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Weak forms of the Regularity Principle in the presence of equation image

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Abstract

We study the strength of weak forms of the Regularity Principle in the presence of equation image (induction on bounded existential formulas) relative to other subsystems of equation image. In particular, the Bounded Weak Regularity Principle is formulated, and it is shown that when applied to E1 formulas, this principle is equivalent over equation image to equation image.

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