Slider comments on Impossibility results for unbounded utilities

[Slider]

I am having trouble trying to translate between infinity-hiding style and explicit infinity style. My grievance with might be stupid.

$\frac{1}{2}{X}_0$

split X_0 into equal number parts to final form

$\frac{1}{2}(ϵX_0+ϵX_0+ϵX_0+...)$

move the scalar in

$\frac{1}{4}ϵX_0+\frac{1}{8}ϵX_0+\frac{1}{16}ϵX_0+...$

combine scalars

$\frac{ϵ}{4}{X}_0+\frac{ϵ}{8}{X}_0+\frac{ϵ}{16}{X}_0+...$

Take each of these separately to the rest of the original terms

$(\frac{ϵ}{4}{X}_0+\frac{1}{4}{X}_1)+(\frac{ϵ}{8}{X}_0+\frac{1}{8}{X}_2)+(\frac{ϵ}{16}{X}_0+\frac{1}{16})+...$

Combine scalars to try to hit closest to the target form

$\frac{1}{2}(\frac{ϵ}{2}{X}_0+\frac{1}{2}{X}_1)+\frac{1}{4}(\frac{ϵ}{2}{X}_0+\frac{1}{2}{X}_1)+\frac{1}{8}(\frac{ϵ}{2}{X}_0+\frac{1}{2}{X}_1)+...$

$\frac{ϵ}{2}{X}_0+\frac{1}{2}{X}_1$is then quite far from $\frac{1}{2}{X}_0+\frac{1}{2}{X}_1$

Within real precision a single term hasn’t moved much $\frac{ϵ}{2}{X}_0+\frac{1}{2}{X}_1\sim \frac{1}{2}{X}_1$

This suggests to me that somewhere there are “levels of calibration” that are mixing levels corresponding to members of different archimedean fields trying to intermingle here. Normally if one is allergic to infinity levels there are ways to dance around it /​ think about it in different terms. But I am not efficient in translating between them.

[Slider]

New attempt

$X_{\infty }=\frac{1}{2}{X}_0+\frac{1}{4}{X}_1+\frac{1}{8}{X}_2+\frac{1}{16}{X}_4+\dots$

I think I now agree that $X_0$ can be written as $\frac{1}{2}{X}_0+\frac{1}{4}{X}_0+\frac{1}{8}{X}_0...$

However this uses a “de novo” indexing and gets only to

$\frac{1}{2}$ ($\frac{1}{2}{X}_0+\frac{1}{4}{X}_0+\frac{1}{8}{X}_0...$)$+\frac{1}{4}{X}_1+\frac{1}{8}{X}_2+\frac{1}{16}{X}_4+\dots$

taking terms out form the inner thing crosses term lines for the outer summation which counts as “messing with indexing” in my intuition. The suspect move just maps them out one to one

$(\frac{1}{4}{X}_0+\frac{1}{4}{X}_1)+(\frac{1}{8}{X}_0+\frac{1}{8}{X}_2)+(\frac{1}{16}{X}_0+\frac{1}{16}{X}_4)+...$

But why is this the permitted way and could I jam the terms differently in say apply to every other term

$(\frac{1}{4}{X}_0+\frac{1}{4}{X}_1)+\left(\frac{1}{8}{X}_2\right)+(\frac{1}{8}{X}_0+\frac{1}{16}{X}_4)+\frac{1}{32}{X}_8+(\frac{1}{16}{X}_0+\frac{1}{64}{X}_{1}6)+...$

If I have $\left(\sum _{i=0}^a{x}_i\right)+\left(\sum _{j=0}^a{y}_j\right)$ I am more confident that they “index at the same rate” to make $\sum _{u=0}^c{x}_u+y_u$. However if I have $\left(\sum _i^a{x}_i\right)+\left(\sum _j^b{y}_j\right)$ I need more information about the relation of a and b to make sure that mixing them plays nicely. Say in the case of b=2a then it is not okay to think only of the terms when mixing.