If you're having math problems, I feel bad for you, son...

[napoleon_in_rags]

32. Just answer me one question Joseph8th.

Forget about pi for now, forget about everything. Lets just talk about the natural numbers. They can be defined as a set that contains 1, and contains n+1 for every n in N. So they are defined inductively.

Now I think I argued briefly about whether there was an infinitely long number in the natural numbers with somebody above. I argued that there was not. My proof (in my head) looked like this:

a) 1 is a finite natural number.
b) 1 added on to any finite natural number is also finite.

Why? By contradiction. Suppose it is not the case that 1 added on to any finite number was finite. Than it must be the case that there is some finite natural number which when one is added to it, becomes infinite. By which I mean there is some number with a finite amount of digits that attains an infinite number of digits when one is added to it. Absurd.

Therefore the set of natural numbers doesn't include a natural number with infinite length.

Now I will argue the opposite is true, and an infinite number is in the naturals. A one-to-one mapping between any set of natural numbers and a single natural number can be defined as follows:

To convert a set of natural numbers to a single natural number, take the sum of 2^n for each number n in the set. So {1,2,3} = 2^1+2^2+2^3 = 14.

To invert that and construct the set for any natural number m, get the first number s1 by taking floor(log2(m)), then subtract s1 from m to get m2, take floor(log2(m2)) to get the next number, repeat until down to 1. So for 14, floor(log2(14)) = 3. 2^3 = 8. 14-8 =6. floor(log2(6)) = 2 2^2 = 4. 6-4 = 2. floor(log2(2)) = 1. Done. {1,2,3} reconstructed from a natural number by a recursive function.

That defines a 1-to-1 mapping between the naturals and all the sets of the naturals. So by induction, for every n+1 starting at 1, the number representing the set of all numbers up to and including n+1 (call it the set number) can be gotten by adding 2^(n+1) to the set number that corresponded to n. So clearly, for every set of natural numbers, there is a set number that is also a natural number. So because the set of ALL natural numbers is itself a set of natural numbers, there must be a set number for that too, also contained in n. Clearly, the length of this number is infinite, so there must be an infinitely long number in n.

Which of these two arguments is right, the first stating there is no infinite number in N, or the second stating that there is?

Just curious what you think, I bumped into this in my thoughts and its tripping me out.