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I'm looking for a reference for the fact that if $\kappa$ is supercompact and $Q$ is a poset of cardinality less than $\kappa$, then $V^Q\models \check{\kappa} \ \text{is supercompact}$.

Apparently there's one in Jech, but I can't find it. Any pointers?

Thanks.

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  • $\begingroup$ I changed the notation to $\check\kappa$, which is the one used by Jech (and in most modern texts about forcing, to my knowledge). $\endgroup$ – Asaf Karagila Apr 11 '13 at 22:33
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This is Theorem 21.2, p. $390$ at the bottom.

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  • $\begingroup$ This seems to be on page 390 in the Third Millennium edition. $\endgroup$ – Zhen Lin Apr 11 '13 at 20:12
  • $\begingroup$ @Zhen Lin: Yes, thank you. I was looking at the wrong page number. :-) $\endgroup$ – Asaf Karagila Apr 11 '13 at 20:38
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There is a standard approach to these results, that dates back to Lévy and Solovay. The point is that, for any $\lambda$, given an embedding witnessing $\lambda$-supercompactness of $\kappa$, there is a canonical way of lifting the embedding to a $\lambda$-supercompact embedding in $V^Q$ (using simply that $Q$ is "small"). Lévy and Solovay present this in terms of extending ultrafilters in the ground model to ultrafilters in $V^Q$ in canonical ways. Their paper is

Azriel Lévy, Robert M. Solovay. Measurable cardinals and the continuum hypothesis, Israel J. Math. 5, (1967), 234–248. MR0224458 (37 #57).

Besides the Lévy-Solovay paper or the argument in Jech's book, a modern presentation can be seen, for example, in Cummings's paper in the Handbook. There are also several results by Hamkins and others extending these techniques to other contexts, see for example

Joel David Hamkins, W. Hugh Woodin. Small forcing creates neither strong nor Woodin cardinals. Proc. Amer. Math. Soc. 128 (10), (2000), 3025–3029. MR1664390 (2000m:03121).

Very briefly, given $j:V\to M$, define $\hat j:V[G]\to M[G]$ by $\hat j(\dot x_G)=j(\dot x)_G$. Standard arguments verify that this is well-defined, elementary, etc. Note that here we are using that $Q\in M$. If $Q$ has size $\kappa$ or larger, there are cases where the extension is still possible, but we need to be more careful. For example, $j(Q)$ is in general strictly larger than $Q$, so the existence of a $j(Q)$-generic over $M$ is no longer for free. The study of this situation leads to what we now call Silver's master conditions.

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