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To complement Asaf's answer, let me point out that you don't need any large cardinals at all to get embeddings like this; they exist as long as there is an uncountable transitive model of ZFC+"there are no inaccessible cardinals". If $N$ is such an uncountable model we can get an embedding by taking a countable elementary substructure $X\prec N$ and ...


Note that in Asaf's answer, the elementary embedding $j: M\rightarrow N$ does not "live" (that is, is not definable in) $M$. By contrast, if we have an elementary embedding $j: M\rightarrow N$ which is definable in $M$ (from parameters in $M$), then $crit(j)$ is inaccessible, in fact measurable, in $M$: letting $\kappa=crit(j)$, we form the ultrafilter ...


No, not necessarily. Suppose $\kappa$ is measurable in $W$, let $M$ be $W[G]$ the model obtained after forcing with $\operatorname{Col}(\omega,<\kappa)$. If $j\colon W\to W'$ is an ultrapower embedding with critical point $\kappa$, then we can force over $M$ to obtain a model in which $j$ can be extended from $W$ to $M$. Another important example is the ...


Your argument is correct but the point is that you do not have to appeal to Godel's 2nd incompleteness theorem to prove this and Jech does that.

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