What is the solution to prime sum problem, is it a surreal number ?
If not, then what is it ?
Let there be a set of natural numbers : $$1, 2, 3, 4, 5, 6, 7, 8, 9, 10, \dots$$
Half of this set are divided by firs prime : $$ \frac{1}{2} $$
Considering unique divisions by second prime : $$ \frac{1}{2} + \left(\frac{1}{3} - \frac{1}{2\cdot3}\right) $$
Considering unique division by third prime : $$ \frac{1}{2} + \left(\frac{1}{3} - \frac{1}{2\cdot3}\right) + \left(\frac{1}{5}-\frac{1}{2\cdot5}-\frac{1}{3\cdot5}+\frac{1}{2\cdot3\cdot5}\right) $$
And so on $$ \dots $$
This problem can be written as : $$ S = \frac{1}{p_1}\!+\!\frac{1}{p_2}\left(1\!-\!\frac{1}{p_1}\right)\!+\! \frac{1}{3}\left(1\!-\!\frac{1}{p_2}\right)\left(1\!-\!\frac{1}{p_1}\right)\!+\!\ldots\!=\!1\!-\!\delta $$
This sum follows to $1$ but it never be, because we never take in account $1$ from this set.
What value then $\delta$ have, is it infinitely small ?