User blog comment:TheKing44/Ordinal Definable System of Fundamental Sequences/@comment-35470197-20191201030851

> However, no other system of fundamental sequences definable in first order set theory could include a limit ordinal $ \alpha $ that odsfs does not, since then $ \alpha $'s fundamental sequence in that system would be ordinal definable, contradicting the fact that is not in odsfs.

Could you tell me the reason? Are you stating that the limit \(\alpha\) of a strictly increasing countable sequence \(\alpha_n\) of ordinals admitting ordinal definable fundamental sequences \(\alpha_n[m]\) again admits an ordinal definale fundamental sequences?