Every countable ordinal $\alpha$ can be written uniquely in Cantor canonical form as a finite arithmetical expression, say $C(\alpha)$. We thus have the 1-1 correspondence between the countable ordinals and their corresponding finite Goedel (ordinal) numbers:
$C(\alpha) \leftrightarrow \lceil C(\alpha) \rceil$
Since the set of Goedel numbers $\{\lceil C(\alpha) \rceil\}$ is denumerable, shouldn't the set $\{C(\alpha)\}$ of countable ordinals also be denumerable, with cardinality $\aleph_{0}$?
Let me re-phrase my query:
Conventional wisdom argues (correctly if the set theory ZF is consistent) that the cardinality of the countable ordinals---which is the set of all the ordinals below the first uncountable ordinal $\varepsilon_{0}$, each of whom can be written in Cantor canonical (not normal) form as a finite arithmetical expression---is $\aleph_{1}$.
However, the equally well-defined set-theoretic 1-1 correspondence that I detailed above seems to imply that this cardinality is $\aleph_{0}$.
So, which is it?
For the meaning of 'Cantor canonical form' as used here and another, albeit more convoluted, 1-1 correspondence, see http://alixcomsi.com/45_Countable_Ordinals_Update.pdf