User blog comment:Ecl1psed276/Question about standard notation/@comment-30754445-20180808072912/@comment-35470197-20180809215620

Maybe it is better for us to show an example when enumerating actual weakly inaccessible cardinals is necessary.

Let I(a,b) denote the b-th weakly a-inaccessible cardinal.

When we work with Rathjen's OCF and its associated ordinal notation, we need to determine the normal form expression (a kind of a standard form in an ordinal notation system) of a given ordinal. If an ordinal a is expressed as Veblen varphi(b,c), then we need b,c < a in order to ensure that it is a normal form expression. On the other hand, if we have a = chi(d,e), then the normal form expression of a is never varphi(b,c). So we can uniquely determine which to use varphi or chi in the normal form, i.e. a normal form expression is unique.

On the other hand, if we use I(a,b) in the way allowing limits, then this property does not hold. You may have two distinct normal form expressions of a given ordinal.

The uniqueness of a normal form expression is essentially used in the definition of Rathjen's OCF. Roughly speaking, he defined a set C_{kappa}(a) as the set of ordinals which have normal form expressions consisting of combinations of "sufficiently small ordinals below kappa", 0, +, varphi, Phi, chi, and psi_d(a') with a' < a. This works by all of his delicate arguments on uniqueness expressions, and gives the full strength of Mahlo level, because he defined psi_{kappa}(a) by using not only a but also the expression of a.

Skipping the uniqueness of normal form expressions of a given ordinal a, we always suffer the problem on ill-definedness when we define psi_{kappa}(a) using expression of a. Or just we can define psi_{kappa}(a) without using specific expression of a, as Deedlit did. Then it is not so easy for us to understand that the resulting OCF actually achieves Mahlo-level.