Perhaps your intuition changes if you think about the line (in a plane) coinciding with the x-axis. Don't you think it is reasonable to define that line as the set of points (in the plane) having y=0? Also, should that line not be identical (isomorphic) to the set of real numbers - which is just a set of numbers? And should not all other lines in the plane be identical (isomorphic) to the first line?
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Perhaps your intuition changes if you think about the line (in a plane) coinciding with the x-axis. Don't you think it is reasonable to define that line as the set of points (in the plane) having y=0? Also, should that line not be identical (isomorphic) to the set of real numbers - which is just a set of numbers? And should not all other lines in the plane be identical (isomorphic) to the first line?