I entertain no doubts as to the truths of the tranfinites, which I recognized with God's help and which, in their diversity, I have studied for more than twenty years; every year, and almost every day brings me further in this science.

My beautiful proof lies all in ruins.

The transfinite numbers are in a certain sense themselves new irrationalities and in fact in my opinion the best method of defining the finite irrational numbers is wholly disimilar to, and I might even say in priciple the same as, my method described above of introducing trasfinite numbers. One can say unconditionally: the transfinite numbers stand or fall with the finite irrational numbers; they are like each other in their innermost being; for the former like the latter are definite delimited forms or modifications of the actual infinite.

I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers.

There is no doubt that we cannot do without variable quantities in the sense of the potential infinite. But from this very fact the necessity of the actual infinite can be demonstrated.

The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type.

I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author. Thus I believe that there is no part of matter which is not - I do not say divisible - but actually divisible; and consequently the least particle ought to be considered as a world full of an infinity of different creatures.