Georg Cantor and Infinity

1
Georg Cantor and Infinity

• An introduction to set theory and transfinite
  numbers

2
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.

3
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.

4
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.

5
Bibliography

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

6
Banach-Tarski Paradox, assuming the Axiom of
Choice
7
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.