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Hanul Jeon
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The relation between $\Pi_1$-Foundation and $\Sigma_1$-Foundation over Kripke-Platek set theory

Let $\mathsf{KP\omega}_0$ be Kripke-Platek set theory with Infinity but Foundation (or $\in$-Induction) restricted to $\Delta_0$-formulas. $\mathsf{ZF}$ proves $\in$-Induction holds for arbitrary formulas, but it is because we have Full Separation which is unavailable over $\mathsf{KP\omega}_0$, so $\in$-Induction on more complex formulas become a non-trivial statement over $\mathsf{KP\omega}_0$.

I wonder whether the following is known:

Question. Working over $\mathsf{KP\omega}_0$,

  1. Does $\Pi_1$-Foundation prove $\Sigma_1$-Foundation or vice versa?
  2. What about their consistency strength? Does $\Pi_1$-Foundation have a higher consistency strength than that of $\Sigma_1$-Foundation?
Hanul Jeon
  • 3.1k
  • 2
  • 14
  • 44