There is an explanation how a supposed paradox about flipping infinite amounts of coins is not a no-go result for infinidesimal properties if one understand the significance of using cardinals or ordinals (numerosity). The repaired that is that if you append flipping a coin and proceeding only if heads to flipping infinite amount of coins that makes the probability go down. The “broken” take is that one flip added keeps the series the same so those need to have the same probability.
That kind of logic is used in a lot of places. One place is Library of Babel. Have library with books of infinite length that are in lexiographical order using up all the letters at all the positions. A supposed paradox: Take only the books that begin with “A” then remove the leading “A” as it is not needed. you get the “full library again”. This fun is stopped by the books having been of length ω and now being of length ω−1. But if you would use cardinality for size you could be fooled that you arrived exactly where you left off (and with numerocity you don’t). There seems to be diffulty judging the infinite length to be definetely literally ω. But the transfinite amount that the book has should stay constant within the scenario even if the exact value of this variable transfinite amount floats between takes on the scenario.
It seemed that the same pattern is also at work in sun from pea paradox. Express each point of a sphere with infinidesimal rotations from a starting point. Chop it up into “starts with left rotation”, “start with rigth rotation”, “starts with up rotation”, “starts with down rotation” and “do not rotate”.
The group F 2
can be “paradoxically decomposed” as follows: Let S(a) be the set of all non-forbidden strings that start with a and define S(a−1), S(b) and S(b−1) similarly. Clearly,
F 2 = { e } ∪ S ( a ) ∪ S ( a − 1 ) ∪ S ( b ) ∪ S ( b − 1 )
but also
F 2 = a S ( a − 1 ) ∪ S ( a ) ,
and
F 2 = b S ( b − 1 ) ∪ S ( b ) ,
where the notation aS(a−1) means take all the strings in S(a−1) and concatenate them on the left with a.
Like adding a flip before infinite set of flips, adding a rotation before an infinite set of rotations actually gets you somewhere else. The “annihilating rotations” of aa−1 actually shortens it to ω−1.
So there is an infinidesimal error going around (two or four missing points adjacent to the opposite side of the ball?). Which means in the end you get two “almost balls” which is a lot less pressing than getting two exact balls.
But it is a rather technical thing and I can’t actually follow it in the needed detail. In case it ends up being a thing taking the quality of infinity to be the amount of infinity (mixing cardinal and ordinal matters) is the core of it.
Ended up looking at https://www.journals.uchicago.edu/doi/10.1093/bjps/axw013
There is an explanation how a supposed paradox about flipping infinite amounts of coins is not a no-go result for infinidesimal properties if one understand the significance of using cardinals or ordinals (numerosity). The repaired that is that if you append flipping a coin and proceeding only if heads to flipping infinite amount of coins that makes the probability go down. The “broken” take is that one flip added keeps the series the same so those need to have the same probability.
That kind of logic is used in a lot of places. One place is Library of Babel. Have library with books of infinite length that are in lexiographical order using up all the letters at all the positions. A supposed paradox: Take only the books that begin with “A” then remove the leading “A” as it is not needed. you get the “full library again”. This fun is stopped by the books having been of length ω and now being of length ω−1. But if you would use cardinality for size you could be fooled that you arrived exactly where you left off (and with numerocity you don’t). There seems to be diffulty judging the infinite length to be definetely literally ω. But the transfinite amount that the book has should stay constant within the scenario even if the exact value of this variable transfinite amount floats between takes on the scenario.
It seemed that the same pattern is also at work in sun from pea paradox. Express each point of a sphere with infinidesimal rotations from a starting point. Chop it up into “starts with left rotation”, “start with rigth rotation”, “starts with up rotation”, “starts with down rotation” and “do not rotate”.
Like adding a flip before infinite set of flips, adding a rotation before an infinite set of rotations actually gets you somewhere else. The “annihilating rotations” of aa−1 actually shortens it to ω−1.
So there is an infinidesimal error going around (two or four missing points adjacent to the opposite side of the ball?). Which means in the end you get two “almost balls” which is a lot less pressing than getting two exact balls.
But it is a rather technical thing and I can’t actually follow it in the needed detail. In case it ends up being a thing taking the quality of infinity to be the amount of infinity (mixing cardinal and ordinal matters) is the core of it.