How to Count Up to the First Epsilon

Note: Several boring passages in the following sequence are omitted. The way it should proceed at these spots is, however, quite clear. Similar cuts would be necessary even if presenting only the series up to one billion or less.

 0
  1
  2
  3
  4
  ...
 omega
    omega + 1
    omega + 2
    omega + 3
    omega + 4
    ...
   omega * 2
    omega * 2 + 1
    omega * 2 + 2
    omega * 2 + 3
    omega * 2 + 4
    ...
   omega * 3
    omega * 3 + 1
    omega * 3 + 2
    omega * 3 + 3
    omega * 3 + 4
    ...
   omega * 4, ...
   ...
  omega^2
     omega^2 + 1
     omega^2 + 2
     omega^2 + 3
     omega^2 + 4
     ...
    omega^2 + omega
     omega^2 + omega + 1
     omega^2 + omega + 2
     omega^2 + omega + 3
     omega^2 + omega + 4
     ...
    omega^2 + omega * 2
     omega^2 + omega * 2 + 1
     omega^2 + omega * 2 + 2
     omega^2 + omega * 2 + 3
     omega^2 + omega * 2 + 4
     ...
    omega^2 + omega * 3
     omega^2 + omega * 3 + 1
     omega^2 + omega * 3 + 2
     omega^2 + omega * 3 + 3
     omega^2 + omega * 3 + 4
     ...
    omega^2 + omega * 4, ...
    ...
   omega^2 * 2
     omega^2 * 2 + 1
     omega^2 * 2 + 2
     omega^2 * 2 + 3
     omega^2 * 2 + 4
     ...
    omega^2 * 2 + omega
     omega^2 * 2 + omega + 1
     omega^2 * 2 + omega + 2
     omega^2 * 2 + omega + 3
     omega^2 * 2 + omega + 4
     ...
    omega^2 * 2 + omega * 2
     omega^2 * 2 + omega * 2 + 1
     omega^2 * 2 + omega * 2 + 2
     omega^2 * 2 + omega * 2 + 3
     omega^2 * 2 + omega * 2 + 4
     ...
    omega^2 * 2 + omega * 3
     omega^2 * 2 + omega * 3 + 1
     omega^2 * 2 + omega * 3 + 2
     omega^2 * 2 + omega * 3 + 3
     omega^2 * 2 + omega * 3 + 4
     ...
    omega^2 * 2 + omega * 4, ...
    ...
   omega^2 * 3
     omega^2 * 3 + 1
     omega^2 * 3 + 2
     omega^2 * 3 + 3
     omega^2 * 3 + 4
     ...
    omega^2 * 3 + omega
     omega^2 * 3 + omega + 1
     omega^2 * 3 + omega + 2
     omega^2 * 3 + omega + 3
     omega^2 * 3 + omega + 4
     ...
    omega^2 * 3 + omega * 2
     omega^2 * 3 + omega * 2 + 1
     omega^2 * 3 + omega * 2 + 2
     omega^2 * 3 + omega * 2 + 3
     omega^2 * 3 + omega * 2 + 4
     ...
    omega^2 * 3 + omega * 3
     omega^2 * 3 + omega * 3 + 1
     omega^2 * 3 + omega * 3 + 2
     omega^2 * 3 + omega * 3 + 3
     omega^2 * 3 + omega * 3 + 4
     ...
    omega^2 * 3 + omega * 4, ...
    ...
   omega^2 * 4, ...
   ...
  omega^3, ...
     omega^3 + omega, ...
     omega^3 + omega * 2, ...
     omega^3 + omega * 3, ...
     omega^3 + omega * 4, ...
     ...
    omega^3 + omega^2, ...
     omega^3 + omega^2 + omega, ...
     omega^3 + omega^2 + omega * 2, ...
     omega^3 + omega^2 + omega * 3, ...
     omega^3 + omega^2 + omega * 4, ...
     ...
    omega^3 + omega^2 * 2, ...
     omega^3 + omega^2 * 2 + omega, ...
     omega^3 + omega^2 * 2 + omega * 2, ...
     omega^3 + omega^2 * 2 + omega * 3, ...
     omega^3 + omega^2 * 2 + omega * 4, ...
