Question 1: Is the set of Natural Numbers
N, {
y ∊ N | 1 ≤ y} demonstrably numerable?
On the surface, it seems perfectly reasonable to assume that the natural numbers are demonstrably numerable. If so, then the natural numbers can be used as a yardstick to compare other infinite sets against in size comparisons.
To list the elements of
N one need only devise an algorithm such as:
Let
n0 = 1 then
n1 = n0 + 1… ni = ni-1 + 1…
and let it iterate forever.
This algorithm can be coded in Visual Basic to generate an endless list of natural numbers:
Module VBModule
Sub Main()
Dim nnum As ULong = 1
For i As ULong = 1 To ∞
nnum = nnum + 1
Next
End Sub
End Module
If the program is left to run it will generate successive natural numbers indefinitely which implies by mathematical induction that the natural numbers are demonstrably numerable. However, closer inspection reveals that even with infinite iterations of the formula not all natural numbers can be listed, even in theory. This shall be demonstrated in the proof of Theorem 1.
Theorem 1 will be proved by constructing a natural number
Y from a listing of the elements of
N that will itself never appear in the listing of
N, contradicting the assumption that the natural numbers can be enumerated with no exceptions. Modifying the code of the VB program we can generate a natural number from the enumeration of
N that will be an element of
N but will never appear in the enumeration of
N. This is similar to Cantor’s use of the Diagonal Method to show that an enumeration of the real numbers on the (0, 1) interval will permit the construction of a real number on that same interval which will never appear in the enumeration.
Algorithm 1: Let
n0 = 1 then
n1 = n0 + 1, n2 = n1 + 1… ni = ni-1 + 1…
Program 1: Generate the set of Natural Numbers
N, {
y ∊ N |
1 ≤ y}
Module VBModule
Sub Main()
Dim nnum As ULong = 1
For i As ULong = 1 To ∞
nnum = nnum + 1
Next
End Sub
End Module
Definition 6: A
straggler is a number of the same order that can be constructed from numbers in an enumeration that will never itself appear in the enumeration.
Canon 3: A straggler is a single number. It is not a series of separate numbers as might be inferred from looking at its history as the enumeration of a set grows. As a set is enumerated, time passes and as time passes the straggler changes according to the rules of its construction. For finite sets with a constructible straggler, the straggler in the last row in the enumeration is the straggler that differs from all elements in the enumeration. For infinite sets with a constructible straggler, there is no last row in the enumeration. The straggler is deemed different from all elements in the enumeration if it can be inferred by mathematical induction.
Canon 4: The ability to construct a straggler of the same order from an enumeration of an infinite set implies that the set is not numerable.
Program 2: Construct the straggler Y from the enumeration of N,
Module VBModule
Sub Main()
Dim nnum As ULong = 1
Dim Y As String = "1"
Dim lnum As String = "1"
Dim snum as string = ""
For i As ULong = 1 To ∞
nnum = nnum + 1
snum = Cstr(nnum)
lnum = lnum & snum
lnum = lnum.Replace(" ", "")
Y = lnum
Next
End Sub
End Module
Given 1: The set of natural numbers
N, {
y ∊ N |
1 ≤ y}
