I Can the area be understood as the "number of points"?

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WWGD

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The only model of this I can think of is that of n-squares, a line broken into dx_i ,lengths a volume broken into cubes ##dxdyf(x_k,y_k) ## assuming converfence,,etc. Volume is the number of such units/scaled cubes. Edit I remember a contest a while back on counting the number of gumballs in a jar. This was , essentially the volume in gumball units instead of cc.
 
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WWGD

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A problem is that both, e.g., a square of side 1, side 2, any finite size will have areas ##1,4,.., n^2 ## respectively but each will have uncountably-many points, so measure and cardinality are different concepts.
 
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I recommend you to look into the field called "nonstandard analysis." They invented clever ways to think about infinities. Perhaps you can even define "the number of points in a line" using their methods? Not sure.

Eugene.
 

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