First, consider the following infinite collection of real numbers. Describe in y

Question

First, consider the following infinite collection of real numbers. Describe in your own words how these numbers are constructed (that is, describe the procedure for generating this list of numbers). Then, using Cantor’s diagonalization argument, find a number that is not on the list. Give at least the first 10 digits of the number and explain how to find the rest. How do you know that your number is not on the list? 0.123456789101112131415161718 . . . 0.2468101214161820222426283032 . . . 0.369121518212427303336394245 . . . 0.4812162024283236404448525660 . . . 0.510152025303540455055606570 . . .

Explanation / Answer

"The real numbers between 0 and 1 can be listed in some order, say, r1,r2,r3,...r1,r2,r3,...Let the decimal representation of these real numbers be

r1=0.d11r1=0.d11d12d12d13d13d14d14... r2=0.d21r2=0.d21d22d22d23d23d24d24... r3=0.d31r3=0.d31d32d32d33d33d34d34... r4=0.d41r4=0.d41d42d42d43d43d44d44...

Where dijdij is an element of {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Then for a new real number with decimal expansion r = d1d1d2d2d3d3d4d4... where the decimal digits are determined by the following rule:

didi = {4 if diidii does not equal 4, 5 if diidii = 4}.

And I'm sorry but..what? What in the world is any of this trying to get at? What is the whole r1, r2, r3 thing even mean? Why do we have to create a "new real number"? What is the point? Why? Why are we doing any of this? I don't understand any of the process behind it and I don't understand how it all leads to the conclusion that the real numbers are uncountable. I have absolutely no idea what is going on here.

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