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3-strategy absolute logic admissible set Ai-definable approximation atomic sentences axiom branch choose Claim constant symbols construction defined denote disjoint element elementary submodel enumeration example existential quantification exists expansion game extensional finite first-order logic first-order relative following conditions hold formula G Str(r game G game quantifier game sentence implies isomorphic language rule leaf Lemma len(w Logics absolute relative Loow maximal element Moreover Mostowski collapsing moves need to show ordinal P-name p.r. mapping pair phantom game pick Player 3 wins predicate primitive recursive mapping Proof rank(x regular cardinal relative to KP relative to KP+Inf result semantic game sentence f sequence set-theoretical similarly standard set theory structure 971 sublogic subset Suppose syntax tree TC(r Theorem transitive set trivial urelements VK(x vocabulary well-founded tree winning strategy wins a play wins EG wins EG(97t wins the play X-expander