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The Levy-Solovay theorem says that if $\kappa$ is measurable, then it remains measurable in the extension by a small forcing ($|\mathbb{P}|<\kappa$). Is still true if we replace $|\mathbb{P}|<\kappa$ with "$\mathbb{P}$ has the $\lambda-\textrm{c.c.}$ for some $\lambda<\kappa$"? Or even, can a c.c.c. forcing destroy measurability of $\kappa$?

This conjecture seems to check out with all the natural examples of forcings that I know.


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up vote 8 down vote accepted

The forcing to add $\kappa$ many Cohen reals is c.c.c and thereby preserves all cardinals but destroys the fact that $\kappa$ is even strongly inaccessible. (See for example Kunen "Set Theory: An Introduction to Independence Proofs" North-Holland, Ch. VII.)

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