Timeline for Why surreal numbers cannot be extended further in this way using measure approach?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 17, 2023 at 10:06 | vote | accept | Anixx | ||
| Sep 15, 2023 at 17:13 | comment | added | Anixx | This is why it is pity: mathoverflow.net/questions/454646/… | |
| Apr 23, 2023 at 23:43 | comment | added | Anixx | One thing that bothers me is that whatever means we use to extend the numbers with such quantities, we go beyond surreal numbers. And surreal numbers are the maximal ordered field... How is it possible? Also, how to incorporate a) countable but dense sets, b) uncountable sets with dimension 0? | |
| Apr 23, 2023 at 22:41 | comment | added | Anixx | Cantor's cardinality is translation-invariant. Numerocity, on the other hand, is not. Volume of an infinite body (expressed as numerocity of unit volumes) changes as we shift it. I think, the both measures have their own pros but numerocity is definitely more precise. | |
| Apr 23, 2023 at 22:38 | comment | added | Joel David Hamkins | You call it a pity, but to me it appears to be an extremely illuminating and enormously successful theory. Numerosity, of course, is simply a bit of vocabulary meant to suggest "number", but the fact remains that it violates the Cantor-Hume principle. Galileo had thrown up his hands in confusion at the conflict between Euclid's principle and Cantor-Hume, opting ultimately to keep Euclid. It seems numerosity is motivated perhaps by the same idea. | |
| Apr 23, 2023 at 22:28 | comment | added | Anixx | If we talk about the non-dense subsets of reals, we can equate the numerocity of the set of roots of a function $f(x)$ the following numerocity: $\int_{-\infty}^\infty \delta(f(x))|f'(x)| dx$. This means, we even can write down the expression for numerocity of any subset of reals as an integral, but the exact meaning of such integral would need clarification. | |
| Apr 23, 2023 at 22:15 | comment | added | Anixx | "Meanwhile, one can consider various nonstandard counting measures, which provide a nonstandard measure similar to what you mention." - this is exactly what I was interested in, because it seems to me that measure theory may be powerful enough to handle non-countable numerocities. | |
| Apr 23, 2023 at 22:13 | comment | added | Anixx | "which will mean for many people that you are not speaking of "the number of" any longer" - this is pity that people equate cantorian cardinality to the "number of elements". | |
| Apr 23, 2023 at 22:11 | comment | added | Anixx | Thanks for pointing out at the Hume principle, but my question is not about cardinality but rather about numerocity (Lebesque measure of dimension $0$) or, about generalization of Lebesgue measures in general. The concept of numerocity is discussed for instance, here: mathenchant.wordpress.com/2018/09/16/a-new-game-with-infinity The numerocities of non-dense subsets of reals are quite easy: they can be equated to divergent series, thus to Hardy fields elements or by surreal numbers. The mistery appears when we want to represent numerocities of dense sets, and especially of continuum. | |
| Apr 23, 2023 at 18:01 | history | answered | Joel David Hamkins | CC BY-SA 4.0 |