User blog comment:TheKing44/Ordinal Definable System of Fundamental Sequences/@comment-35470197-20191201030851/@comment-35470197-20191203031058

Thank you!

> A slightly easier way to argue the last paragraph is to say that ...

I guess that you are giving an alternative proof of the statement "$ \alpha $ has a ordinal definable bijection with the natural numbers" in the last paragraph, but I could not understand it.

First, concerning your explation "$ \alpha $ has cofinality $ \omega $ according to HOD", α is of cofinality ω by the fact "$ g(\alpha) $ is an ordinal definable fundamental sequence for $ \alpha $", and it looks irrelevant to HOD isn't it? Here you are considering the case where V can be different from OD, and hence I could not understand why you are refering to HOD. Are you relativise the statement "$ \alpha $ has a ordinal definable bijection with the natural numbers" at HOD⊂V?

Secondly, concerning your explanation "all the ordinals before it are countable according to HOD", I again could not understand why you are referring to HOD.

Thirdly, concerning your explanation "and therefore we can argue from within HOD that it is countable since HOD satisfies ZFC", proof of the relativisation of the countability of α at HOD just implies the consistency of the countability of α with ZFC, doesn't it? Or is there a well-known absoluteness between HOD and V for a specific class of formulae including the statement?