Learn Hardcore Greek: Δεν μπορώ να κοιμηθώ, γιατί βήχω όλο το βράδυ. - I can't sleep, because I cough all night.

Because Modern Greek does not use an infinitive the way English does. After verbs like μπορώ (I can), Greek normally uses να + a verb form called the subjunctive.

So:

μπορώ να κοιμηθώ = I can fall asleep / I can get to sleep
literally, something like I can that I fall asleep

That sounds strange in English, but it is completely normal in Greek.

Why is it κοιμηθώ and not κοιμάμαι?

This is an aspect question.

κοιμάμαι = I sleep / I am sleeping
κοιμηθώ = I fall asleep / I get to sleep

In this sentence, the idea is not really I cannot be in a sleeping state, but rather I cannot manage to fall asleep. That is why Greek uses κοιμηθώ, the aorist subjunctive form.

So:

Δεν μπορώ να κοιμηθώ = I can’t get to sleep
Δεν μπορώ να κοιμάμαι would sound odd in most contexts, unless you meant something more like I’m not able to be sleeping or I can’t sleep as a general state, which is less natural here.

Why is δεν used here?