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    $\begingroup$ Why are people so afraid of inaccessibles? Sure they have higher consistency strength, thus "might be" inconsistent from the perspective of ZFC, but in the grand scheme of things from the set theorists' perspective, they are relatively innocuous. $\endgroup$ Commented Jun 23, 2022 at 12:20
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    $\begingroup$ @MonroeEskew yes, but most mathematicians are not set theorists for whom measurable cardinals and larger are no big deal. It's not about being "afraid", but wanting to know if the de facto standard axioms for maths are actually sufficient to do generic mathematics. $\endgroup$ Commented Jun 23, 2022 at 13:27
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    $\begingroup$ The reason for my question is to understand the motivations and philosophy. Although RM is "a thing", my impression is that most mathematicians are not Reverse but Forward, meaning they primarily care about proving new results and reaching a new understanding of a topic and knowing what's true, rather than investigating what axioms or fragments of schema are needed for a result. So when people start using universes to prove things but then get worried about going beyond ZFC, this seems like this is a relatively isolated concern rather than part of a pattern of being an RMer. $\endgroup$ Commented Jun 24, 2022 at 7:27
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    $\begingroup$ I would be less perplexed if the same people also were cataloguing exactly how complex of formulae they were using with Replacement and Comprehension, or whether the full Powerset axiom was needed, etc. To illustrate my point, it is rare to find a mathematician who is deeply interested in Harvey Friedman's RM result about Borel determinacy, but much more common to find one who wants to know a proof of Borel determinacy, or apply it. $\endgroup$ Commented Jun 24, 2022 at 7:36
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    $\begingroup$ @MonroeEskew perhaps you should ask a fresh MO question about all this, rather than me, on my question, which is deliberately rather narrow in scope :-) $\endgroup$ Commented Jun 24, 2022 at 12:42