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5

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 $V_\kappa$ ...


2

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 ...



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