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The axiom schema of replacement seems, to me, the least intuitive of the axioms of ZFC. Further, it seems that, at least according to the wikipedia page, most of the applications are in set theory proper. As such it doesn't seem like we'd lose much in "ordinary mathematics" if we dropped it. Further, the previous page seems to suggest that (ZFC - replacement) is "more consistent" than ZFC (i.e. ZFC implies consistency of (ZFC - replacement)), so this seems like a clear advantage. What is the states of (ZFC - replacement) as a foundational system? What would we lose in "ordinary mathematics" if we didn't have replacement?

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  • $\begingroup$ I do not agree with your idea that removing Replacement does not harm much mathematics. For example, we cannot access ordinals greater than $\omega+\omega$ without Replacement. Moreover, Borel determinacy is known to be equivalent to Replacement over the remaining axioms of ZFC. $\endgroup$ – Hanul Jeon Sep 13 at 17:38
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    $\begingroup$ @HanulJeon That last sentence isn't accurate - since Borel determinacy is a statement about a particular level of the $V$-hierarchy, we can always have replacement fail "further up" without affecting it. Friedman's result is more subtle. $\endgroup$ – Noah Schweber Sep 13 at 18:11
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    $\begingroup$ To the OP, it may help to demystify replacement to note that it is in fact equivalent to the more concrete principle of transfinite recursion - see here. In a precise sense, what you get by adding transfinite recursion to $\mathsf{Z}$ or $\mathsf{ZC}$ is the ability to perform "unboundedly long" iterative constructions - consider e.g. Cantor-Bendixson derivatives or Ulm invariants. $\endgroup$ – Noah Schweber Sep 13 at 18:15
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    $\begingroup$ @KeeferRowan Well, that's why I put it as a comment instead of an answer. (Although that said I disagree with your definition of "ordinary mathematics," since things like Ulm invariants are in fact pretty standard fare within mathematics - I think something like "concrete mathematics" would be better.) $\endgroup$ – Noah Schweber Sep 13 at 18:54
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    $\begingroup$ Also related: karagila.org/2019/in-praise-of-replacement $\endgroup$ – Asaf Karagila Sep 13 at 18:55
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This is an issue of two different perspectives "talking past each other."

$\mathsf{ZFC}$ is indeed massively overshooting what we actually need for "concrete" mathematics (I disagree with the use of "ordinary" mathematics in this context). But that's part of the point (well, that's ahistorical; rather, it has emerged as part of the appeal). By settling on such a strong theory as $\mathsf{ZFC}$ as our "default" theory for mathematics, we save mathematicians a lot of effort: it's easy to convince oneself that a given "natural-language" proof actually translates into $\mathsf{ZFC}$ - or more accurately, that if there is a serious issue re: fully formalizing that natural-language proof, it's not related to $\mathsf{ZFC}$ but rather reflects a genuine ambiguity/gap/error in the natural-language argument itself.

The question of what foundations are actually needed for various parts of mathematics is however an extremely interesting one. The relevant topic is reverse mathematics, and broadly speaking I'd say that the theory $\mathsf{ACA_0}$ is the "right" one for most contexts. For example, despite its extreme complexity it's generally believed that the proof of Fermat's Last Theorem can be modified to go through in $\mathsf{ACA_0}$. And this is well below $\mathsf{Z}$ (= $\mathsf{ZFC}$ without choice or replacement) in power.

That said, there are arguably concrete results which require serious axiomatic strength - this has been most intensively studied by Harvey Friedman (e.g. with Boolean relation theory). The relevant statements are fairly innocuous-seeming combinatorial principles. Now contra Friedman I don't actually find these statements particularly natural, and I think this is a common stance, but certainly his work points towards a real possibility that we may eventually find ourselves grappling with set-theoretic principles - at least up to consistency strength - in even very concrete questions.

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  • $\begingroup$ I accepted the answer since I think it basically clarifies things. I will note though that it seems weird that choice is considered the controversial axiom in ZFC, while it seems that replacement is less natural, less necessary, and more of jump (in terms of consistency and whatnot). I think I might have over shot with "concrete math" as the standard; choice has some nice results outside of set theory (every vector space has a basis, etc.) and though they aren't concrete, I would say they are more in the "mainstream" than stuff purely in set theory (1/2) $\endgroup$ – Keefer Rowan Sep 13 at 19:29
  • $\begingroup$ (2/2) This isn't a dig at set theory, it just seems like there are a lot of axioms one could choose to accept or not, and that will deeply affect what your resulting set theory is, and it seems like one should be basically ambivalent about what axioms to take. In something like linear algebra though, it comes up as a concrete field and then gets more abstract. Choice let's you take something that works in the concrete (finite dimensional) case and take it to the abstract case. It doesn't seem like replacement is doing as much, though that is probably just my lack of knowledge about set theory $\endgroup$ – Keefer Rowan Sep 13 at 19:32
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Disclaimer: I'm not qualified to answer this question. Having said that, this paper seems relevant; I especially found the following quote stimulating, although it only addresses whether replacement is "intuitive", rather than its relevance to "ordinary mathematics".

Replacement can be seen as a crucial bulwark of indifference to identification, in set theory and in modern mathematics generally. To describe a prominent example, several definitions of the real numbers as generated from the rational numbers have been put forward — in terms of the geometric continuum, Dedekind cuts, and Cauchy sequences — yet in mathematical practice there is indifference to actual identification with any particular objectification as one proceeds to work with the real numbers. In set theory, one opts for a particular representation for an ordered pair, for natural numbers, and so forth. What Replacement does is to allow for articulations that these representations are not necessary choices and to mediate generally among possible choices. Replacement is a corrective for the other axioms, which posit specific sets and subsets, by allowing for a fluid extensionalism.

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  • $\begingroup$ Thanks for the reference, though I don't really see how that's true at all. I mean I can construct isomorphisms of the different representations of $\mathbb R$ using set functions (which really just relies on specification on Cartesian product, which we can get from the power set). Replacement is about class functions, and as such doesn't seem at all necessary or even helpful in being indifferent between isomorphic representations. $\endgroup$ – Keefer Rowan Sep 13 at 18:50

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