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^{a_i} * n_i < epsilon₀ is the Cantor normal form of a limit ordinal (i.e., a_t > 0) and n is a natural number, then a(n) is defined as follows:
if a_t=b+1 is isolated, then
a(n) = sum{i=1..t-1} omega^{a_i} * n_i + omega^{a_t} * (n_t - 1) + n * omega^{b}
if a_t is a limit, then
a(n) = sum{i=1..t-1} omega^{a_i} * n_i + omega^{a_t} * (n_t - 1) + omega^{a_t(n)}
If a < epsilon₀, then a_t < a and so the definition is correct. We can further define
epsilon₀(0) = 0
epsilon₀(n+1) = omega^{epsilon₀(n)}
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:
the depth of the recursion (when reached, the ordinal was just output and not decomposed any further),
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