A new approach to the theory of numbers "Mathematics considers natural numbers as points on a number line. I think it is a bit lopsided. Natural numbers are not just a series of points on a number line with an interval of 1. Let's try to build another series of intervals 3 ^ 0.5. ......" Andrey Shvets: The proof (?) of Fermat's theorem 
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Your discourse suffers a fatal flaw. Your argument is circular. You have defined numbers in general and natural numbers in particular using addition. So you need a definition of addition to have definition of numbers. But you need numbers to be able to have a definition of addition. Which argument is therefore shown to be circular and invalid. A suitable set of axioms for the natural numbers is N1 : 1 is a natural number as its successor number N2 : If a set of objects contains 1 and the successor of each of its members, then it contains every natural number N3 : There is no natural number for which 1 is a successor This set allows the deduction of the standard laws of arithmetic. Does yours? Note there is no need to introduce undefined concepts like number lines, intervals etc. 
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N1 : 1 is a natural number as is its successor number. Actually my post was meant as an incentive to explore this development more fully. Of course axioms follow definitions, so one and successor (if any) are defined first, But axioms are often shorter than definitions so I omitted all this at first presentation. 
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