Lets consider a line of length 1m.

If space is continuous, it means there is an infinity of points in this line.

Now, size of each point=1/infinity=0

 

This implies that our line is of zero size because its constituents are all of zero size.

 

How to resolve this contradiction?

How to resolve this contradiction?

 

Don't try and use infinities, or division by zero, in arithmetic.

1/infinity = infinitesimal (approaching zero but not zero)

The cardinality of the number of points in the interval is that of the continuum. Summation rule doesn't work in this case. Contrast with the total length of all the rational points (countable). Here the total "length" (measure) is 0.

I am of the opinion that it is the "length of all definable points (countable)". Therefore the "length" once the limit has reached "undefined value" is then zero. That is the nature of zero is undefined value not the absence of value. Thus the continuum.