# Another question on collection and replacement in Sigma(n)-KP

Does Sigma(n)-KP with Sigma(n)-replacement instead of Sigma(n)-collection have Sigma(n)-collection as a theorem?

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I am aware of the following work where the related result that ZFC minus powerset with replacement is weaker than ZFC minus powerset with collection: arxiv.org/abs/1110.2430. But that does not answer my question as it pertains to weaker systems. –  Frode Bjørdal Jul 20 '13 at 21:58
I am interested in versions of KP with infinity. –  Frode Bjørdal Jul 20 '13 at 22:03

In your comment, you linked to my paper V. Gitman, J. D. Hamkins, T. A. Johnstone, What is the theory ZFC without power set? (see it also at the arXiv). And although your theories are much weaker, in fact the main result of that paper does answer this question.

The paper shows that even if you have $\text{ZFC}-\text{P}$, axiomatized using $\Sigma_n$-replacement for every $n$--and this includes your desired theory $\text{KP}+\Sigma_n$-replacement--you still cannot prove collection, and it seems to me that the counterexample models show that you cannot prove even $\Pi_1$-collection, which is essentially equivalent to $\Sigma_2$-collection (by my answer to your question on $\Sigma_n$ admissible ordinals yesterday).

Thus, for $n\geq 2$, one cannot prove $\Sigma_n$-collection from $\text{KP}+\Sigma_n$-replacement, nor even from the much stronger theory $\text{ZFC}-\text{P}$, axiomatized via the replacement axiom.

The main idea is that there are highly homogeneous models of these theories, which satisfy the replacement axiom only because it is seldom the case except in comparatively trivial cases to have unique witnesses for a property, making this axiom less powerful than you might think. And with the power set axiom, one cannot build the von Neumann hierarchy in order to form a collecting set of the form $V_\alpha$, which is how one proves that replacement implies collection in ZFC.

But meanwhile, it appears that if one adds $V=L$ to the theory, or considers models of your theory only of the form $L_\alpha$, then indeed you will deduce $\Sigma_n$-collection, since you can use the $L$-hierarchy for collecton instead of the von Neumann hierarchy. That is, if you only care about identifying ordinals $\alpha$ for which $L_\alpha$ satisfies $\text{KP}+\Sigma_n$-collection, then it suffices that they satisfy $\text{KP}+\Sigma_n$-replacement.

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Your last paragraph answers my main concern in this. I have been interested in what level of the constructible hierarchy which is needed to model KP with Sigma(n)-replacement. From your last sentence I infer that the answer is the least Sigma(n)-admissible ordinal. Is that right? –  Frode Bjørdal Jul 20 '13 at 23:14
Yes, I think that if one is looking only at $L_\alpha$ models, it doesn't matter whether you require $\Sigma_n$-replacement or $\Sigma_n$-collection. –  Joel David Hamkins Jul 20 '13 at 23:36