Timeline for Literature that helps explain what the theory of numerosities contributes with
Current License: CC BY-SA 4.0
18 events
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| Jul 5 at 0:12 | comment | added | Anixx | @MikeBattaglia When we are talking about uncountable numerosities, the finite part is the Euler's characteristic: math.stackexchange.com/questions/4934215/… | |
| Jul 5 at 0:11 | comment | added | Anixx | @MikeBattaglia definitely. The basic idea behind numerosities is Euclid's principle, but I think, the construction via ultrafilters is suboptimal. If we talk about subsets of naturals, then numerosity is this (generally, divergent) sum: $\sum_{k=1}^\infty p_S(k)$, sum of the indicator function over the whole set. And this sum has regularized value, which is equal to the finite part of numerosity. Here is my recent answer to the James Propp's question, compring his and my methods: mathoverflow.net/questions/248994/… | |
| Jul 5 at 0:04 | comment | added | Mike Battaglia | Thanks for explaining - are these related in some way to the other things called "numerosities?" | |
| Jun 19 at 0:20 | comment | added | Anixx | @MikeBattaglia My numerosities have finite part equal to the regularized sum of the indicator function and it can be easily calculated by the James Propp's method: mathenchant.wordpress.com/2018/09/16/a-new-game-with-infinity I recall now that when we discussed this with James Propp, he gave me a link to the paper you inked and we both agreed that the difference between these two numbers is 1/2 and not 1 or 0. I consider this discrepancy a bug or mistake in their paper as this contradicts their own axioms. | |
| Jun 19 at 0:20 | comment | added | Anixx | @MikeBattaglia their page 63 says that their system has arbitrariness and depending on ultrafilter, $num(odd)=num(even)$ or $num(odd)=num(even)+1$ where $even$ and $odd$ are positive even and odd integers. In my system there is no such arbitrariness, $N(even)=\omega/2-1/2$ and $N(odd)=\omega/2$, so their difference is $1/2$, which is logical, because the first one is $N(\mathbb Z)/4$ and the later satisfies $x+(x-1)=N(\mathbb Z)/2$. | |
| Jun 19 at 0:20 | comment | added | Anixx | @MikeBattaglia philosohically the both papers have the same vibes as my numerisity and the both cite Euclid's principle. It seems the first paper just uses partial sums as you suggested. I also notice that they denote N the non-negative integers and introduce consant α which is equivalent to my ω+1/2. Some principles behind my ideas I explained here: philosophy.stackexchange.com/a/112762/796 | |
| Jun 18 at 22:04 | comment | added | Mike Battaglia | Aren't you talking about these numerosities? And these ones? If so, the thing I am saying is explained pretty simply here - plato.stanford.edu/entries/infinity/numerosities.html | |
| Jun 18 at 16:15 | comment | added | Anixx | @MikeBattaglia I have no idea about what those ultrafilters are. This concept is not needed. | |
| Jun 18 at 15:59 | comment | added | Mike Battaglia | I thought these numerosities were equivalent to the existence of some kind of special ultrafilter? | |
| Jun 18 at 5:55 | comment | added | Anixx | @MikeBattaglia hyperreals have nothing to do with this at all. Hardy fields have canonical embedding into surreals, which we use. Hyperreals do not. I do not know what they are useful for at all. | |
| Jun 18 at 5:52 | comment | added | Anixx |
@MikeBattaglia not quite. The partial sum is discrete, so, first we need to remove discreteness. Implicitely this is done by Newton interpolation (hidden in Sum operator). The opposite process can be done with this code: S = Log[\[Omega]]; DifferenceDelta[Integrate[Normal[SolveValues[S == k, \[Omega]]], k], k] /. C[1] -> 0 //Last // FullSimplify // Expand. In this case we represent surreal number as a germ, then the germ as divergent integral, then divide that integral into segments of area 1, and the centers of mass of these segments are the sequence of desired numerosity.
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| Jun 18 at 2:56 | comment | added | Mike Battaglia | That's very interesting. Is this the same as just taking the partial sum of the indicator function of your set, and then evaluating it at the nonstandard hypernatural $(0, 1, 2, 3, ...)$? | |
| Jun 7 at 23:32 | history | edited | Anixx | CC BY-SA 4.0 |
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| Jun 7 at 21:34 | history | edited | Anixx | CC BY-SA 4.0 |
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| Jun 7 at 21:21 | history | answered | Anixx | CC BY-SA 4.0 |