     ...
    omega^3 + omega^2 * 3, ...
     omega^3 + omega^2 * 3 + omega, ...
     omega^3 + omega^2 * 3 + omega * 2, ...
     omega^3 + omega^2 * 3 + omega * 3, ...
     omega^3 + omega^2 * 3 + omega * 4, ...
     ...
    omega^3 + omega^2 * 4, ...
    ...
   omega^3 * 2, ...
     omega^3 * 2 + omega, ...
     omega^3 * 2 + omega * 2, ...
     omega^3 * 2 + omega * 3, ...
     omega^3 * 2 + omega * 4, ...
     ...
    omega^3 * 2 + omega^2, ...
     omega^3 * 2 + omega^2 + omega, ...
     omega^3 * 2 + omega^2 + omega * 2, ...
     omega^3 * 2 + omega^2 + omega * 3, ...
     omega^3 * 2 + omega^2 + omega * 4, ...
     ...
    omega^3 * 2 + omega^2 * 2, ...
     omega^3 * 2 + omega^2 * 2 + omega, ...
     omega^3 * 2 + omega^2 * 2 + omega * 2, ...
     omega^3 * 2 + omega^2 * 2 + omega * 3, ...
     omega^3 * 2 + omega^2 * 2 + omega * 4, ...
     ...
    omega^3 * 2 + omega^2 * 3, ...
     omega^3 * 2 + omega^2 * 3 + omega, ...
     omega^3 * 2 + omega^2 * 3 + omega * 2, ...
     omega^3 * 2 + omega^2 * 3 + omega * 3, ...
     omega^3 * 2 + omega^2 * 3 + omega * 4, ...
     ...
    omega^3 * 2 + omega^2 * 4, ...
    ...
   omega^3 * 3, ...
     omega^3 * 3 + omega, ...
     omega^3 * 3 + omega * 2, ...
     omega^3 * 3 + omega * 3, ...
     omega^3 * 3 + omega * 4, ...
     ...
    omega^3 * 3 + omega^2, ...
     omega^3 * 3 + omega^2 + omega, ...
     omega^3 * 3 + omega^2 + omega * 2, ...
     omega^3 * 3 + omega^2 + omega * 3, ...
     omega^3 * 3 + omega^2 + omega * 4, ...
     ...
    omega^3 * 3 + omega^2 * 2, ...
     omega^3 * 3 + omega^2 * 2 + omega, ...
     omega^3 * 3 + omega^2 * 2 + omega * 2, ...
     omega^3 * 3 + omega^2 * 2 + omega * 3, ...
     omega^3 * 3 + omega^2 * 2 + omega * 4, ...
     ...
    omega^3 * 3 + omega^2 * 3, ...
     omega^3 * 3 + omega^2 * 3 + omega, ...
     omega^3 * 3 + omega^2 * 3 + omega * 2, ...
     omega^3 * 3 + omega^2 * 3 + omega * 3, ...
     omega^3 * 3 + omega^2 * 3 + omega * 4, ...
     ...
    omega^3 * 3 + omega^2 * 4, ...
    ...
   omega^3 * 4, ...
   ...
  omega^4, ...
  ...
 omega^omega, ...
     omega^omega * 2, ...
     omega^omega * 3, ...
     omega^omega * 4, ...
     ...
    omega^(omega + 1), ...
     omega^(omega + 1) * 2, ...
     omega^(omega + 1) * 3, ...
     omega^(omega + 1) * 4, ...
     ...
    omega^(omega + 2), ...
     omega^(omega + 2) * 2, ...
     omega^(omega + 2) * 3, ...
     omega^(omega + 2) * 4, ...
     ...
    omega^(omega + 3), ...
     omega^(omega + 3) * 2, ...
     omega^(omega + 3) * 3, ...
     omega^(omega + 3) * 4, ...
     ...
    omega^(omega + 4), ...
    ...
   omega^(omega * 2), ...
     omega^(omega * 2) * 2, ...
     omega^(omega * 2) * 3, ...
     omega^(omega * 2) * 4, ...
