User blog comment:P進大好きbot/List of common mistakes on formal logic appearing in googology/@comment-30279966-20200118174447/@comment-35470197-20200119074326

When one talks "X is a comuptable number", then I interprete it as "the defining formula of X associated to its expression is a valid defining formula of a computable number" by the reason above.

In this case, Σ(20) is not an expression which gives a valid defining formula, because it has no information of codes of computable functions. But if we consider another interpretation such as "There is a meta-theoretic number (which is computable in any reasonable formulation) n such that Σ(20) = n is provable under ZFC", then it might be true. On the other hand, Σ(10^100) is computable in neither sense.

If we actually interested in the value of a number, then the boundary of the computability is highly ambiguous. That is why I usually consider the validity of the defining formula given by its expression.