We define the smallest positive real number as the number which is explicitly greater than zero and yet less than all other positive real numbers except itself.
The smallest positive real number, if exists, implies the existence of the second greater positive real number after it, which subtracts the smallest positive real number equals the smallest positive real number. The difference between the second greater positive real number and smallest positive real number could not be any other positive real number greater than the smallest positive real number, otherwise there must be a number with the magnitude of twice the smallest positive real number between the smallest positive real number and the second greater positive real number, which contradicts to the definition of the second greater positive real number, that is there is no number between it and the the smallest positive real number. Follow the same meaning, one could define the third greater positive real number which subtracts the second greater positive real number equals the smallest positive real number, then the 4th greater positive real number, 5th, ...this would finally make the set of positive real number countable, while Cantor already proved the set of positive real number is uncountable using the diagonal argument.
The smallest positive real number, if exists, also implies the existence of the indivisible unit. The smallest positive real number is not legitimate to divide, otherwise one would get numbers less than the smallest positive real number, which contradicts to the definition of the smallest positive real number. N.B. the conclusion is conducted out in terms of assuming the existence of the smallest positive real number. In such a number system, the ultimate unit of measurement would be the smallest positive real number, based on such idea one would be eventually led to the world of atomism.
The Greek scientist Democritus (about 460– 380 B.C.) apparently considered solids as "sums" of a tremendous number of extremely small "indivisible" atoms (don't get confused with that in chemistry). Democritus held that his atoms, being not only very small but the smallest possible particles of matter, were not only too small to be divided physically but also logically indivisible. In such a system, the ultimate unit of measurement would be the size of an atom.
Obviously, the atom unit size is equal to the smallest positive real number. However, Euclidean geometry, in particular, the Pythagorean theorem denies the existence of such indivisible atom size, therefore denied the existence of the indivisible unit-the smallest positive real number.
Consider any geometrical figures (e.g., squares, triangles, etc.) with line segments as sides from atomism, then the length of each side will be measured in atoms, and each side will be assigned an integer as its measure. (Each side will be n atomic units long, where n is a positive integer.) Now consider an isosceles right triangle with side composed of 100 atoms, how many atoms its hypotenuse includes? Using the Pythagorean theorem \(\sqrt{100^2 + 100^2}=\sqrt{2\times 100^2}=100\sqrt{2}\), the hypotenuse includes \(100\sqrt{2}\) atoms, while \(100\sqrt{2}\) is not a whole number. And notice that this is true irrespective of the size of the side, so the situation does not change if we suppose that the side of the isosceles right triangle are composed of a very large number of very small “space atoms”. Even if the sides are billions of atoms long, the length of the hypotenuse will still be an irrational number of such atoms. Let \(a\), the whole number of atoms in the side of a isosceles right triangle, be as large as you like, and let \(c\) be the number of atoms in the hypotenuse; \(c\) will still be an irrational number, for \(c = a\sqrt{2}\) . This means that there is no integer \(c\) such that the hypotenuse of an isosceles right triangle is \(c\) space atoms long if its side is some integer \(n\) space atoms long. To put it another way, the diagonal and the side of a square cannot both be measured atomistically.
In one word, the smallest positive real number doesn't exist !