     ...
    omega^(omega * 2 + 1), ...
     omega^(omega * 2 + 1) * 2, ...
     omega^(omega * 2 + 1) * 3, ...
     omega^(omega * 2 + 1) * 4, ...
     ...
    omega^(omega * 2 + 2), ...
     omega^(omega * 2 + 2) * 2, ...
     omega^(omega * 2 + 2) * 3, ...
     omega^(omega * 2 + 2) * 4, ...
     ...
    omega^(omega * 2 + 3), ...
     omega^(omega * 2 + 3) * 2, ...
     omega^(omega * 2 + 3) * 3, ...
     omega^(omega * 2 + 3) * 4, ...
     ...
    omega^(omega * 2 + 4), ...
    ...
   omega^(omega * 3), ...
     omega^(omega * 3) * 2, ...
     omega^(omega * 3) * 3, ...
     omega^(omega * 3) * 4, ...
     ...
    omega^(omega * 3 + 1), ...
     omega^(omega * 3 + 1) * 2, ...
     omega^(omega * 3 + 1) * 3, ...
     omega^(omega * 3 + 1) * 4, ...
     ...
    omega^(omega * 3 + 2), ...
     omega^(omega * 3 + 2) * 2, ...
     omega^(omega * 3 + 2) * 3, ...
     omega^(omega * 3 + 2) * 4, ...
     ...
    omega^(omega * 3 + 3), ...
     omega^(omega * 3 + 3) * 2, ...
     omega^(omega * 3 + 3) * 3, ...
     omega^(omega * 3 + 3) * 4, ...
     ...
    omega^(omega * 3 + 4), ...
    ...
   omega^(omega * 4), ...
   ...
  omega^(omega^2), ...
     omega^(omega^2 + 1), ...
     omega^(omega^2 + 2), ...
     omega^(omega^2 + 3), ...
     omega^(omega^2 + 4), ...
     ...
    omega^(omega^2 + omega), ...
     omega^(omega^2 + omega + 1), ...
     omega^(omega^2 + omega + 2), ...
     omega^(omega^2 + omega + 3), ...
     omega^(omega^2 + omega + 4), ...
     ...
    omega^(omega^2 + omega * 2), ...
     omega^(omega^2 + omega * 2 + 1), ...
     omega^(omega^2 + omega * 2 + 2), ...
     omega^(omega^2 + omega * 2 + 3), ...
     omega^(omega^2 + omega * 2 + 4), ...
     ...
    omega^(omega^2 + omega * 3), ...
     omega^(omega^2 + omega * 3 + 1), ...
     omega^(omega^2 + omega * 3 + 2), ...
     omega^(omega^2 + omega * 3 + 3), ...
     omega^(omega^2 + omega * 3 + 4), ...
     ...
    omega^(omega^2 + omega * 4), ...
    ...
   omega^(omega^2 * 2), ...
     omega^(omega^2 * 2 + 1), ...
     omega^(omega^2 * 2 + 2), ...
     omega^(omega^2 * 2 + 3), ...
     omega^(omega^2 * 2 + 4), ...
     ...
    omega^(omega^2 * 2 + omega), ...
     omega^(omega^2 * 2 + omega + 1), ...
     omega^(omega^2 * 2 + omega + 2), ...
     omega^(omega^2 * 2 + omega + 3), ...
     omega^(omega^2 * 2 + omega + 4), ...
     ...
    omega^(omega^2 * 2 + omega * 2), ...
     omega^(omega^2 * 2 + omega * 2 + 1), ...
     omega^(omega^2 * 2 + omega * 2 + 2), ...
     omega^(omega^2 * 2 + omega * 2 + 3), ...
     omega^(omega^2 * 2 + omega * 2 + 4), ...
     ...
    omega^(omega^2 * 2 + omega * 3), ...
     omega^(omega^2 * 2 + omega * 3 + 1), ...
     omega^(omega^2 * 2 + omega * 3 + 2), ...
     omega^(omega^2 * 2 + omega * 3 + 3), ...
     omega^(omega^2 * 2 + omega * 3 + 4), ...
