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