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The Square Root and the Relations of Order

1906
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Transactions of the American Mathematical Society
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One of the most obvious discriminations between positive and negative numbers is that the former possess square roots in the field of reals while the latter do not. This distinction, however, has not yet been used in any of the current systems of postulates. On the contrary, an order relation, <, is usually introduced as an undefined symbol. Though the existence of a square root cannot be deduced from order relations without the use of a continuity assumption of some sort, it turns out to be

doi:10.2307/1986433
fatcat:4qfpyegiqvek3gfqaqz2it4as4