So, apparently, 1/ω ≠ 1/(ω+1) in surreal numbers. But if I understand correctly, which is unlikely, we still don't have a definition of integration for surreal numbers. So, I'd hesitate to rely on that as an authority. I now wonder if all infinitesimals have the same size in the hyperreals? And even if they have the same size, are they the *same number*?
In my ignorance, it seems like we have 2 examples with which to form a (perhaps false but useful) dichotomy: https://en.wikipedia.org/wiki/Nonstandard_analysis, where it seems like infinitesimals are distinguishable and https://en.wikipedia.org/wiki/Synthetic_differential_geometry, where they are not (or not all of them ... or ... something). I have a lot of homework to do, I guess. On 7/23/20 10:40 AM, uǝlƃ ↙↙↙ wrote: > Thanks for putting in a little more effort. So, in your definitions, 1/aleph0 > = 1/aleph1. That's tightly analogous, if not identical, to saying a point is > divisible because point/2 = point. But before you claimed a point is > indivisible. So, if you were more clear about which authority you were citing > when you make your claims, we wouldn't have these discussions. > > On 7/23/20 10:35 AM, Frank Wimberly wrote: >> I am aware of the hierarchy of infinities. Aleph0 is the cardinality of the >> integers. Aleph1 is the cardinality of the power set of the integers which >> is the cardinality of the real numbers (that's a theorem which is easy but I >> don't feel like typing it on a cellphone keyboard). Aleph2 is the >> cardinality of the power set of aleph1, etc. >> >> In my definition of 1/infinity, assume infinity means aleph0. But I believe >> it works for any infinite number. That last word is important. > -- ↙↙↙ uǝlƃ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . FRIAM Applied Complexity Group listserv Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/
