[FOM] Tarski's Axiom of Inaccessibles

[Gregory Taylor]

Zermelo, in his 1930 paper concerning models of the theory of urelements and sets, urged ``the existence of an unbounded sequence of [strongly inaccessible cardinals] as a new axiom for the metatheory of sets.'' See p. 429 of Zermelo, Collected Works, vol. 1, Ebbinghaus and Kanamori, eds., 2010.

It seems that Zermelo's remark by itself implies no more than an $\omega$-sequence of inaccessibles.  How is that to be compared with what Tarski's Axiom of Inaccessibles requires?

We think we see an argument showing that, in any model of ZF + Tarski's Axiom that does contain a hyperinaccessible of the first kind (alternately, a 1-inaccessible), the strongly inaccessibles number at least $\kappa$ with $\kappa$ the least hyperinaccessible of the first kind.  Is this correct?  (Thanks in advance.)

Gregory Taylor (subscriber)
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