     ...
    omega^(omega^2 * 2 + omega * 4), ...
    ...
   omega^(omega^2 * 3), ...
     omega^(omega^2 * 3 + 1), ...
     omega^(omega^2 * 3 + 2), ...
     omega^(omega^2 * 3 + 3), ...
     omega^(omega^2 * 3 + 4), ...
     ...
    omega^(omega^2 * 3 + omega), ...
     omega^(omega^2 * 3 + omega + 1), ...
     omega^(omega^2 * 3 + omega + 2), ...
     omega^(omega^2 * 3 + omega + 3), ...
     omega^(omega^2 * 3 + omega + 4), ...
     ...
    omega^(omega^2 * 3 + omega * 2), ...
     omega^(omega^2 * 3 + omega * 2 + 1), ...
     omega^(omega^2 * 3 + omega * 2 + 2), ...
     omega^(omega^2 * 3 + omega * 2 + 3), ...
     omega^(omega^2 * 3 + omega * 2 + 4), ...
     ...
    omega^(omega^2 * 3 + omega * 3), ...
     omega^(omega^2 * 3 + omega * 3 + 1), ...
     omega^(omega^2 * 3 + omega * 3 + 2), ...
     omega^(omega^2 * 3 + omega * 3 + 3), ...
     omega^(omega^2 * 3 + omega * 3 + 4), ...
     ...
    omega^(omega^2 * 3 + omega * 4), ...
    ...
   omega^(omega^2 * 4), ...
   ...
  omega^(omega^3), ...
     omega^(omega^3 + omega), ...
     omega^(omega^3 + omega * 2), ...
     omega^(omega^3 + omega * 3), ...
     omega^(omega^3 + omega * 4), ...
     ...
    omega^(omega^3 + omega^2), ...
     omega^(omega^3 + omega^2 + omega), ...
     omega^(omega^3 + omega^2 + omega * 2), ...
     omega^(omega^3 + omega^2 + omega * 3), ...
     omega^(omega^3 + omega^2 + omega * 4), ...
     ...
    omega^(omega^3 + omega^2 * 2), ...
     omega^(omega^3 + omega^2 * 2 + omega), ...
     omega^(omega^3 + omega^2 * 2 + omega * 2), ...
     omega^(omega^3 + omega^2 * 2 + omega * 3), ...
     omega^(omega^3 + omega^2 * 2 + omega * 4), ...
     ...
    omega^(omega^3 + omega^2 * 3), ...
     omega^(omega^3 + omega^2 * 3 + omega), ...
     omega^(omega^3 + omega^2 * 3 + omega * 2), ...
     omega^(omega^3 + omega^2 * 3 + omega * 3), ...
     omega^(omega^3 + omega^2 * 3 + omega * 4), ...
     ...
    omega^(omega^3 + omega^2 * 4), ...
    ...
   omega^(omega^3 * 2), ...
     omega^(omega^3 * 2 + omega), ...
     omega^(omega^3 * 2 + omega * 2), ...
     omega^(omega^3 * 2 + omega * 3), ...
     omega^(omega^3 * 2 + omega * 4), ...
     ...
    omega^(omega^3 * 2 + omega^2), ...
     omega^(omega^3 * 2 + omega^2 + omega), ...
     omega^(omega^3 * 2 + omega^2 + omega * 2), ...
     omega^(omega^3 * 2 + omega^2 + omega * 3), ...
     omega^(omega^3 * 2 + omega^2 + omega * 4), ...
     ...
    omega^(omega^3 * 2 + omega^2 * 2), ...
     omega^(omega^3 * 2 + omega^2 * 2 + omega), ...
     omega^(omega^3 * 2 + omega^2 * 2 + omega * 2), ...
     omega^(omega^3 * 2 + omega^2 * 2 + omega * 3), ...
     omega^(omega^3 * 2 + omega^2 * 2 + omega * 4), ...
