Book by Georg Cantor, 1888. Book by Georg Cantor, 1883. The essence of mathematics lies precisely in its freedom. The pure mathematician knows that pure mathematics has an end in itself which is more allied with philosophy. Improve yourself, find your inspiration, share with friends. "Foundations of a General Theory of Aggregates". 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. 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. 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. 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. The old and oft-repeated proposition "Totum est majus sua parte" [the whole is larger than the part] may be applied without proof only in the case of entities that are based upon whole and part; then and only then is it an undeniable consequence of the concepts "totum" and "pars". His theory today is known as the ‘Cantor’s Theorem’. 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. "Out of the Mouths of Mathematicians: A Quotation Book for Philomaths". 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. 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. Great innovation only happens when people aren't afraid to do things differently. Every transfinite consistent multiplicity, that is, every transfinite set, must have a definite aleph as its cardinal number. 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! "Modern Mathematicians". Source. 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. The old and oft-repeated proposition "Totum est majus sua parte" [the whole is larger than the part] may be applied without proof only in the case of entities that are based upon whole and part; then and only then is it an undeniable consequence of the concepts "totum" and "pars". To ask the right question is harder than to answer it. A set is a Many that allows itself to be thought of as a One. Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed too great a demand! There is no doubt that we cannot do without variable quantities in the sense of the potential infinite. The transfinite numbers are in a sense the new irrationalities [ ... they] stand or fall with the finite irrational numbers. "'Over the different views with regard to the actual infinite numbers' ('Ueber die verschiedenen Ansichten in Bezug auf die actualunendlichen Zahlen')". In order for there to be a variable quantity in some mathematical study, the domain of its variability must strictly speaking be known beforehand through a definition. 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! Infinity, in its first form (the improper-infinite) presents itself as a variable finite [veranderliches Endliches]; in the other form (which I call the proper infinite [Eigentlich-unendliche]) it appears as a thoroughly determinate [bestimmtes] infinite. Every transfinite consistent multiplicity, that is, every transfinite set, must have a definite aleph as its cardinal number. 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. The potential infinite means nothing other than an undetermined, variable quantity, always remaining finite, which has to assume values that either become smaller than any finite limit no matter how small, or greater than any finite limit no matter how great. 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. 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. A set is a Many that allows itself to be thought of as a One. Don't always blindly follow guidance and step-by-step instructions; you might run into something interesting. Georg Cantor. Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all because I have followed its roots, so to speak, to the first infallible cause of all created things.

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