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What allows us to do the construction found in Cantor's diagonal argument? Is there an axiom we must adopt to allow for such infinite constructions?
Are you asking about math without induction, like Robinson arithmetic?What I mean by "infinite" construction is that we are allowed to select the next digit of the number ad infinitum - we allow ourselves to say that the construction "ends". Is this notion independent of others in mathematics; i.e. if we conduct mathematics without its use, do we get contradictions?