     ...
    omega^(omega^3 * 2 + omega^2 * 3), ...
     omega^(omega^3 * 2 + omega^2 * 3 + omega), ...
     omega^(omega^3 * 2 + omega^2 * 3 + omega * 2), ...
     omega^(omega^3 * 2 + omega^2 * 3 + omega * 3), ...
     omega^(omega^3 * 2 + omega^2 * 3 + omega * 4), ...
     ...
    omega^(omega^3 * 2 + omega^2 * 4), ...
    ...
   omega^(omega^3 * 3), ...
     omega^(omega^3 * 3 + omega), ...
     omega^(omega^3 * 3 + omega * 2), ...
     omega^(omega^3 * 3 + omega * 3), ...
     omega^(omega^3 * 3 + omega * 4), ...
     ...
    omega^(omega^3 * 3 + omega^2), ...
     omega^(omega^3 * 3 + omega^2 + omega), ...
     omega^(omega^3 * 3 + omega^2 + omega * 2), ...
     omega^(omega^3 * 3 + omega^2 + omega * 3), ...
     omega^(omega^3 * 3 + omega^2 + omega * 4), ...
     ...
    omega^(omega^3 * 3 + omega^2 * 2), ...
     omega^(omega^3 * 3 + omega^2 * 2 + omega), ...
     omega^(omega^3 * 3 + omega^2 * 2 + omega * 2), ...
     omega^(omega^3 * 3 + omega^2 * 2 + omega * 3), ...
     omega^(omega^3 * 3 + omega^2 * 2 + omega * 4), ...
     ...
    omega^(omega^3 * 3 + omega^2 * 3), ...
     omega^(omega^3 * 3 + omega^2 * 3 + omega), ...
     omega^(omega^3 * 3 + omega^2 * 3 + omega * 2), ...
     omega^(omega^3 * 3 + omega^2 * 3 + omega * 3), ...
     omega^(omega^3 * 3 + omega^2 * 3 + omega * 4), ...
     ...
    omega^(omega^3 * 3 + omega^2 * 4), ...
    ...
   omega^(omega^3 * 4), ...
   ...
  omega^(omega^4), ...
  ...
 omega^(omega^omega), ...
     omega^(omega^omega * 2), ...
     omega^(omega^omega * 3), ...
     omega^(omega^omega * 4), ...
     ...
    omega^(omega^(omega + 1)), ...
     omega^(omega^(omega + 1) * 2), ...
     omega^(omega^(omega + 1) * 3), ...
     omega^(omega^(omega + 1) * 4), ...
     ...
    omega^(omega^(omega + 2)), ...
     omega^(omega^(omega + 2) * 2), ...
     omega^(omega^(omega + 2) * 3), ...
     omega^(omega^(omega + 2) * 4), ...
     ...
    omega^(omega^(omega + 3)), ...
     omega^(omega^(omega + 3) * 2), ...
     omega^(omega^(omega + 3) * 3), ...
     omega^(omega^(omega + 3) * 4), ...
     ...
    omega^(omega^(omega + 4)), ...
    ...
   omega^(omega^(omega * 2)), ...
     omega^(omega^(omega * 2) * 2), ...
     omega^(omega^(omega * 2) * 3), ...
     omega^(omega^(omega * 2) * 4), ...
     ...
    omega^(omega^(omega * 2 + 1)), ...
     omega^(omega^(omega * 2 + 1) * 2), ...
     omega^(omega^(omega * 2 + 1) * 3), ...
     omega^(omega^(omega * 2 + 1) * 4), ...
     ...
    omega^(omega^(omega * 2 + 2)), ...
     omega^(omega^(omega * 2 + 2) * 2), ...
     omega^(omega^(omega * 2 + 2) * 3), ...
     omega^(omega^(omega * 2 + 2) * 4), ...
     ...
    omega^(omega^(omega * 2 + 3)), ...
     omega^(omega^(omega * 2 + 3) * 2), ...
     omega^(omega^(omega * 2 + 3) * 3), ...
     omega^(omega^(omega * 2 + 3) * 4), ...
     ...
    omega^(omega^(omega * 2 + 4)), ...
    ...
   omega^(omega^(omega * 3)), ...
     omega^(omega^(omega * 3) * 2), ...
     omega^(omega^(omega * 3) * 3), ...
     omega^(omega^(omega * 3) * 4), ...
