Given $ZFC^{-}$, that is, ZFC-Powerset+Collection+Separation, is there a set of alternative axioms $X$ (other than the trivial one, namely, {Powerset}) that, when added to $ZFC^{-}$, allow one to derive Powerset as a theorem of $ZFC^{-}$+$X$ and recover full $ZFC$? (Thanks to Prof. Hamkins for setting me straight on the correct formulation of $ZFC^{-}$.)
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asked Dec 13, 2014 at 1:14
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2$\begingroup$ Replacement is a consequence of ZFC-. $\endgroup$– François G. DoraisCommented Dec 13, 2014 at 1:48
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$\begingroup$ @FrançoisG.Dorais: ZFC- is ZFC-Powerset, correct? If so then this is different than $ZFC^{-}$. $\endgroup$– Thomas BenjaminCommented Dec 13, 2014 at 2:06
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$\begingroup$ I was using your definition. $\endgroup$– François G. DoraisCommented Dec 13, 2014 at 2:08
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2$\begingroup$ Replacement is basically a special case of collection. Depending on how it is formulated, collection might give you a set that is larger than the range of your function, but you can always trim it down using comprehension. $\endgroup$– François G. DoraisCommented Dec 13, 2014 at 14:04
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1$\begingroup$ In that case, the powerset axiom is really hard to beat! $\endgroup$– François G. DoraisCommented Dec 22, 2014 at 2:41
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1 Answer
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By compactness, if a sentence $\sigma$ is equivalent to the powerset axiom over $\mathrm{ZFC}^-$, then this equivalence is provable in Kripke-Platek set theory ($\mathrm{KP}$) with infinity, $\Sigma_n$-separation and $\Sigma_n$-collection for some $n$. Due to the connection between $\mathrm{KP}$ and admissible ordinals, a good place to look for this is in the $\alpha$-recursion theory literature. The only thing that came to mind were results by Evangelos Kranakis on partition relations in $\alpha$-recursion theory. I don't recall the specifics of Kranakis's work offhand, but that thought did lead me to an example.
Consider the following abstract extenson of Ramsey's Theorem to infinite cardinals.
For every cardinal $\kappa$ there is a cardinal $\lambda$ such that $\lambda\to(3)^2_\kappa$.
In $\mathrm{ZFC}$, one can show that $\lambda = (2^\kappa)^+$ works. In $\mathrm{ZFC}^-$, one can show that if $\lambda \leq 2^\kappa$ then $\lambda\not\to(3)^2_\kappa$. Thus, if there is a $\lambda$ such that $\lambda\to(3)^2_\kappa$ then $\lambda$ must be so large that $2^\kappa \lt \lambda$.
answered Dec 13, 2014 at 3:59
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$\begingroup$ @FrancoisGDorais: Regarding the abstract extension of Ramsey's Theorem and its relation to $ZFC^{-}$--very interesting. Thanks. Since I only require that $ZFC^{-}$+$\sigma$$\vdash$Powerset and not full equivalence, is it correct to infer that by compactness, this implication is provable in $KP$ with infinity, $\Sigma_{n}$-separation and $\Sigma_{n}$-collection for some n? What I am really after is what $\sigma$ might be. $\endgroup$ Commented Dec 13, 2014 at 10:31
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1$\begingroup$ @ThomasBenjamin: Ah, you don't require $\sigma$ to be a theorem of $\mathrm{ZFC}$? Yes, compactness still applies: if $\mathrm{ZFC}^- \vdash \sigma\to\mathrm{Powerset}$ then this is provable in $\Sigma_n\mbox{-}\mathrm{KP}$ for some $n$. $\endgroup$ Commented Dec 13, 2014 at 14:12