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A building of the actual numbers utilizing $log mathbb{P}$ Answer

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A building of the actual numbers utilizing $log mathbb{P}$

Given the clique of prime numbers $mathbb{P}$, it’s identified that the weather of $log mathbb{P}$ are dimensionally unbiased $p_{i neq j}, p_j in mathbb{P}, log p_{i neq j} perp log p_j$ within the sense that:

commence{equation}
exists alpha_i in mathbb{Z}, sum_{i=1}^infty alpha_i log p_i = log p_j iff alpha_j = 1 land alpha_{i neq j} = 0 tag{1}
aim{equation}

Furthermore, if we outline the infinite-dimensional vector area:

commence{equation}
textual content{span}(log mathbb{P}) = Big{sum_{i=1}^infty alpha_i log p_i lvert p_i in mathbb{P}, alpha_i in mathbb{Z}Big} tag{2}
aim{equation}

then it’s identified that:

commence{equation}
log mathbb{Q}_+ subset textual content{span}(log mathbb{P}) tag{3}
aim{equation}

Now, it not too long ago occurred to me that utilizing the Riemann rearrangement theorem we could deduce that:

commence{equation}
forall alpha in mathbb{R} exists lambda_i in mathbb{Q}, alpha = sum_{i=1}^infty lambda_i log p_i tag{*}
aim{equation}

Might this specific building of the actual numbers breathe identified by a specific designation? If so, energy there breathe a sublime proof that does not trust upon the Riemann rearrangement theorem?

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