Stronger Systems
Much as we have defined Σn and Πn (or, more accurately, Σ0n and Π0n) formulae, we can define Σ1n and Π1n formulae in the following way: a Δ10 (or Σ10 or Π10) formula is just an arithmetical formula, and a Σ1n, respectively Π1n, formula is obtained by adding existential, respectively universal, class quantifiers in front of a Π1n−1, respectively Σ1n−1.
It is not too hard to see that over a not too weak system, any formula of second-order arithmetic is equivalent to a Σ1n or Π1n formula for all large enough n. The system Π11-comprehension is the system consisting of the basic axioms, plus the ordinary second-order induction axiom and the comprehension axiom for every Π11 formula φ. It is an easy exercise to show that this is actually equivalent to Σ11-comprehension (on the other hand, Δ11-comprehension, defined by the same trick as introduced earlier for Δ01 comprehension, is actually weaker).
Read more about this topic: Second-order Arithmetic
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