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There are a couple examples I'm aware of where choosing to axiomatize $\mathsf{ZF(C)}$ using collection instead of replacement results in a much nicer (or at least less surprising) picture:

  • Let $\mathsf{ZFC}-$ and $\mathsf{ZFC}^-$ be "$\mathsf{ZFC}$-without-powerset" presented using replacement and collection respectively. Then $\mathsf{ZFC}-$ is weak in a number of annoying ways, while $\mathsf{ZFC}^-$ behaves "as it should;" see Gitman/Hamkins/Johnstone.

  • Parallel to the above, let $\mathsf{ZF}\ominus$ and $\mathsf{ZF}^\ominus$ be "$\mathsf{ZF}$-without-extensionality" presented using replacement and collection respectively. Then $\mathsf{ZF}^\ominus$ is equiconsistent with $\mathsf{ZF}$, while $\mathsf{ZF}\ominus$ is all the way down at the level of $\mathsf{Z}$. These facts were proved respectively by H. Friedman and Scott (see this old answer of Andres Caicedo). While low consistency strength is in one sense a tameness property, its appearance here is quite surprising (to me at least).

Overall, these results suggest to me that collection rather than replacement makes more sense in terms of presenting various set theories.

Are there notable exceptions to this theme - that is, situations where using replacement rather than collection results in a "tamer" picture in some sense?

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    $\begingroup$ Collection implies Replacement straightforwardly---something like Pairing + a tiny fragment of Separation should be enough to prove the implication. Doesn't that suggest that the answer to your question is no? Namely, the collection of models of $T$ + Collection is in general smaller than the collection of models of $T$ + Replacement, so we should expect the latter to be less tame. $\endgroup$
    – Julia Williams
    Commented Feb 20 at 21:10
  • $\begingroup$ This is perhaps not the kind of thing you're thinking about but in the context of constructive set theory I'd say that replacement is arguably 'more natural' than collection. This makes intuitive sense, given that replacement has a more 'explicit' flavor, but is also evidenced by the fact that explicitly named constructive set theories with collection (such as IZF) essentially all fail to have the existence property (i.e., that every provable existential statement has a uniquely definable witness). $\endgroup$
    – James E Hanson
    Commented Feb 20 at 22:27
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    $\begingroup$ @KamerynWilliams Why not post an answer? Although I think you are assuming separation in the base theory. Although separation is a standard axiom of Zermelo set theory, it is also common to find axiomatizations of ZFC that omit separation and deduce it as a consequence of replacement. That is historically anachronistic, since Z had separation and then replacement was added to form ZFC, but the resulting axiomatization is redundant. $\endgroup$
    – Joel David Hamkins
    Commented Feb 20 at 23:21
  • $\begingroup$ To be precise the comparison is between replacement on one hand and "separation + collection" on the other hand. Collection without separation is weaker than replacement. $\endgroup$
    – Zuhair Al-Johar
    Commented Feb 22 at 18:39
  • $\begingroup$ @user3840170 It's not my notation, it's from the Gitman/Hamkins/Johnstone paper. (OK, the circle thing is mine.) More significantly, removing powerset is not the same as saying that only first-order objects exist; ZF is a first-order theory, after all. To be honest I find your suggestions significantly worse than what I used. $\endgroup$
    – Noah Schweber
    Commented Nov 30 at 12:05

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