User blog comment:Flitri/An ordinal Collapsing up to the Least weakly Mahlo Cardinal/@comment-35470197-20190409053305/@comment-35470197-20190411040639

> Is this enough or is anything syntactically ambiguous?

No. There are many errors. For example, you have not corrected mistakes like N = Enum{x | x < ω}, which I have already pointed out.

As I emphasised, you are using many non-traditional conventions without specifying all precise definitions. For example, what is @? It is undefined in your manuscript, and highly depends on the context. I rephrase my reply above. Please read back your manuscript more carefully. I am certain that it costs several days. Please do not give up reading it back before completing to define undefined stuffs. Statements containing undefined stuffs can never be estimated to be sufficient.
 * It is very hard to point out "all" problems. I hope that you will fix all such problems. After then, I can point out more (non-syntax theoretic but set theoretic) problems.

> cof(x) = min{μ|Sup{β < μ} & α[β] = x for some μ-indexed sequence α with α[δ] < α[λ] ⇌ δ < λ}

Typo. Remove the "&".

> There is one more case: What about ordinals satisfying additional functions like ξ = ω^ξ?

Functions are not conditions, and hence "ordinals satisfying additional functions like ξ = ω^ξ" does not make sense.

> To get these consider the original ordinal, for example ψ(0[1]), and append a pair of ordinals (β, α), making it ψ(0[1]β, α). This ordinal is the α-th ordinal below Ω(ψ(0[1])+1+β) satisfying ξ = ω^ξ. It well ordered due to ψ(β, α) well ordered. As a result all higher ordinals are well ordered recursively downward, getting their well-orderedness from its lower order cousins well ordered ness.

Ordinals are trivially well-ordered. What an ordinal notation is required is a primitive recursive well-ordering. What you wrote is nothing to do with it. Generally, a recursive well-ordering is not so easy to define in several lines. For example, see the argument in p.261 of Rathjen's paper on the standard ordinal notation with weakly Mahlo. In order to intreprete the set-theoretic order into a primitive recursive well-ordering, Rathjen needed to use all of Propostions 2.2(vi), 2.4, 3.14, 5.5(v), (vi), 5.10, 5.13, 7.2, 7.4, 7.5.