In this series of blog posts I will present a proof
that the set of Natural Numbers, N, is not countable. Intuitively it
seems self-evident that the elements of N can be enumerated simply by
listing them 1, 2, 3, 4… n, … To demonstrate that N is uncountable I will construct a
natural number, Y (called a straggler), derived from the process of listing N
such that Y will never appear in the list.
The tools employed in proving the theorem are proof by construction, which I will use to derive Y from the listing of N, mathematical induction, which will show that Y will not be in the enumeration of N for P(1), P(n) and P(n+1) iterations of the computer program used to generate N, and lastly, contradiction wherein I will show that the creation of Y contradicts the assumption that N is numerable.
Next I will review Cantor's Diagonal method to show how employing it creates a straggler in the list of real numbers on the (0, 1) interval.
The tools employed in proving the theorem are proof by construction, which I will use to derive Y from the listing of N, mathematical induction, which will show that Y will not be in the enumeration of N for P(1), P(n) and P(n+1) iterations of the computer program used to generate N, and lastly, contradiction wherein I will show that the creation of Y contradicts the assumption that N is numerable.
Next I will review Cantor's Diagonal method to show how employing it creates a straggler in the list of real numbers on the (0, 1) interval.
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