User blog comment:Hyp cos/TON, stable ordinals, and my array notation/@comment-31580368-20180927133742/@comment-31580368-20180927150541

You are sure that such ordinals exist? Just not always between the large cardinals and the large ordinals there are correlations. For example:

For expression very large order of hyper-inaccessible recursive terms-ordinal are used. (M^M)- hyper-inaccessible or (1,0)-inaccessible, (M^M^2)- hyper-inaccessible or (1,0,0)-inaccessible, e.t.c. similar to Bahman hierarchy on Veblen hierarchy. We can make a stronger recursion to express even greater cardinals. But Δ11-set (hyperrecursive) of term-ordinal on hyper-inaccessible is greatly inaccessible cardinal and is equal Mahlo cardinal (Carmody Theorem 10). But Δ11-set of term-ordinal on hyper-Mahlo cardinal is greatly Mahlo cardinal and not equal Weakly compact cardinal. Nevertheless Δ11-set of term-ordinal on recursively hyper-Mahlo ordinal equal П3-reflecting. And so on. Δ11-set of term-ordinal on hyper-П3-reflecting equal П4-reflecting, e.t.c.

I think you can give a definition of a 1-П1-reflecting is Δ11-set of term-ordinal on hyper-1-stable.