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    Countable linear ordering isomorphic to subset of rationals

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    Show that every countable linear ordering is isomorphic to some subset of the rationals under their usual order, but that omega_1 (the least uncountable ordinal) with its well order is not isomorphic to any set of reals under their usual ordering. The solution may use any algebraic facts about the reals.

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    https://brainmass.com/math/linear-transformation/countable-linear-ordering-isomorphic-subset-rationals-301414

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    Solution Summary

    Given any countable linear ordering, it is shown how to define an order isomorphism of that linear ordering with some subset of the rationals (under the usual ordering of the rationals). Also, it is proved that omega_1 with its well order is not isomorphic to any set of reals under their usual ordering.

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