If number is greater than zero, smaller number added to itself enough times can equal the number {axiom of Archimedes, number} {Archimedes axiom, number}.
Number-theory axioms {calculation axiom} can be about calculation, such as associative law, commutative law, and distributive law.
Adding anything to real numbers makes all preceding axioms untrue, so real-number system cannot be larger {axiom of completeness} {completeness axiom}.
Number-theory axioms {connection axiom} can be about operations, such as closure, uniqueness, and identity.
Number-theory axioms {continuity axiom} can be about continuity, such as axiom of Archimedes and axiom of completeness.
Number-theory axioms {order axiom} can be about order, such as transitive law. For two different numbers, one number is greater and one number is smaller. If first number is greater than second number, then first number plus third number is greater than second number plus third number. If first number is greater than second number, then first number times third number is greater than second number times third number.
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Date Modified: 2022.0225