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Georg Cantor Biography

Georg Ferdinand Ludwig Philipp Cantor (/ˈkæntɔːr/ KAN-tor; German: [ˈɡeɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfɪlɪp ˈkantɔʁ]; March 3 [O.S. February 19] 1845 – January 6, 1918) was a German mathematician. He invented set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. In fact, Cantor's method of proof of this theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.

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Georg Cantor Quotes

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.

Finite, Limit, Matter, Mean, Smaller, Variables

The transfinite numbers are in a sense the new irrationalities [. . . they] stand or fall with the finite irrational numbers.

Fall, Irrational, Irrational Numbers, Mathematics, Sense

In mathematics, the art of asking questions is more valuable than solving problems.

Art, Asking, Education, Math, Mathematics, Problems

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.

Author, Faith, God, Hold, Hope, Ought

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.

Development, Ideas, Immanent, Obligation, Reality

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.

Discovery, Diversity, Recognized, Science, Year, Years

Use Campaign link tagging labels all for specifying slight differences in content for split testing.

Campaigns, Content, Differences, Labels, Specifying

Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed too great a demand!

1984, Demand, Should Have, Wait, Waiting, Years

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.

Defended, Numbers, Opinion, Opposition, Undertaking, Views

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.

Doubt, Facts, Necessity, Quantities, Variable, Variables

To ask the right question is harder than to answer it.

Answer, Answers, Harder, Math, Question, Right

Don't always blindly follow guidance and step-by-step instructions; you might run into something interesting.

Guidance, Interesting, Might, Run, Running

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.

Independent, Numbers, Finite Number, Type, Deo, Otherworldly

In mathematics the art of proposing a question must be held of higher value than solving it.

Art, Higher, Math, Mathematics

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 lader 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.

Believe, Discovery, Distinguishable, Finite Number, Infinite, Unbounded

Mathematics is entirely free in its development, and its concepts are only linked by the necessity of being consistent, and are co-ordinated with concepts introduced previously by means of precise definitions.

Definitions, Development, Introduced, Linked, Math, Mean

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". 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.

Hand, Hands, Innumerably, Numbers, Parte, Reality

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.

Accepted, Power, Science, Tenaciously, Understood, Widely

Every transfinite consistent multiplicity, that is, every transfinite set, must have a definite aleph as its cardinal number.

Aleph, Cardinals, Consistent, Numbers, Set