Consider the interval (0, 1). This set of real numbers contains no smallest element. It is not true that all sets of real numbers have or should have a least

Consider the interval (0, 1). This set of real numbers contains no smallest element. It is not true that all sets of real numbers have or should have a least element. It is a property of the standard model of the nonzero integers that all subsets have a least element. It should be noted that there are models of ZFC in which each real number is definable. You may be interested in reading the top answer on this mathoverflow.net question: https://mathoverflow.net/questions/44102/is-the-analysis-as-taught-in-universities-in-fact-the-analysis-of-definable-numb https://mathoverflow.net/questions/44102/is-the-analysis-as-...