User blog comment:MilkyWay90/Math number/@comment-35392788-20180902214346/@comment-35470197-20180906033401

In order to construct a computable large number, we need to give a step-by-step method to compute it.

For example, "add all numbers definable by formulae in a given finite set" gives an uncomputable large number, because there is no step-by-step way to check whther a given formular consisting of 10^100 or less letters defines a natural number or not. (This is related to Busy Beaver function and Rayo's function.)

On the other hand, "add all numbers definable by formulae F in a formula in a given finite set such that there is a proof of length smaller than or equal to 10^100 that F defines a natural number" gives a computable large number, because there is a step-by-step way to check whether a finite seuqence of formulae is a proof or not. (This is related to Verification/Falsification game and the least transcendental integer.)