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The diagonal argument is a powerful technique most famously used by mathematician Georg Cantor to show that certain sets are not countable, or 'denumerable'. The essence of the argument is to assume that you have listed all elements of a set and then, by constructing a new element that differs from each listed element in at least one aspect, you show that your list cannot be complete.

To envision this, imagine we have a list of real numbers between 0 and 1, written in their decimal expansion form. By changing the diagonal elements of this list—to be precise, changing the nth digit of the nth number—we can always create a number not on the list. The key here is the construction of a new number by ensuring that it differs from each of the listed numbers at some decimal place.

The diagonal argument is elegant because it doesn't rely on the size or specifics of the list, which makes it a universal refutation of any attempt to enumerate real numbers in an interval or set. It applies not just to the interval (0,1), but to any infinite set of real numbers.

Enumeration, in the context of set theory, refers to a process by which elements of a set are listed or 'counted'. For a set to be denumerable, or countable, it must be possible to establish a one-to-one correspondence between the set's elements and the natural numbers (1, 2, 3, ...).

In practice, this means that if you can pair each element of your set with a distinct natural number without missing any element or natural number, your set is denumerable. For example, the set of all natural numbers is clearly denumerable since each element is it’s own pair. However, when applying such enumeration to the set of real numbers within an interval, as in our exercise, complications arise. Unlike natural numbers, real numbers cannot be listed in such a neat, simple pairing with natural numbers due to their uncountably infinite nature.

Apart from natural numbers, rational numbers too are denumerable, as there is a systematic way to list all fractions. However, the real numbers have 'gaps' between them, known as irrational numbers, which can't be expressed as fractions. Together, they form a continuum of numbers, making their enumeration impossible, which is a concept that often challenges our intuitive understanding of infinity.

Cantor's theorem is a fundamental result established by Georg Cantor in 1891, which states that for any given set, the set of all its subsets (its power set) has a strictly greater number of elements than the set itself. This applies even if the original set is infinite.

One of the most startling implications of Cantor's theorem is related to the real numbers. It implies that even though the set of natural numbers is infinite, the set of real numbers is a 'larger' kind of infinity, known as uncountable infinity. In the context of our exercise, Cantor's theorem indirectly supports the notion that real numbers cannot be enumerated, because it points to the existence of different sizes of infinity.

Cantor's diagonal argument, as applied in the exercise, actually originated from the proof of this theorem. It beautifully illustrates the concept that just being infinite does not imply that a set can be matched up with the natural numbers. These ideas laid the foundation for the entire field of set theory and our modern understanding of infinity.