Archimedean group
The Archimedean property of any ordered algebraic structure, such as a linearly ordered group, and in particular of the system of real numbers, is the property of lacking (non-zero) infinitesimals. Such structures that lack infinitesimals are called Archimedean; those that possess infinitesimals are non-Archimedean. A number x would be infinitesimal if the inequality
continues to hold no matter how large the finite cardinal number n of terms in this sum.
The non-existence of nonzero infinitesimal real numbers follows from the least-upper-bound property of the real numbers, as follows. If nonzero infinitesimals exist, then the set of all of them has a least upper bound c. Either c is infinitesimal or it is not. If c is infinitesimal, then so is 2c, but that contradicts the fact that c is an upper bound of the set of all infinitesimals (unless c is 0, so that 2c is no bigger than c). If c is not infinitesimal, then neither is c/2, but that contradicts the fact that among all upper bounds, c is the least (unless c is 0, so that c/2 is no smaller than c).
Archimedes of Syracuse stated that for any two line segments, laying the shorter end-to-end only a finite number of times will always suffice to create a segment exceeding the longer of the two in length. Nonetheless, Archimedes used infinitesimals in mathematical arguments, although he denied that those were finished mathematical proofs.
Referenced By
Construction of real numbers | Construction of the real numbers | Constructions of the real numbers | List of mathematical topics | List of mathematical topics (A-C) | List of mathematics topics | Real Numbers | Real number
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