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# Extending Woodin’s definition of large cardinal axiom to all axioms

Let $$\phi(\kappa)$$ be a local large cardinal axiom if and only if it is of the form $$\exists\kappa_0…\kappa_n(\phi(\kappa,\kappa_0…))\land\forall i\le n(\kappa_i\in C^{(\alpha_i)})$$, such that $$V_{\kappa_i}\prec_{\alpha_{i+1}} V_{\kappa_{i+1}}$$.