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This is also known as "$\rm Ord$ is Mahlo". Namely, every class which is closed and unbounded has a regular cardinal in it. It is certainly weaker than a Mahlo cardinal: If $\kappa$ is Mahlo, then $\langle V_\kappa,\in\rangle$ is a model of this theory. But this is weaker. We only need every definable class to have a regular cardinal, and in $V_\kappa$ ...
Let $\kappa$ be a Mahlo cardinal in $V$, i.e. let $\kappa$ be inaccessible such that $S:= \{ \alpha \in \kappa \mid \alpha \text{ is regular} \}$ is stationary in $\kappa$. First, note that $L \models \kappa \text{ is inaccessible}$. Indeed, if $L \models \kappa \text{ is not a cardinal}$, then there is some $\mu < \kappa$ and some $f \in L$ such that $L ... 3 End-extensions are not at all trivial to obtain. First of all, note that if$X$is an end-extension of$V_\kappa$, then$X$is a transitive model of$\sf ZFC$of height$>\kappa$. This is not to be taken lightly. Suppose$V=L$and there exists a single inaccessible cardinal$\kappa$. Moreover assume that$\sf\operatorname{Con}(ZFC+I)$is false in our ... 2 The functional calculus of first-order is FOL. For a mathematical logic textbook of early '60s, see: Alonzo Church, Introduction to Mathematical Logic (1956), page 281: We shall speak of the pure functional calculus of first order with equality if the primitive symbols include all propositional and functional variables and no functional constants ... 2 The question is what do you mean by large cardinals and what do you mean by useful? For example, it is consistent that there exists a set$A$such that the cofinite filter on$A\$ is an ultrafilter. We can even show that such ultrafilter is closed under intersection of ordinal-indexed sequences (mainly because under this assumption every such sequence is ...