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. Thought Set Many Itself. This view [of the infinite], which I consider to be the sole correct one, is held by only a few. 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. "Journey Through Genius". The essence of mathematics lies precisely in its freedom. "Infinity and the Mind". 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. A false conclusion once arrived at and widely accepted is not easily dislodged and the less it is understood the more tenaciously it is held. 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. While possibly I am the very first in history to take this position so explicitly, with all of its logical consequences, I know for sure that I shall not be the last! Georg Cantor. Unfortunately, however, this "axiom" is used innumerably often without any basis and in neglect of the necessary distinction between "reality" and "quantity", on the one hand, and "number" and "set", on the other, precisely in the sense in which it is generally false. While possibly I am the very first in history to take this position so explicitly, with all of its logical consequences, I know for sure that I shall not be the last! 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. Despite the criticism he faced, he continued publishing papers and stated his ‘diagonal argument’ and proved the existence of uncountable set. Soon after that he started working on trigonometric series by Bernhard Riemann, a German mathematician and his work later came to be known as the set theory. My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. Don't always blindly follow guidance and step-by-step instructions; you might run into something interesting. In Mathematics the art of proposing a question must be held of higher value than solving it. Cantor was born into a family of musicians and even had slight interest in music during his early age but that later changed. A set is a Many that allows itself to be thought of as a One. The essence of mathematics lies entirely in its freedom. 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. A set is a Many that allows itself to be thought of as a One. 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. “In Mathematics the art of proposing a question must be held of higher value than solving it.” ― … There is no doubt that we cannot do without variable quantities in the sense of the potential infinite. Georg Cantor quotes The totality of all alephs cannot be conceived as a determinate, well-defined, and also a finished set. The essence of mathematics lies in its freedom. "From Kant to Hilbert: A Source Book in the Foundations of Mathematics". Book by Georg Cantor, 1886. The essence of mathematics lies precisely in its freedom. Book by Rudy Rucker, 1987. Unfortunately, however, this "axiom" is used innumerably often without any basis and in neglect of the necessary distinction between "reality" and "quantity", on the one hand, and "number" and "set", on the other, precisely in the sense in which it is generally false. There is no doubt that we cannot do without variable quantities in the sense of the potential infinite. And you can get it if you try. A collection of Georg Cantor Quotes about mathematics, logic, power, conclusion, innovation, people, art, mind, matter, beauty, education, development, numbers, facts, discovery etc. Georg Cantor — As quoted in Journey Through Genius (1990) by William Dunham Tags: theory , stands , firm , rock , arrow , directed , against , return , quickly Mathematics, in the development of its ideas, has only to take account of the immanent reality of its concepts and has absolutely no obligation to examine their transient reality.

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