limited, the sizes of the former would be smaller, even though both collections are infinite,” it would be clear that this counterargument is more balanced and more deserving of acceptance than the other.

The second point:

If numbers, their successive levels, and all other magnitudes are doubled without limit, this doubling occurs only in the mind. Everything the mind conceives and supposes in this way is finite, and the mind continues doubling until it can no longer do so. Likewise, when someone utters the names of numbers or other words, he continues speaking until he can no longer do so. Even if we suppose that he never becomes incapable, but that the mind continues conceiving and the tongue continues speaking, all of this belongs to existence in the mind and in words, inwardly and on the tongue. Whatever of this comes into existence is finite and has a definite beginning: a starting point in a person's thought and speech. Whatever of it comes into being successively constitutes its endpoint. This, however, demonstrates the possibility of something continuing endlessly into the future, and that something may have a beginning without having an end. For whatever occurs to the mind and is uttered by the tongue has a beginning, yet it is possible for it to continue without end. The breaths, words, and movements of the people of Paradise fall into this category: they are inspired to glorify Allah just as they are inspired to breathe. The angels' perpetual glorification also falls into this category.

Thus, this doubling of numbers in thought and speech demonstrates the existence of something that continues endlessly into the future while having a definite beginning. As for differences in magnitude within it—whether this means doubling in the mind, on the tongue, or both—it is clear that, if one says:

“Double one, double that result, double that result again, and so on,”