     ...
    omega^(omega^(omega * 3 + 1)), ...
     omega^(omega^(omega * 3 + 1) * 2), ...
     omega^(omega^(omega * 3 + 1) * 3), ...
     omega^(omega^(omega * 3 + 1) * 4), ...
     ...
    omega^(omega^(omega * 3 + 2)), ...
     omega^(omega^(omega * 3 + 2) * 2), ...
     omega^(omega^(omega * 3 + 2) * 3), ...
     omega^(omega^(omega * 3 + 2) * 4), ...
     ...
    omega^(omega^(omega * 3 + 3)), ...
     omega^(omega^(omega * 3 + 3) * 2), ...
     omega^(omega^(omega * 3 + 3) * 3), ...
     omega^(omega^(omega * 3 + 3) * 4), ...
     ...
    omega^(omega^(omega * 3 + 4)), ...
    ...
   omega^(omega^(omega * 4)), ...
   ...
  omega^(omega^(omega^2)), ...
     omega^(omega^(omega^2 + 1)), ...
     omega^(omega^(omega^2 + 2)), ...
     omega^(omega^(omega^2 + 3)), ...
     omega^(omega^(omega^2 + 4)), ...
     ...
    omega^(omega^(omega^2 + omega)), ...
     omega^(omega^(omega^2 + omega + 1)), ...
     omega^(omega^(omega^2 + omega + 2)), ...
     omega^(omega^(omega^2 + omega + 3)), ...
     omega^(omega^(omega^2 + omega + 4)), ...
     ...
    omega^(omega^(omega^2 + omega * 2)), ...
     omega^(omega^(omega^2 + omega * 2 + 1)), ...
     omega^(omega^(omega^2 + omega * 2 + 2)), ...
     omega^(omega^(omega^2 + omega * 2 + 3)), ...
     omega^(omega^(omega^2 + omega * 2 + 4)), ...
     ...
    omega^(omega^(omega^2 + omega * 3)), ...
     omega^(omega^(omega^2 + omega * 3 + 1)), ...
     omega^(omega^(omega^2 + omega * 3 + 2)), ...
     omega^(omega^(omega^2 + omega * 3 + 3)), ...
     omega^(omega^(omega^2 + omega * 3 + 4)), ...
     ...
    omega^(omega^(omega^2 + omega * 4)), ...
    ...
   omega^(omega^(omega^2 * 2)), ...
     omega^(omega^(omega^2 * 2 + 1)), ...
     omega^(omega^(omega^2 * 2 + 2)), ...
     omega^(omega^(omega^2 * 2 + 3)), ...
     omega^(omega^(omega^2 * 2 + 4)), ...
     ...
    omega^(omega^(omega^2 * 2 + omega)), ...
     omega^(omega^(omega^2 * 2 + omega + 1)), ...
     omega^(omega^(omega^2 * 2 + omega + 2)), ...
     omega^(omega^(omega^2 * 2 + omega + 3)), ...
     omega^(omega^(omega^2 * 2 + omega + 4)), ...
     ...
    omega^(omega^(omega^2 * 2 + omega * 2)), ...
     omega^(omega^(omega^2 * 2 + omega * 2 + 1)), ...
     omega^(omega^(omega^2 * 2 + omega * 2 + 2)), ...
     omega^(omega^(omega^2 * 2 + omega * 2 + 3)), ...
     omega^(omega^(omega^2 * 2 + omega * 2 + 4)), ...
     ...
    omega^(omega^(omega^2 * 2 + omega * 3)), ...
     omega^(omega^(omega^2 * 2 + omega * 3 + 1)), ...
     omega^(omega^(omega^2 * 2 + omega * 3 + 2)), ...
     omega^(omega^(omega^2 * 2 + omega * 3 + 3)), ...
     omega^(omega^(omega^2 * 2 + omega * 3 + 4)), ...
     ...
    omega^(omega^(omega^2 * 2 + omega * 4)), ...
