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Having examined the infinity of the real numbers, I now want to present a proof and verification that I conceived just after going to bed on Friday, 2007 Dec 21, one more elegant than what I presented in the previous essay. I now present the simplest possible proof and verification of the proposition that the set of the natural numbers and the set of the real numbers contain the same quantity of elements.

Recall that the set of the natural numbers consists of all of the positive integers, the counting numbers. The set of the real numbers consists of all possible combinations of one natural number with one decimal fraction. Because both the natural numbers and the decimal fractions constitute infinite sets, we expect their combinations to comprise a set even more infinite than the set of the natural numbers; that is, we intuitively expect the quantity of elements comprising the set of the real numbers to exceed Aleph-Null, the infinity of the counting numbers in Georg Cantor's nomenclature. Of course, we know that we cannot trust our intuitions about infinity: we must put every proposition pertaining to infinite sets to a strict logical proof.

To Prove: That the set of the real numbers and the set of the natural numbers contain the same quantity of elements.

Define the interleaving collapse of a real number as follows: for an upward collapse we take the fractional part of the number and put its first digit between the first and second digits of the integer part of the number, put its second digit between the second and third digits of the integer part of the number, and so on. Thus, for example, the upward interleaving collapse of the real number 3981.35 yields the natural number 3095831. Note that we must include the implicit zeroes to the right of the fractional part of the number. Further, for a downward collapse we take the integer part of the number and put its first digit between the first and second digits of the fractional part, and so on. In our example the downward interleaving collapse of 3981.35 yields the fraction 0.31580903.

There exists no real number that we cannot subject to an upward interleaving collapse. And every real number yields a unique natural number when so subjected: no other real number can yield the same natural number. Thus we can associate every real number uniquely with a natural number.

Next define the interleaf unpacking of a natural number as the process of creating a real number by taking the natural number's second digit and putting it in the first place to the right of the decimal point, taking the natural number's fourth digit and putting it in the second place to the right of the decimal point, and so on. As an example of that process we use it to unpack 1836151527 into 86557.21131.

There exists no natural number that we cannot subject to an interleaf unpacking. And every natural number yields a unique real number when so subjected: no other natural number can yield the same real number. Thus we can associate every natural number uniquely with a real number.

Under collapse no two real numbers will yield the same natural number, so all of the real numbers will yield the same quantity of natural numbers. And under unpacking no two natural numbers will yield the same real number, so all of the natural numbers will yield the same quantity of real numbers. So now we know that we can create a unique, bijective, one-to-one match between the complete set of the real numbers and the complete set of the natural numbers. That fact necessitates that both sets contain the same quantity of elements. Q.E.D.

Thus we prove and verify the proposition equivalent to stating that the infinity of the real numbers is identical to the infinity of the natural numbers.

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