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In that page there presented several definitions of Standard model of ZF.

Now the page was speaking of ZF which contains an axiom of Regularity of course, and so it appears that definitions 2,3,4 of standard models of ZF all would imply that the model is externally well founded.

Now what would constitute a definition of a standard model of ZF-Regularity?

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    $\begingroup$ I don't think it has an established meaning in that case; I would certainly avoid using it without defining it explicitly first. (I vaguely recall the term "ordinals-standard" being used to describe a model with well-founded ordinals, and I think that term is clear enough to use, if not common.) $\endgroup$ – Noah Schweber Jun 16 '18 at 17:00
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I don't recall ever hearing such term being used. One reason might be that most of the uses of ZF-Regularity that I have seen were to produce choiceless constructions à la Fraenkel–Mostowski–Specker models. So you define an inner model, rather than an extension. In that case, well-foundedness is less interesting (compared to forcing, that is).

If I were pressed to the wall to define what a standard model of ZF-Regularity should be, I would say a model whose ordinals are well-founded. That way, when you move to the inner model of ZF given by the von Neumann hierarchy, you get a standard model of ZF.

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