The standard way to define the sum of an uncountable number of reals is to set $$ \sum_{\alpha \in A} x_{\alpha} = \sup \left(\left\{ \sum_{\alpha \in S} x_{\alpha} ~|~ S \subset A ~\text{and}~ |S| ~\text{finite} \right\}\right) .$$
I feel uneasy doing this, as we are using finite sums to approximate an uncountable sum, meaning we skipped a whole cardinality.
To this end, would it be unreasonable to set $$ \sum_{\alpha \in A} x_{\alpha} = \sup \left(\left\{ \sum_{i = 0}^\infty x_{n_i} ~|~ n_i \in S ~\text{are all distinct and the sum is convergent} \right\}\right) ?$$