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Countable set
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==Total orders== Countable sets can be [[total order|totally ordered]] in various ways, for example: *[[Well-order]]s (see also [[ordinal number]]): **The usual order of natural numbers (0, 1, 2, 3, 4, 5, ...) **The integers in the order (0, 1, 2, 3, ...; β1, β2, β3, ...) *Other (''not'' well orders): **The usual order of integers (..., β3, β2, β1, 0, 1, 2, 3, ...) **The usual order of rational numbers (Cannot be explicitly written as an ordered list!) In both examples of well orders here, any subset has a ''least element''; and in both examples of non-well orders, ''some'' subsets do not have a ''least element''. This is the key definition that determines whether a total order is also a well order.
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