User blog comment:Moooosey/dco + decord sequence/@comment-35470197-20191009222350/@comment-35470197-20191011003835

Please feel free to ask questions. The point is that in order to define a system of fundametal sequences up to \(\Gamma_0\), you need to define a map \begin{eqnarray*} \{\textrm{limit ordinals up to } \Gamma_0\} \times \mathbb{N} & \to & \{\textrm{ordinals below } \Gamma_0\} \\ (\alpha,n) & \mapsto & \alpha[n] \end{eqnarray*} satisfying several conditions. In order to define the value \(\alpha[n]\), you are allowed to use only \(\alpha\), \(n\), variables which are quantified (i.e. "for all balah-blah" or "there exists blah-blah"), and stuffs which have already been completely defined using them.

You tried to define \(\alpha[n]\) for the case \(\alpha\) is expressed as \(a+b\). If such a pair \((a,b)\) of ordinals were unique, then it would make sense because \(a\) and \(b\) would be well-defined. However, such a pair is actually is not unique.

Therefore you need to set several assumptions on the expression \(\alpha = a + b\) so that \((a,b)\) will be uniquely determined from them. For example, you set the condition \(b < \varphi_a(b)\) in Rule 9. It is really good, because the condition (known as the condition for the standard expressions with respect to Veblen hierarchy) characterise \((a,b)\) from the information of \(\alpha = \varphi_a(b)\). So please remember why you needed this condition. Then you will find why you need additional assumptions for Rule 14.