"My wife is not eating cheese."
Translation:Meine Frau isst keinen Käse.
Nicht = not and keine is not any.
My german teacher explained it easiest like this: Keine applies to a noun. Ich habe kein Käse - I have no cheese or I have no cheese. Ich weiß nicht - I do not know
As you can see no noun in the second sentence as it applies to negation. Don't know if this is 100% correct, but makes sense to me.
Interesting observation Dom_Hyde. I never thought of that. Even when used with "some" though I still think of it as singular. I wouldn't say "Some cheese are yellow and some are white". Same with liquids. I would say "do you want some water?" But I wouldn't consider water plural - I wouldn't say "Some water have too much chlorine" I would still use the singular form of the verb. Nonetheless, with respect to the German, I think we are supposed to learn that it's "der Käse".