    ...
   omega^(omega^(omega^2 * 3)), ...
     omega^(omega^(omega^2 * 3 + 1)), ...
     omega^(omega^(omega^2 * 3 + 2)), ...
     omega^(omega^(omega^2 * 3 + 3)), ...
     omega^(omega^(omega^2 * 3 + 4)), ...
     ...
    omega^(omega^(omega^2 * 3 + omega)), ...
     omega^(omega^(omega^2 * 3 + omega + 1)), ...
     omega^(omega^(omega^2 * 3 + omega + 2)), ...
     omega^(omega^(omega^2 * 3 + omega + 3)), ...
     omega^(omega^(omega^2 * 3 + omega + 4)), ...
     ...
    omega^(omega^(omega^2 * 3 + omega * 2)), ...
     omega^(omega^(omega^2 * 3 + omega * 2 + 1)), ...
     omega^(omega^(omega^2 * 3 + omega * 2 + 2)), ...
     omega^(omega^(omega^2 * 3 + omega * 2 + 3)), ...
     omega^(omega^(omega^2 * 3 + omega * 2 + 4)), ...
     ...
    omega^(omega^(omega^2 * 3 + omega * 3)), ...
     omega^(omega^(omega^2 * 3 + omega * 3 + 1)), ...
     omega^(omega^(omega^2 * 3 + omega * 3 + 2)), ...
     omega^(omega^(omega^2 * 3 + omega * 3 + 3)), ...
     omega^(omega^(omega^2 * 3 + omega * 3 + 4)), ...
     ...
    omega^(omega^(omega^2 * 3 + omega * 4)), ...
    ...
   omega^(omega^(omega^2 * 4)), ...
   ...
  omega^(omega^(omega^3)), ...
     omega^(omega^(omega^3 + omega)), ...
     omega^(omega^(omega^3 + omega * 2)), ...
     omega^(omega^(omega^3 + omega * 3)), ...
     omega^(omega^(omega^3 + omega * 4)), ...
     ...
    omega^(omega^(omega^3 + omega^2)), ...
     omega^(omega^(omega^3 + omega^2 + omega)), ...
     omega^(omega^(omega^3 + omega^2 + omega * 2)), ...
     omega^(omega^(omega^3 + omega^2 + omega * 3)), ...
     omega^(omega^(omega^3 + omega^2 + omega * 4)), ...
     ...
    omega^(omega^(omega^3 + omega^2 * 2)), ...
     omega^(omega^(omega^3 + omega^2 * 2 + omega)), ...
     omega^(omega^(omega^3 + omega^2 * 2 + omega * 2)), ...
     omega^(omega^(omega^3 + omega^2 * 2 + omega * 3)), ...
     omega^(omega^(omega^3 + omega^2 * 2 + omega * 4)), ...
     ...
    omega^(omega^(omega^3 + omega^2 * 3)), ...
     omega^(omega^(omega^3 + omega^2 * 3 + omega)), ...
     omega^(omega^(omega^3 + omega^2 * 3 + omega * 2)), ...
     omega^(omega^(omega^3 + omega^2 * 3 + omega * 3)), ...
     omega^(omega^(omega^3 + omega^2 * 3 + omega * 4)), ...
     ...
    omega^(omega^(omega^3 + omega^2 * 4)), ...
    ...
   omega^(omega^(omega^3 * 2)), ...
     omega^(omega^(omega^3 * 2 + omega)), ...
     omega^(omega^(omega^3 * 2 + omega * 2)), ...
     omega^(omega^(omega^3 * 2 + omega * 3)), ...
     omega^(omega^(omega^3 * 2 + omega * 4)), ...
     ...
    omega^(omega^(omega^3 * 2 + omega^2)), ...
     omega^(omega^(omega^3 * 2 + omega^2 + omega)), ...
     omega^(omega^(omega^3 * 2 + omega^2 + omega * 2)), ...
     omega^(omega^(omega^3 * 2 + omega^2 + omega * 3)), ...
     omega^(omega^(omega^3 * 2 + omega^2 + omega * 4)), ...
