Georg Cantor was a German mathematician, born on March 3, 1845 at St. Petersburg, Russia. He created 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.

An important exchange of letters with Richard Dedekind, Mathematician at the Brunswick Technical Institute, who was his lifelong friend and colleague, marked the beginning of Cantor’s ideas on the theory of sets. Both agreed that a set, whether finite or infinite, is a collection of objects that share a particular property while each object retains its own individuality.

But when Cantor applied the device of the one-to-one correspondence (e.g., {a, b, c} to {1, 2, 3}) to study the characteristics of sets, he quickly saw that they differed in the extent of their membership, even among infinite sets.

In 1873, Cantor demonstrated that the rational numbers, though infinite, are countable because they may be placed in a one-to-one correspondence with the natural numbers. He showed that the set (or aggregate) of real numbers (composed of irrational and rational numbers) was infinite and uncountable.

Even more paradoxically, he proved that the set of all algebraic numbers contains as many components as the set of all integers and that transcendental numbers (those that are not algebraic, as π), which are a subset of the irrationals, are uncountable and are therefore more numerous than integers, which must be conceived as infinite.

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