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How can the Grothendieck-Tarski axiom seen to be true in the cumulative hierarchy of sets?

What are intuitions that would convince us that this axiom is true?

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  • $\begingroup$ Why downvote? I would be happy about any suggestion how to improve this question. $\endgroup$
    – user99371
    Commented Oct 6, 2016 at 17:14
  • $\begingroup$ You might get better votes by including a link, and asking: "Does the G-T axiom follow from our intuitions abut the cumulative hierarchy of sets, and if so how? What intuitions around the axiom or what ways of using the axiom might persuade us to adopt it as part of our usual axioms for set theory?" $\endgroup$
    – user44143
    Commented Oct 8, 2016 at 3:51

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My intuition supporting a belief in a proper class of (strongly) inaccessible cardinals (equivalently, Grothendieck universes) is that, since the cumulative hierarchy is intended to continue "forever", we should not expect to see some phenomenon in the full universe $V$ that didn't occur at some earlier partial universe $V_\kappa$. We should continue the hierarchy beyond that occurrence. Since $V$ satisfies full, second-order replacement (and power set), I'd expect some $V_\kappa$ to do likewise, which makes $\kappa$ inaccessible. Similarly, if there were only a set of inaccessible cardinals, say of order-type $\alpha$, then $V$ would be "describable" as the $\alpha$-th place where full second-order replacement and power set hold, which seems suspicious when $\alpha$ is some tiny (i.e.,set-sized) order-type.

Bernays wrote a very nice paper, "Zur Frage der Unendlichkeitsschemata in der Axiomatischen Mengenlehre", about getting large-cardinal consequences (far stronger than a proper class of inaccessibles) from reflection principles.

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