     ...
    omega^(omega^(omega^3 * 2 + omega^2 * 2)), ...
     omega^(omega^(omega^3 * 2 + omega^2 * 2 + omega)), ...
     omega^(omega^(omega^3 * 2 + omega^2 * 2 + omega * 2)), ...
     omega^(omega^(omega^3 * 2 + omega^2 * 2 + omega * 3)), ...
     omega^(omega^(omega^3 * 2 + omega^2 * 2 + omega * 4)), ...
     ...
    omega^(omega^(omega^3 * 2 + omega^2 * 3)), ...
     omega^(omega^(omega^3 * 2 + omega^2 * 3 + omega)), ...
     omega^(omega^(omega^3 * 2 + omega^2 * 3 + omega * 2)), ...
     omega^(omega^(omega^3 * 2 + omega^2 * 3 + omega * 3)), ...
     omega^(omega^(omega^3 * 2 + omega^2 * 3 + omega * 4)), ...
     ...
    omega^(omega^(omega^3 * 2 + omega^2 * 4)), ...
    ...
   omega^(omega^(omega^3 * 3)), ...
     omega^(omega^(omega^3 * 3 + omega)), ...
     omega^(omega^(omega^3 * 3 + omega * 2)), ...
     omega^(omega^(omega^3 * 3 + omega * 3)), ...
     omega^(omega^(omega^3 * 3 + omega * 4)), ...
     ...
    omega^(omega^(omega^3 * 3 + omega^2)), ...
     omega^(omega^(omega^3 * 3 + omega^2 + omega)), ...
     omega^(omega^(omega^3 * 3 + omega^2 + omega * 2)), ...
     omega^(omega^(omega^3 * 3 + omega^2 + omega * 3)), ...
     omega^(omega^(omega^3 * 3 + omega^2 + omega * 4)), ...
     ...
    omega^(omega^(omega^3 * 3 + omega^2 * 2)), ...
     omega^(omega^(omega^3 * 3 + omega^2 * 2 + omega)), ...
     omega^(omega^(omega^3 * 3 + omega^2 * 2 + omega * 2)), ...
     omega^(omega^(omega^3 * 3 + omega^2 * 2 + omega * 3)), ...
     omega^(omega^(omega^3 * 3 + omega^2 * 2 + omega * 4)), ...
     ...
    omega^(omega^(omega^3 * 3 + omega^2 * 3)), ...
     omega^(omega^(omega^3 * 3 + omega^2 * 3 + omega)), ...
     omega^(omega^(omega^3 * 3 + omega^2 * 3 + omega * 2)), ...
     omega^(omega^(omega^3 * 3 + omega^2 * 3 + omega * 3)), ...
     omega^(omega^(omega^3 * 3 + omega^2 * 3 + omega * 4)), ...
     ...
    omega^(omega^(omega^3 * 3 + omega^2 * 4)), ...
    ...
   omega^(omega^(omega^3 * 4)), ...
   ...
  omega^(omega^(omega^4)), ...
  ...
 omega^(omega^(omega^omega)), ...
 ...
epsilon_0.

The sequence was produced by a recursive computation of the ordinal function a(n) that is used e.g. when extending the hierarchy of Ackermann functions beyond omega.

If a = sum{i=1..t} omega^{ai} * ni < epsilon0 is the Cantor normal form of a limit ordinal (i.e., at > 0) and n is a natural number, then a(n) is defined as follows:

If a < epsilon0, then at < a and so the definition is correct. We can further define

Clearly sup{a(n)} = a, and given a and n, the recursion stops after a finite number of steps.

The above sequence was generated by a short recursive procedure computing a(n), n=0,1,2,..., first for a = epsilon0, then for the calculated values etc. Two parameters were given:

  1. the depth of the recursion (when reached, the ordinal was just output and not decomposed any further),
  2. a natural number approximating omega (the numbers a(n) were computed for n only up to this number, then three dots were output).
Both parameters were set to 5 for the sequence above (though 6 would be more fun). The level of the recursion is indicated by tabbing.

The sequence shows the abilities of the arithmetical module for the Ordinal Calculator, since the procedure itself was just about twenty lines of C++ code.


Libor Behounek, July 1999