Georg Cantor and Infinity An introduction to set theory and transfinite numbers

Carl F. Gauss in a letter to H. C. Schumacher, 12 July 1831 “As to your proof, I must protest most vehemently against your use of the infinite as something consummated, as this is never permitted in mathematics. The infinite is but a figure of speech: an abridged form for the statement that limits exist which certain ratios may approach as closely as we desire, while other magnitudes may be permitted to grow beyond all bounds.”

Richard Dedekind on cuts: “If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions.”

Jacques Hadamard in a letter to Emile Borel, 1905 “From the infinitesimal calculus to the present, it seems to me, the essential progress in mathematics has resulted from successively annexing notions which, for the Greeks or the Renaissance geometers or the predecessors of Riemann, went ‘outside’ mathematics because it was impossible to define them.”

Bibliography John Stillwell, Yearning for the Impossible: The Surprising Truths of Mathematics (2006). John Stillwell, Yearning for the Impossible: The Surprising Truths of Mathematics (2006). John D. Barrow, Pi in the Sky: Counting, Thinking, and Being (1993). John D. Barrow, Pi in the Sky: Counting, Thinking, and Being (1993).

Banach-Tarski Paradox, assuming the Axiom of Choice

Georg Cantor “My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? 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.”