Abstract: 

The Barwise-Schlipf theorem connects model theory of first-order arithmetic (recursively saturated models of PA) with second-order arithmetic (expansions satisfying Δ¹₁-comprehension). A key step in its proof is to refute Δ¹₁-comprehension by exhibiting a proper Δ¹₁-definable cut. The nonexistence of such cuts is an induction principle much weaker than Δ¹₁-comprehension. To articulate the gap, we formulate and study several related induction principles, drawing analogies with well-studied principles in first-order arithmetic. In contrast to the first-order setting, we show that these principles are inequivalent over the base theory of arithmetical comprehension. The separating models are constructed using Steel forcing in nonstandard models, sometimes with saturation assumptions. Our results refine the existing framework for calibrating the inductive strength of principles above arithmetical comprehension. This is joint work with C. T. Chong, T. Slaman, Y. Yang.

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