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The answer is no, one can have models of ZFC set theory with a definable truth predicate for first-order truth in $L$, but without having $0^\sharp$. One way to build such a model is like this. In Kelly-Morse KM set theory, you can prove the existence of a truth predicate for first-order truth for the whole universe $V$, and then by forcing you can code ...
Well assuming that $\lambda^A$ is always the first recursively admissible which is bigger than $\omega_1^A$, which I think should be true, I think my question is after all not so interesting: Either there is a largest recursively inaccessible smaller than $\omega_1^A$, in which case $\lambda^A$ is a successor in the recursively inaccessible ordinals, or \$\...