"The girl likes to eat cheesecake."
Translation:Του κοριτσιού του αρέσει να τρώει τσίζκεικ.
του κοριτσιού is article + noun (both in the genitive case). The next του is the personal pronoun, in the genitive case.
A literal translation might be "To the girl, to her it is pleasing that she eats cheesecake"; a slightly less literal one, "The girl, she likes to eat cheesecake".
I'm not sure whether the second του is necessary but it sounds better to me that way.
Or you could rearrange the sentence and use a preposition + accusative case instead of the genitive case: Αρέσει στο κορίτσι να τρώει τσίζκεϊκ.
Ancient Greek used to have a dative case; as in some other European languages (e.g. Latin or German), the dative was used for recipients of giving and also for more metaphorical "recipients" of feelings as here (she likes the cheesecake = the cheesecake gives 'good feelings' to her).
When the dative case was lost, the genitive case took over some of these uses in modern Greek -- especially in personal pronouns.
With nouns, I think it's more common to use σε + accusative to render the old dative, but plain genitive is also often possible.
So the genitive case is not only about possession, but can also be used for a recipient, and that's more or less what is happening in this sentence.
(There are also other uses of the genitive case, e.g. after certain prepositions.)
Can one assume that the dative case markers in Modern Greek happen to be exactly the same as the genitive case markers? Thinking of the "genitive" markers like μου, σου, του etc. as associtated with indirect objects (as in "μου έδωσε το βιβλίο") actually appears rather strange to me.
There are no dative case markers in Modern Greek because Modern Greek has no dative case, except in some fossilised expressions retained from Ancient Greek, e.g. Δόξα τω Θεώ "Praise (be) to God" or συν Αθηνά και χείρα κίνει "God helps those who help themselves" or τοις εκατό "percent".
Sort of, but no. Modern Greek has no dative case, but that's because early sound changes rendered the dative indistinguishable from the accusative (καλῷ and καλόν both sound like "kaló"), so the dative functions got redistributed to other cases. In some dialects it merged with the accusative, but in standard modern Greek its functions were taken over by the genitive, probably because the accusative was overburdened already.
I was about to answer "yes", but that's not necessarily the case. It's also ok if the second "του" is omitted (not as common, maybe a bit less natural as well, but it's ok).
"Του κοριτσιού αρέσει να τρώει τσίζκεϊκ=Του κοριτσιού του αρέσει να τρώει τσίζκεϊκ=Στο κορίτσι αρέσει να τρώει τσίζκεϊκ"
I had never wondered about such a "rule", but, if it proves helpful to you, then stick to it ;)
Both should be accepted as correct. The "like + ing" and "like to + infinitive" forms have small differences in meaning, but both could apply here.
Check this out for more information and relevant examples: https://www.grammaring.com/to-infinitive-or-gerund-like-hate-prefer-cant-bear
Indeed, that is correct!
A sentence like “I want to eat/have eggs today” would be translated as «Θέλω να φάω αυγά σήμερα», because we’re talking about an one-off action in the future.
On the contrary, a sentence like “I like eating fruit in the summer” would be translated as «Μου αρέσει να τρώω φρούτα το καλοκαίρι», because this time we’re talking about a habit, a continuous action.
I hope this helps :)