My argument is about the rigor of theorems. ZFC may be incomplete, but every existing theorem that builds up the entirety of modern mathematics is (in theory, a

My argument is about the rigor of theorems. ZFC may be incomplete, but every existing theorem that builds up the entirety of modern mathematics is (in theory, admittedly) built upon a rigorous chain of truth, traced back to those axioms and definitions. Most mathematicians who don't study metamathematics or philosophy of math have no reason to ever think about formalism or computability. Godel's completeness theorem stipulates that there are self-contained consistent systems within the language wherein all of their nice proofs are true and universal, and where plenty of truth still needs to be discovered.