This is a nice question which I think will interest many people. I recently wrote a short article on this very topic. read more
Real numbers are not infinite, because they are just a concept real numbers is just two words, a number will become real when you start writing it, the real set of real numbers are the numbers that has been written which is finite. read more
There are also plenty of transfinite numbers less than [math]\omega[/math] but larger than any [math]n\in\mathbb N[/math], including [math]\omega-1,\frac{\omega}{3},\sqrt{\omega}[/math], and so on. There is no single concept of “infinity” and the more general concept of transfinite numbers is a rich and beautiful area to explore. read more
Continuity and the Real Numbers. Consider the set of all irrational numbers, a subset of the real numbers excluding any whole or rational numbers. The set of all irrational numbers is not denumerable. This means that we finally have an infinite set whose cardinality is not . This can be shown by using another diagonalization technique. read more