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Definition 1.15. A proof of second order arithmetic in the tree form formulation with G L C as its logical basis (see [5] for the precise definition) (Including substitution as one of the rules of inference) is called (3-) reducible if it satisfies the following.

japanese journal of mathematics

The ordinals of the systems of second order arithmetic with the provably &lt;i&gt;Δ&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;&lt;sup&gt;1&lt;/sup&gt;-comprehension axiom and with the &lt;i&gt;Δ&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;&lt;sup&gt;1&lt;/sup&gt;-comprehension axiom respectively