User blog comment:Nayuta Ito/faketest/e0/@comment-30754445-20180805015451/@comment-30754445-20180807011435

"I just meant that it conflicts the convention in Rathjen's OCF. I mind it just because Rathjen's OCF is the origin of Deedlit's OCF. "

Well, Deedlit also used the "I(ω) is the ωth inaccesible" version in his own post, when he wrote the formula relating χ and I up to χ(MM,0). So he had already committed to this "inconsistency" (which is, really, no inconsistency at all. Just because Rathjen wrote things in a given way, doesn't mean that you can't be inspired by his notation and still write some things a teeny-bit differently).

Also, remember that Rathjen already did the hard work of proving that his notation actually works. I'm guessing he used the definition-with-closure because it facilitated his proof. And since we aren't required to redo his work, we can afford some simplifications. Unlike Rathjen, we can concentrate on making the notation as friendly to the user as possible.