So different infinity levels are actually fields which we do not know how far apart they are and thus can not mix them?
That propabilities are always comparable is a pretty used property so taking that away is not trivial at all.
When you “substitute in” standard values in order to get appriciables-only field, then something that was 1−e can drop below 0.9. This change of order seems really disruptive. Althought being able to determine where it happens might be a plus.
1−e and 0+e already exist as “almost surely” and “almost never”. Those that require more than real precision do use it but do not like to express it as numbers. For example a dart is almost surely going to land favoring some side of the target. The only way it doesn’t is the dead center bulleye, but since it is a set of measure 0 (translation of “infinidesimal” of infinity-averse crowd) that is fine.
There is the question of hitting the horizontal or vertical axis of the target. Since lines are in a way lexigraphically smaller than areas, treating areas, lines and points as three levels of infinity might be a bit more expressive of designating that areas get reals and others are 0. I do not know how a infinity-averse person would dance around that. That I know that if you compare areas to areas, lines to lines, and points to points having a single real measure is sufficient. So where the usage case would loom just sectioning the activity to 3 less formal sections allows the formalism within each section brrr in the usual way.
I have been intrigued by surreal probabilities before. Mostly what new things they could provide doesn’t accomplish that much outside of theorethising about itself. There is also the issue that infinidesimal doubt can not be disspelled by finite evidence. And no finite repetition of a infinidesimal chance can result in an appriciable total probability.
So different infinity levels are actually fields which we do not know how far apart they are and thus can not mix them?
That propabilities are always comparable is a pretty used property so taking that away is not trivial at all.
When you “substitute in” standard values in order to get appriciables-only field, then something that was 1−e can drop below 0.9. This change of order seems really disruptive. Althought being able to determine where it happens might be a plus.
1−e and 0+e already exist as “almost surely” and “almost never”. Those that require more than real precision do use it but do not like to express it as numbers. For example a dart is almost surely going to land favoring some side of the target. The only way it doesn’t is the dead center bulleye, but since it is a set of measure 0 (translation of “infinidesimal” of infinity-averse crowd) that is fine.
There is the question of hitting the horizontal or vertical axis of the target. Since lines are in a way lexigraphically smaller than areas, treating areas, lines and points as three levels of infinity might be a bit more expressive of designating that areas get reals and others are 0. I do not know how a infinity-averse person would dance around that. That I know that if you compare areas to areas, lines to lines, and points to points having a single real measure is sufficient. So where the usage case would loom just sectioning the activity to 3 less formal sections allows the formalism within each section brrr in the usual way.
I have been intrigued by surreal probabilities before. Mostly what new things they could provide doesn’t accomplish that much outside of theorethising about itself. There is also the issue that infinidesimal doubt can not be disspelled by finite evidence. And no finite repetition of a infinidesimal chance can result in an appriciable total probability.