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My understanding is that the proofs that CH and not-CH are consistent with ZFC are both about ZFC and in ZFC. Is it possible to do these proofs about ZFC but in a weaker axiomatic system?

(It is also my understanding that there are multiple methods of proving "CH is independent of ZFC". I'm interested in the question of if there is any way of proving this in a weaker system, and I'm interested in either direction (ZFC+CH or ZFC+not-CH).)

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    $\begingroup$ Yes, you can do the entire thing in $\sf PA$, and even weaker systems. Namely, $\sf PA\vdash\operatorname{Con}(ZFC)\rightarrow\operatorname{Con}(ZFC+CH) \land\operatorname{Con}(ZFC+\lnot CH)$. $\endgroup$
    – Asaf Karagila
    Commented Mar 23, 2015 at 14:43
  • $\begingroup$ Cool. Where can I learn more? $\endgroup$
    – jbapple
    Commented Mar 23, 2015 at 18:35
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    $\begingroup$ Probably the old Kunen book. I know that through Boolean-valued models you can also prove this, although I'm not 100% sure as to how this translates to a proof from within $\sf PA$. I saw some M.Sc. thesis once that had the details written out. $\endgroup$
    – Asaf Karagila
    Commented Mar 23, 2015 at 18:40
  • $\begingroup$ Thank you. I'm going to leave the question open in the hopes that someone will drop in with a solid reference. $\endgroup$
    – jbapple
    Commented Mar 23, 2015 at 20:10

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These independence proofs only use elementary finitistic reasoning. Finitistic reasoning itself could be formalized as PRA (primitive recursive arithmetic). PRA is a much weaker system than PA let alone ZFC. The proof theoretic ordinal of PRA is just $\omega^\omega$. Most mathematicians regard statements provable in PRA as really true (with the exception of perhaps ultrafinitists).

In a sense, the independence results really show in a strictly finitistic and constructive way how one could transform a proof of $0=1$ from ZFC+(not)CH into a proof of $0=1$ from ZFC alone.

Some remarks about this can be found in the book Set Theory (2013 by Kenneth Kunen) (Chapter II.1 Informal Remarks on Consistency Proofs).

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