# It would be possible to define an uniform distribution on $\Bbb N$ using infinitesimals?

In standard analysis it is clear that it is impossible to define an uniform probability distribution on $$\Bbb N$$ because there is no constant $$c\in\Bbb R$$ such that $$\sum_{k=1}^\infty c=1$$.

Using Robinson's hypernaturals it doesn't seem feasible either because $$\Bbb N$$ is an external set and so there is no way that $$\Bbb N$$ appear as a set in an internal formula.

Another way to do this could be to try to define an external function of the kind $$f:\{1,\ldots, N\}\to\Bbb N$$ here $$N$$ is an unlimited hypernatural and $$f$$ is surjective and linear. However again this doesn't seem feasible because the structure of the hyperfinite set $$\{1,\ldots, N\}$$ seems very different that the structure of $$\Bbb N$$.

Someone knows some attempt like this, maybe using a different kind of infinitesimals different than those of Robinson's theory?

Another way to say the same: someone knows some attempt to extend the field of reals such that there is some constant $$c\neq 0$$ such that $$\sum_{k=1}^\infty c\in\Bbb R\setminus\{0\}$$?

If $$F$$ is a field with a topology such that addition is separately continuous and infinite sums are defined as limits of partial sums then $$\sum c$$ diverges for every non-zero $$c\in F$$.
Say $$s=\sum c$$ and let $$s_n$$ be the $$n$$th partial sum. Then $$s_{n+1}=s_n+c$$, so continuity of addition shows that $$s=s+c$$, hence $$c=0$$.