I am tripping over the notation a little bit. I was representing eps times 1 as (0,1)(1,0) so (eps, 1)(1, 0) and (1, 0)(eps, 1) both evaluate to 2eps in my mind which would make the tail relevant.
If we just have pure lexiographics then it can be undefined whether we can product them together. In my mind I am turning the lexiographic to surreals by using them as weights in a Cantor normal form and then using surreal multiplication.
So in effect I have something like (a,b)(c,d)=(ac,ad+bc,bd).
I guess I know about “approximately equal” when two numbers would round out to the same nearest real number. The picture might also be complicated on whether immense chances exist. That is if you have a coin that has a finite chance to come up with something and I give you more than finite amount of tries to get it there is only a neglible chance of failure. Then ordinarily if a option has no finite chances of paying out it could be ignored but an immense exploration of a “finite-null” coin which has neglible chances of paying out could matter even at the finite level. And I guess on the other direction are the pascal wagers. Neglible chances of immense rewards. So there are sources other than aligning the finite multipliers to get effects that matter on the finite level.
I am tripping over the notation a little bit. I was representing eps times 1 as (0,1)(1,0) so (eps, 1)(1, 0) and (1, 0)(eps, 1) both evaluate to 2eps in my mind which would make the tail relevant.
Surely this would be your representation for infinitesimals, not for eps? By eps, I mean epsilon in the sense of standard real numbers, i.e. a small but positive real number. The issue is that in the real world, there will always be complexity, noise and uncertainty that will introduce small-but-positive real gaps, and these gaps will outweigh any infinitesimal preference or probability that you might have.
ah, I think I am starting to follow. It is a bit ambigious whether it is supposed to be two instances of one arbitraliy small finite or two (perhaps different) arbitrarily small finites. If it is only one then the tails are again relevant. “Always” is a bit risky word especially in connection with infinite.
I guess the basic situation is that modelling infinidesimal chances has not proven to be handy. But I don’t see that task would be shown to neccesarily frustrate. One could assume that while in theory one could model something in lexiographic way in reality there is some exchange rate between the “lanes” and in that way the blurryness could aid instead of hinder in applicability. Somebody that really likes real-only probablities could insist that unifying the preferences should be done early but there might be benefits in doing it late.
ah, I think I am starting to follow. It is a bit ambigious whether it is supposed to be two instances of one arbitraliy small finite or two (perhaps different) arbitrarily small finites. If it is only one then the tails are again relevant.
I don’t see what you mean. It doesn’t make a different whether you have only one or two small-but-finite quantities (though usually you do have on for each quantity you are dealing with), as long as they are in general position. For instance, in my modification to the example you gave, I only used one: While it’s true that the tail becomes relevant in (0, 1)(1, 0) vs (1, 0)(0, 1) because 0*1=1*0, it is not true that the tail becomes relevant in the slightly modified (eps, 1)(1, 0) vs (1, 0)(0, 1) for any real eps != 0, as eps*1 != 1*0.
So the only case where infinitesimal preferences are relevant are in astronomically unlikely situations where the head of the comparison exactly cancels out.
I thought we were comparing (eps, 1)(1, 0) and (1, 0)(eps, 1). if eps=eps strict equality. if it was (a,1)(1,0) and (1,0)(b,1) and its possible that a!=b it is unsure whether there is equality.
We were comparing epsilon to no-epsilon (what I had in mind with my post).
Anyway, the point is that strict equality would require astronomical consequences, and so only be measure 0. So outside of toy examples it would be a waste to consider lexicographic preferences or probabilities.
I am tripping over the notation a little bit. I was representing eps times 1 as (0,1)(1,0) so (eps, 1)(1, 0) and (1, 0)(eps, 1) both evaluate to 2eps in my mind which would make the tail relevant.
If we just have pure lexiographics then it can be undefined whether we can product them together. In my mind I am turning the lexiographic to surreals by using them as weights in a Cantor normal form and then using surreal multiplication.
So in effect I have something like (a,b)(c,d)=(ac,ad+bc,bd).
I guess I know about “approximately equal” when two numbers would round out to the same nearest real number. The picture might also be complicated on whether immense chances exist. That is if you have a coin that has a finite chance to come up with something and I give you more than finite amount of tries to get it there is only a neglible chance of failure. Then ordinarily if a option has no finite chances of paying out it could be ignored but an immense exploration of a “finite-null” coin which has neglible chances of paying out could matter even at the finite level. And I guess on the other direction are the pascal wagers. Neglible chances of immense rewards. So there are sources other than aligning the finite multipliers to get effects that matter on the finite level.
Surely this would be your representation for infinitesimals, not for eps? By eps, I mean epsilon in the sense of standard real numbers, i.e. a small but positive real number. The issue is that in the real world, there will always be complexity, noise and uncertainty that will introduce small-but-positive real gaps, and these gaps will outweigh any infinitesimal preference or probability that you might have.
ah, I think I am starting to follow. It is a bit ambigious whether it is supposed to be two instances of one arbitraliy small finite or two (perhaps different) arbitrarily small finites. If it is only one then the tails are again relevant. “Always” is a bit risky word especially in connection with infinite.
I guess the basic situation is that modelling infinidesimal chances has not proven to be handy. But I don’t see that task would be shown to neccesarily frustrate. One could assume that while in theory one could model something in lexiographic way in reality there is some exchange rate between the “lanes” and in that way the blurryness could aid instead of hinder in applicability. Somebody that really likes real-only probablities could insist that unifying the preferences should be done early but there might be benefits in doing it late.
I don’t see what you mean. It doesn’t make a different whether you have only one or two small-but-finite quantities (though usually you do have on for each quantity you are dealing with), as long as they are in general position. For instance, in my modification to the example you gave, I only used one: While it’s true that the tail becomes relevant in (0, 1)(1, 0) vs (1, 0)(0, 1) because 0*1=1*0, it is not true that the tail becomes relevant in the slightly modified (eps, 1)(1, 0) vs (1, 0)(0, 1) for any real eps != 0, as eps*1 != 1*0.
So the only case where infinitesimal preferences are relevant are in astronomically unlikely situations where the head of the comparison exactly cancels out.
I thought we were comparing (eps, 1)(1, 0) and (1, 0)(eps, 1). if eps=eps strict equality. if it was (a,1)(1,0) and (1,0)(b,1) and its possible that a!=b it is unsure whether there is equality.
yeah (eps,0) behaves differently from (0,1)
We were comparing epsilon to no-epsilon (what I had in mind with my post).
Anyway, the point is that strict equality would require astronomical consequences, and so only be measure 0. So outside of toy examples it would be a waste to consider lexicographic preferences or probabilities.