User blog comment:MilkyWay90/Help with understanding Veblen array notation/@comment-30754445-20180811202716/@comment-35470197-20180818143038

@Fejfo

I guess that the uncountabele veblen function is not necessarily well-defined even if you fix an FS.

In the successor case, you needa the Scott continuity of \(\varphi_{\alpha}\) in order to ensure the existence of fixed points.

On the other hand, in the uncountable limit case, the diagonalisation \(\beta \mapsto \phi_{A[\beta]}(\beta)\) is not necessarily Scott continuous.

As a result, \(\phi_{A+1}(\beta)\) for an uncountable limit ordinal \(A\) is not necessarily well-defined.

Is there a well-known choice of an FS up to \(\varepsilon_{\Omega+1}+1\) for which the diagonalisation \(\beta \mapsto \phi_{A[\beta]}(\beta)\) is automatically Scott continuous?

> weak-fixed-point

Could you tell me the definition of the notion of a weak-fixed-point? Does it always exists in that setting?