ない and ません nuance

The black board example has ません, which looks like ’ unintentionally not’ to me.
The people’s names example has ない, which also looks like ’ unintentionally not’ to me.

So when do you actually use ない and when ません?
Does the nuance matter much, or is it actually interchangeable?

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They should probably edit that lesson since it does look a bit confusing.

The only difference between ない and ません in terms of nuance is formality.

The reason those examples are confusing is because 見える means “to be visible/to be able to see” but 覚える just means “to remember” (not “to be able to remember”, although it can be implied as in the example). 見えない・見えません literally means “[someone] can’t see [something]” whilst 覚えない・覚えません means “[someone ] doesn’t remember [something]”. Other nuances are possible based on context/verb and that is what the example with 覚える is trying to show. The sentence with 見える is exactly as you would expect it to be with no added layer of nuance. The verbs themselves are the cause of the confusion here - 見える probably shouldn’t be used in this early lesson.

I’ve tried to keep this as short and non-technical as possible but if what I said doesn’t make sense then feel free to ask further questions.

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First of all, thanks for your time! I appreciate the help. I don’t entirely understand it yet, but I think/hope I’m starting to get it.

So while 見えません has only one meaning (I can’t see … (because it’s not visible for me)), the 覚えない could have multiple meanings depending on the sentence (I don’t want to remember, I can’t remember, I forgot)?

Should I just remember that ません is more formal than ない and just ignore the cant/wont/dont parts for now?
If I should try to understand the can’t/won’t/don’t, which ones of those threer are ません and which are ない?

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The main thing you need to remember is that ない・ません both negate the verb they’re attached to. You can think of it as the negative form of the verb (although it is a tiny tiny bit more complex than that). The only difference between them is formality. There is no other difference in nuance between the two. For the moment it is probably safe to forget about any other nuances for ない・ません. The nuances part of the lesson is there to help as Japanese tense is a bit different from English tense but don’t worry too much about it for the moment. Choosing ない vs ません doesn’t have an effect on that sort of nuance. As I said before, the reason those examples are confusing are because of the verb choice. There is actually a lesson on 見える and it can be a cause of confusion for beginners.

For the moment just remember that ない・ません are for the negative form of a verb (not be able to see, not remember, not do, etc) and if/when you progress a bit further or you don’t quite understand what ない or ません is doing in a sentence then revisit the lesson and see if the part about nuances helps.

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Noted, thank you!

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見る → to see (base)
見ない → to not see (negative)
見えない → cannot see (negative potential)

覚える → to remember (base)
覚えない → to not remember (negative)
覚えられない → cannot remember (negative potential)

The confusing part I think is you are mixing up the potential form of 見る with the base form of 覚える they are totally different words, with different conjugations/inflections.