- #1

klabatiba

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Thank you.

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- Thread starter klabatiba
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- #1

klabatiba

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Thank you.

- #2

Preno

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Every infinite set X has a countable subset (this is a theorem of ZFC). We'll construct a function f from an initial set of ordinals to disjoint countably infinite subsets of X such that X its range. Let f(0) be any countably infinite subset of [itex]X = X_0[/itex]. Then define [itex]X_{i+1} = X_i \setminus f(i)[/itex], [itex]X_i = \displaystyle{ \cap_{j<i} X_j}[/itex] for a limit ordinal i and let [itex]f(i)[/itex] is a countably infinite subset of [itex]X_i[/itex] (the possibility of making these choices collectively relies on the Axiom of Choice). The sequence must eventually end with a null set - otherwise, the set X would be larger than all cardinals.

- #3

Dragonfall

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- #4

Elucidus

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--Elucidus

- #5

Preno

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