While reading Lillian Lieber’s book on infinity, I came across an astonishing example of infinite set (on pp. 207). Let’s call the property of existence of a rational number between given two rational number to be “beauty” (a random word introduced by me to make arguments clearer).

The set of rational numbers between 0 and 1 are arranged in ascending order of magnitude, and all of them are coloured **blue**. This is clearly a *beautiful *set. Then another another set of rational numbers between 0 and 1 is taken and arrange in ascending order of magnitude, but all of them are coloured **red**. This is also a *beautiful* set. Now, put these two sets together in such a way that each **blue** number is immediately followed by the corresponding **red** number. For example, **1/2** is immediately followed by **1/2 **etc. It appears that if we interlace two *beautiful* sets, the resulting set should be even more *beautiful*. But since each **blue** number has an immediate successor, namely the corresponding **red** number, so that between these two we can’t find even a single other rational number, **red** or **blue**, the resulting set is NOT *beautiful*.

The set created above is called **Huntington’s Red-Blue set.** It is an ingenious invention, where two *beautiful* sets combined together lead to loss of *beauty*. For more details, read the original paper:

Huntington, Edward V. “The Continuum as A Type of Order: An Exposition of the Modern Theory.” Annals of Mathematics, Second Series, 7, no. 1 (1905): 15-43. doi:10.2307/1967192.