If the Fundamental Theorem of Arithmetic requires proof, then the coNP certificate argument embeds an unacknowledged mathematical assumption that could fail in alternative number-theoretic frameworks.
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coNP certificate(as used in computational complexity theory)
A piece of evidence or information in computer science that can quickly verify a negative answer to a problem (like proving something is NOT true), similar to how a receipt proves you bought something.
knowledge(Distinguished from mere true belief, which may be the product of indoctrination and need not exercise deliberative capacities.)
Justified true belief — true belief that has been arrived at through the exercise of deliberative capacities, including comparison of and deliberation among alternatives.
proof(Frege's formal system; the definition still used by logicians today)
Any finite sequence of statements such that each statement is either an axiom of the formal system or follows from previous members of the sequence by a valid rule of inference.