| 11. | Every union ( = supremum ) of every countable set of countable ordinals is another countable ordinal.
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| 12. | Assuming the axiom of choice, the union of a countable set of countable sets is itself countable.
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| 13. | Assuming the axiom of choice, the union of a countable set of countable sets is itself countable.
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| 14. | Today, countable sets form the foundation of a branch of mathematics called " discrete mathematics ".
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| 15. | So we are talking about a countable union of countable sets, which is countable by the previous theorem.
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| 16. | The coefficients of its logarithm generate the free Lie algebra on a countable set of generators over the rationals.
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| 17. | The rational numbers are countable because the function given by is a surjection from the countable set to the rationals.
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| 18. | In a different language, you have been asked to prove that every ultrafilter on a countable set is principal.
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| 19. | Technically the simplest examples are when " X " is a countable set and ? is a discrete measure.
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| 20. | Every countable set is a strong measure set, and so is every union of countably many strong measure zero sets.
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