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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 $\sum_{k=1}^\infty c\in\Bbb R\setminus\{0\}$?

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Of course we need to specify what properties the extension should have. The answer is no under fairly weak assumptions:

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 $c=0$.

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