@misc{ORDINAL5.MIZ,
  AUTHOR = {Bancerek, Grzegorz},
  TITLE = {Epsilon Numbers and Cantor Normal Form},
  SECTION1 = {Preliminaries},
  SECTION2 = {Tetration (Knuth's Arrow Notation) of Ordinals},
  SECTION3 = {Critical Numbers},
  SECTION4 = {Epsilon Numbers},
  SECTION5 = {Cantor Normal Form},
  ADDRESS1 = {Bia{\l}ystok Technical University},
  DAY = {20},
  MONTH = {October},
  YEAR = {2009},
  SUMMARY = {An epsilon number is a transfinite number which
  is a fixed point of the exponential map: $\omega^\epsilon = \epsilon$.
  The formalization of the concept is done with use of the tetration of ordinals
  (Knuth's arrow notation, $\uparrow\uparrow$).
  Namely, the ordinal indexing of epsilon numbers is defined as follows:
    $\epsilon_0 = \omega\mathop{\uparrow\uparrow}\omega$,
    $\epsilon_{\alpha+1} = \epsilon_\alpha\mathop{\uparrow\uparrow}\omega$,
    and for limit ordinal $\lambda$:
    $\epsilon_\lambda = \lim_{\alpha<\lambda}\epsilon_\alpha
     = \bigcup_{\alpha<\lambda}\epsilon_\alpha$.\par
  Every ordinal number $\alpha$ can be uniquely written as
 $n_1\omega^{\beta_1} + n_2\omega^{\beta_2} + \cdots + n_k\omega^{\beta_k}$,
 where $k$ is a natural number,
 $n_1$, $n_2$, $\ldots$, $n_k$ are positive integers,
 and $\beta_1 > \beta_2 > \ldots > \beta_k$ are ordinal numbers
 ($\beta_k = 0$). This decomposition of $\alpha$ is called
 the Cantor Normal Form of $\alpha$.}}
