I'm sorry but this is just wrong. Since you seem to like Shapiro's book more than traditional set theory books let me quote from page 144 that the existence of an uncountable set is a theorem of ZFC: "Let C be the statement of Cantor's theorem. It entails that the powerset
of the collection of finite ordinals is not countable. Since C is a theorem of
first-order ZFC..."
Also this is not how the metatheory is understood in mathematics, not even in Shapiro's book, who dedicates two whole chapters to the metatheory
I'm sorry but this is just wrong. Since you seem to like Shapiro's book more than traditional set theory books let me quote from page 144 that the existence of an uncountable set is a theorem of ZFC: "Let C be the statement of Cantor's theorem. It entails that the powerset of the collection of finite ordinals is not countable. Since C is a theorem of first-order ZFC..."
Also this is not how the metatheory is understood in mathematics, not even in Shapiro's book, who dedicates two whole chapters to the metatheory