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No, not at all. But it does tell you something about $M$. It tells you that it knows about sufficiently many of the subsets of $X$. So if, for example, $M=V_\kappa$ for a worldly cardinal which is not inaccessible, then this is also true. For obvious reasons: if $X\in M$ then $\mathcal P(X)\in M$, so every possible subset is in $M$ and $M$ and $V$ agree ...


The term "large cardinal" doesn't have an agreed up, concrete definition. When we say that $\kappa$ is a large cardinal we might mean that $\kappa$ is inaccessible and has additional properties; or we might mean that it has properties which imply the consistency of large cardinals. Some people would refer to the former as large cardinals, and the latter as ...

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