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bozar
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Homework Statement
This is actually only related to a problem given to me but I still would like to know the answer. From my understanding, Cantor's Diagonalization works on the set of real numbers, (0,1), because each number in the set can be represented as a decimal expansion with an infinite number of digits. This means 0.5 is not represented only by one digit to the right of the decimal point but rather by the "five" and an infinite number of 0s afterward.
My question then is can the set of natural numbers be represented similarly, ie instead of 5, have an infinite number of 0s preceding it (to the left of it) so the number is represented instead as ...000000005. Then the set of natural numbers can be aligned similarly as done in Cantor's Diagonalization with all numbers represented in its own row with each digit represented as d1, d2, d3, and so on with d being an element of {0,1,2,3,4,5,6,7,8,9}. The difference however would be that the digits in the rows, instead of going from left to right, would go from right to left. d1 would the the rightmost digit (the "ones" digit), d2 would be to the left, and so on. If we create a new number, n = d1d2d3d4... where dI = {1 if dII !=1, 2 if dII = 1}, then this new number will be different than every number in the table which should have represented all the natural numbers. Thus the set of natural numbers is not countable. Sorry if this was confusing, I didn't want to rewrite Cantor's Diagonalization but I didn't want to leave too much out.
Homework Equations
Everything.
The Attempt at a Solution
The only thing I can come up with is that natural numbers cannot be represented with an infinite number of 0s to the left of the number. However, I don't understand then how that differs from representing all decimals as having an infinite number of 0s to the right of the rightmost positive integer.
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