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This was a statement on wikipedia's page on finitely generated module:

Any module is a union of an increasing chain of finitely generated submodules.

But if we look at $\mathbb{R}$ as a $\mathbb{Q}$-module, a chain contains only countably many submodules, and each submodule only contains countably many elements - how can their union by $\mathbb{R}$? Am I getting some definitions wrong?

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    Following the suggestion of @JeremyRickard in his reply (+1), I have changed the sentence in the [Wikipedia page](https://en.wikipedia.org/wiki/Finitely_generated_module#Definition) to *Any module is the union of the directed set of its finitely generated submodules.* – 2017-02-04
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    @AndreasCaranti Thanks for doing that. One day I'll learn how to edit Wikipedia ... – 2017-02-04
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    @JeremyRickard, you're welcome, and thanks for pointing out where the problem was. Editing on the Wikipedia is absolutely straightforward, if you can edit here, you can edit there. I find Maths on the English version of the Wikipedia to be of excellent quality - after all, people like Terence Tao and Tim Gowers use it as a standard reference. But whenever one still finds a mistake, it is easy to correct it. – 2017-02-04

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Wikipedia is wrong here. I suspect that they've just misstated the fact (which is a fact) that every module is the union of a filtered set of finitely generated submodules.

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    Oops yes, I misdiagnosed the mistake that was occurring. It was actually this. – 2017-02-04