[FRIAM] square land math question

[uǝlƃ ↙↙↙]

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.
> 

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↙↙↙ uǝlƃ