1. A number that is smaller than any positive real numbers but greater than 0 is defined as an infinitesimal. An infinitesimal number does not possess the Archimedean property, i.e., if x is infinitesimal, then for every natural number n, n*x < r, where r is any positve real number.
  2. The monad(x) is defined as the set of the numbers that is infinitesimally close to x.
  3. If x .ne. y, then the intersection of monad(x) and monad(y) is an empty set.
  4. George Berkeley (1685-1753) ridiculed infinitestimals as "ghosts of departed quantities".
  5. Issac Newton (1642-1727) gave up the idea of "infinitesimal" while Gottfried Wilhelm Leibnitz (1646-1716) embraced it.
  6. Abraham Robinson (1918–1974) provided a solid logical foundation for the infinitesimals and the infinite numbers. An infinite number is the reciprocal of an infinitesimal.
  7. When the length of a line becomes infinitesimally small, the line will not becomes a point. It is still a line, or a linelet, if you prefer.
  8. A line will be a line. It does not matter how small it becomes.
  9. There is a one-to-one correspondence between every point of a 3-cm line and that of 3-km line. And yet, they are of different length.
  10. Nature does not jump (natura non facit saltus)
  11. But quantum mechanics allows "jump".