to make up the target area. Even the vertical symmetry axis of the board is too narrow, a dart will almost surely that is to say with real probablity 1 land left or right on the board. It is not true that an infinite multiple of an infinidesimal needs to be finite. The vertical axis has more than finite multiples of points compared to the bullseye but it still fails to make a finite area for which a finite money payoff would be appropriate.
I don’t see how probablities for classes of futures neccesiates non-neglible real probabilities. Usually we want to be atleast real smooth but it is not clear to me that real smoothnes is good enough for all cases. This might be connected to the fact that if you integrate the probability density whether you are guaranteed to get a real or might you get something less than a real.
Some of the logic uses that because a chance is positive then it must be finite. In the infinite context infinidesimal chances might be relevant and can break this principle. For expected value calculations it helps that for any transfinite payoff there is a corresponding infinidesimal chance which would make that option ambivalent with a certain finite payout. And for example 4 times the lizards this threshold would be 4 times lower. Mere possiblity giving a finite (even if small) chance seems overgenerous althought I would expected the theory on what kind of evidentiary or reasoning processes can legitimately output infinidesimals seems hard and beyond me atm. But it leaves the door open for me to say “just because its possible doesn’t mean its possible enough”.
In general thinking with surreal numbers has really hammered down to me that “infinite” is an adjective like “finite” and “infinity—inifinity” is undefined for similar reasons that “finite—finite” is undefined. A lot this analysis would go differently if we kept the notion that we need to be asking “how much” even for the transfinite amounts.
(For hyperreals/surreals, I think “finite” means “smaller magnitude than some real,” and so where you say “finite” in your first paragraph I think you mean positive-standard-part, viz., positive and not infinitesimal.)
I’m sympathetic to this point, and I think it’s deep, but I’m pretty sure that it ultimately doesn’t make a difference. We should have positive-standard-part credence in the possibility of infinite utility. We should also have positive-standard-part credence in various subpossibilities. And so they can dominate our expected value calculations. It doesn’t all cancel out (unless you stipulate that it does, which seems to be an incredibly strong empirical assumption that I can’t imagine justifying). I strongly disagree with “for any transfinite payoff there is a corresponding infinidesimal chance which would make that option ambivalent”; this is an empirical claim that we have no reason to believe.
Depending on definitions, you could say that there’s an infinitesimal chance of any particular possible future, but you can’t say that there’s an infinitesimal chance of a class of possible futures such as (some precisification of) the possible futures with exactly one immortal happy lizard.
yes, I mean the primary archimedean field with “finite”. The fields go also “inside” in that 1/ω is in one that 1/ω2 is not in.
I mean the ambivalence point to be a mathematical statement rather than an empirical one about how expected value and infinites work. That is formulas of the form ω∗x=1 have a solution with x=1/ω and that w∗x<a,a=n∗r,n∈N,r∈R can be made true
The empirical part would involve a situation where the correct credence about it was truly infinidesimal. Currently we just designate the nearest real and call it a day. I have a suspicion that events that can happen but have probability 0 would benefit from going beyond real precision.
One can imagine dart competition. $20 for hitting the upper half of the board. $40 for hitting the upper and left half of the board. These two “challenges” or bets would be comparable in expected money if the dart lands uniformly on the board. How much money would be fair for landing squarely on the bullseye? For any small radius around the bullseye such a value is somewhat straighforward to get, yet any finite amount of money is not enough for the center itself. Yet presumably the center is not special and is as likely as any other. So if we are restricted to having a finite money payoff we can’t make an actually arbitrary target but must include infinitely many points to make up the target area. Even the vertical symmetry axis of the board is too narrow, a dart will almost surely that is to say with real probablity 1 land left or right on the board. It is not true that an infinite multiple of an infinidesimal needs to be finite. The vertical axis has more than finite multiples of points compared to the bullseye but it still fails to make a finite area for which a finite money payoff would be appropriate.
I don’t see how probablities for classes of futures neccesiates non-neglible real probabilities. Usually we want to be atleast real smooth but it is not clear to me that real smoothnes is good enough for all cases. This might be connected to the fact that if you integrate the probability density whether you are guaranteed to get a real or might you get something less than a real.