Starting with a transcendental number like $e$, it is known that we can write it as a sum of infinitely many rational numbers of the form :

$e= \sum \limits _{n=0}^{\infty }{\dfrac {1}{n!}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots $

we can see in this form each element here is defined and is related to other elements by a known order.

a question comes to mind here about existance of a sequence of positive rational numbers $\{\omega _n\}\subset\mathbb{Q}^+$ satisfying conditions below:

  1. $ e= \sum \limits _{n=1}^{\infty }\omega_n$

  2. each element $\omega_i$ is not predefined (means it is unknown to us, we just know it is positive rational number).

  3. there be no known order or trend (except the well ordering assortment) between elements of the sequence $\{\omega _n\}$ (means there is no defined relation for example like $\sim$ between elements of any subset of $\{\omega _n\}$).

  4. there be no defined algorithm by which $\{\omega _n\}$ be constructed from $\{\dfrac {1}{n!}\}$ or any other known sequence with sum of $e$.

    note: here conditions 3 and 4 may correlate eachother but I mentioned both for more accurate explanation.

It seems here there are some feelings of ambiguity around how this question is phrased as it was put on hold in mathexchange but also may seems some fundamental issues with it, because of this I think we should first investigate whether this question is undecidable or not under the ZFC axiomatic system but I don't know how?

As two answer to this question here in mathexchange was in a way that represents a few knowledge of their authors about such these kind of problems ,I've decided to also present it in research level community in order to find some knowledge or at least some source related to the topic,thanks.

  • $\begingroup$ You can choose $\omega_1,\omega_3,\omega_5,\dots$ arbitrarily, so long as they approach zero, so there's an uncountable infinity of sequences satisfying all your constraints. $\endgroup$ – Gerry Myerson Nov 18 '18 at 21:26
  • $\begingroup$ @GerryMyerson sorry sir, I didn't get your point ,would you please explain it more? $\endgroup$ – MasM Nov 18 '18 at 22:22
  • $\begingroup$ No. The question doesn't belong here, so I have said all I'm going to say about it. $\endgroup$ – Gerry Myerson Nov 19 '18 at 3:05
  • $\begingroup$ @GerryMyerson at least would you please say where should I ask my question sir? $\endgroup$ – MasM Nov 19 '18 at 13:29
  • $\begingroup$ @MasM , Mathstackexhange would be more appropriate. $\endgroup$ – JoshuaZ Nov 19 '18 at 17:40