What's the difference between these forms?
Can't we use 그 이름은 살것입니다 instead of 그 이름은 살아있을것입니다?
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The form Verb + 아/어/여 있다 is used to indicate the continuance of a state or existence. When you live, you remain living. When you stand, you remain standing.
It is used when a status has commenced and is still continuing on. Usually, this form is used in conjunction with passive voice:
앉아 있다. To be sitting.
누워 있어요. To be lying down.
쌓여 있어요. To be covered.
남아 있어요. To (be) remain(ing).
This form is not to be confused with Verb + 고 있다, which requires an object (so the verb must be transitive). For example, let's look at the difference between "살고 있다" and "살아 있다."
If you say "여동생이 아직 살아 있어요", this means "My little sister is still alive." This is using the definition of 살다 meaning "to be alive, to be living (often in a biological sense)."
On the other hand, if you want to say you live somewhere, you would use 살고 있다. We could for example say "영국에서 살고 있지만, 난 한국인이다." (I am living in England, but I am Korean). This is using the meaning of 살고 있다 to mean "to live (somewhere), to make one's residence (in a place)." This usage requires an object (i.e. where do you live?).