User blog:Edwin Shade/An Injection From The Countable Ordinals To The Reals

To define a function \(\phi\) such that \(\phi :Ord\mapsto\mathbb{R}^+\) we must first define the function \(\varupsilon (\alpha ,n)\) as the reduction-length of the ordinal \(\alpha\) such that the following rules are adhered to:


 * After one reduction, a limit ordinal is reduced to the nth term in it's fundamental sequence.


 * After one reduction, one is subtracted from a successor ordinal.


 * This continues until the ordinal has been reduced to 0.

Now let \(t\) be the base-11 digit for 10; this will function as a separator. \(Ordname(\alpha)\) serves as an injective function to the base-11 reals, such that \(Ordname(\alpha)=Z_{\omega}\), where \(Z_n=Z_{n-1}\ddagger t\ddagger\varupsilon (\alpha ,n)\), \(Z_1=0.\ddagger\varupsilon (\alpha ,1)\); (where \(A\ddagger B\) is the concatenation of symbol strings \(A\) and \(B\)).

Note this function is injective, so though for any given countable ordinals you can compute a unique value of \(Ordname(\alpha)\), you cannot compute an ordinal for any given real number. (Though I have stumbled upon a caveat which if exploitable will allow me to define large countable ordinals from almost any real number.)