Georg Cantor Quotes

georg_cantor3The final problem which Cantor grappled with was the realization that there could be no set containing everything, since, given any set, there is always a larger set — its set of subsets. Furthermore, he believes that infinity actually exists – it is not just a mathematical construct. 

Cantor came to the conclusion that there is an Absolute Infinite that transcends transfinite numbers. It has mathematical properties, and he identified this concept with God. Subsequently, he believed that his new mathematics is actually a form of theology. 

Here are some quotes by Georg Cantor:

I have never proceeded from any ‘Genus supremum’ of the actual infinite. Quite the contrary, I have rigorously proved that there is absolutely no “Genus supremum’ of the actual infinite. What surpasses all that is finite and transfinite is no ‘Genus’; it is the single, completely individual unity in which everything is included, which includes the Absolute, incomprehensible to the human understanding. This is the Actus Purissimus, which by many is called God.

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.

Quoted in Out of the Mouths of Mathematicians, by R. Schmalz. (He is paraphrasing Leibniz, JB)

A set is a Many that allows itself to be thought of as a One.

Quoted in “Infinity and the Mind” by Rudy Rucker.

The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.

Quoted in “Infinity and the Mind” by Rudy Rucker.

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.

Quoted in “Mind Tools: The Five Levels of Mathematical Reality” by Rudy Rucker.

What I assert and believe to have demonstrated in this and earlier works is that following the finite there is a transfinite (which one could also call the supra-finite), that is an unbounded ascending ladder of definite modes, which by their nature are not finite but infinite, but which just like the finite can be determined by well-defined and distinguishable numbers.

Quoted in Understanding the Infinite by Shaughan Lavine.

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 dissimilar to, and I might even say in principle the same as, my method described above of introducing transfinite 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.

Quoted in “Understanding the Infinite” by Shaughan Lavine.

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