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If $0^{\sharp}$ exists, then every uncountabel uncountable cardinal in $V$ is as large as it can be in $L$. |
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If $0^{\sharp}$ exists, then every uncountabel cardinal in $V$ is as large as it can be in $L$.According to Wikipedia, if $0^{\sharp}$ exists, then every uncountable cardinal in $V$ satisfies every large cardinal property in $L$ that can be realized in $L$, e.g. weak compactness, total ineffability, etc. It's easy enough to see why every uncountable in $V$ will be inaccessible, or even Mahlo, in $L$. How can one see that some of the slightly larger large cardinal properties (e.g. weak compactness, total ineffability, etc.) are satisfied in $L$ by the uncountable cardinals in $V$? Is there a good reference for some of these results?
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