Gentzen's ordinals are assigned to finite syntactic objects while PA_ω ordinals are assigned to infinite trees, making them objects of fundamentally different ontological categories.
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ontological categories(Thomasson's system; used to address mutual exclusiveness and exhaustiveness requirements)
Classifications of purported entities distinguished by the relations of dependence they have or lack — specifically dependence on mental states and dependence on spatio-temporally located objects.
ordinals(Proof-theoretic treatment of ordinals, distinct from but related to set-theoretic ordinals)
A central concept in both set theory and proof theory, used by Gentzen to assign measures to proofs in order to demonstrate consistency of PA via well-foundedness