# Is every model of MK (Morse-Kelley-Set-Theory) well-founded?

On page 2 in "The Hyperuniverse Project and Maximality" is written:

The models $$\mathcal{M}$$ of MK are of the form $$\langle M, \in, \mathcal{C} \rangle$$, where $$M$$ is transitive model of ZFC, $$\mathcal{C}$$ the family of classes of $$\mathcal{M}$$ (i.e. every element of $$\mathcal{C}$$ is a subset of $$M$$) and $$\in$$ is the standard $$\in$$ relation.

Why can we assume that $$M$$ is transitive? If we knew, that every model of MK is well-founded, we could use the Mostowski collapse lemma. But is this the case?

Remark: In the book MK is formulated in a two-sorted version (have a look at the link).

• By "model" read instead "standard model", these are the ones we typically want to consider. (But no, arbitrary models need not be well-founded.) – Andrés E. Caicedo Nov 24 '18 at 13:31
• If there are set models at all, you can use compactness to get illfounded ones. – Andrés E. Caicedo Nov 24 '18 at 17:38
• Or you can start with an illfounded model of ZFC+"there is an inaccessible", and show that it contains a model of MK, and that this model is illfounded (from the outside). :-) – Andrés E. Caicedo Nov 24 '18 at 17:41
• Should I delete this question? – Popov Florino Nov 24 '18 at 17:51
• I would suggest instead that you post an answer yourself explaining the situation, so that people who may have a similar question in the future may find this post. – Andrés E. Caicedo Nov 24 '18 at 20:08