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As I study through inaccessible cardinals, I find many examples that show some cardinal being both weakly inaccessible and strongly inaccessible.

So, can anyone show me the example of weakly inaccessible cardinal that is not strongly inaccessible?

Thanks.

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1 Answer

up vote 0 down vote accepted

Suppose that $\kappa$ is a strongly inaccessible cardinal, now force the continuum to be $\kappa^+$.

In the generic extension $\kappa$ is still a limit cardinal and still regular. However it is not a strong limit, so it is just weakly inaccessible.

Note, that if we have a weakly inaccessible cardinal then in some inner model it is strongly inaccessible, so this is really all the examples you can ask for which do not involve even stronger large cardinals.

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I'm a bit puzzled about the tagging of the question. But I guess I'll never understand the distinction between (elementary-set-theory) and (set-theory) :) –  t.b. May 7 '12 at 8:18
    
@t.b. While it is about a rather advance topic, the question itself is quite elementary. After all I could write a complete answer on the bus! –  Asaf Karagila May 7 '12 at 8:19
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