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Robert van Wesep is writing a massive "Foundations of Mathematics" book. It contains several chapters on forcing and large cardinals however due to its current content the materials are not too advanced there.


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For the first question, the elementary embedding is "jumping" at $\kappa$. Namely, the universe of set theory is the same all the way below $\kappa$, and then it get stretched up, somehow. This is the meaning that $\pi(\kappa)>\kappa$. But at the same time, $M$ is a transitive class. So if $\pi(\kappa)>\kappa$, then $\kappa\in M$. The key, I think, ...


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You can find the proof in any set theory book that covers measurable cardinals. For example Jech's "Set Theory" and Kanamori's "The higher infinite". First the idea is to show that the measure is not $\sigma$-additive, but rather $\kappa$-additive. Namely, the union of less than $\kappa$ sets of measure $0$ is still measure $0$. If this is not true, then ...



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