
Re: Cantor's diagonal argument.
Posted:
Oct 3, 2001 2:58 PM


"Giles Redgrave" <g.d.redgrave@elostirion.freeserve.co.uk> wrote in message news://c645a590.0110030525.199817f7@posting.google.com... > I'm having a problem understanding Cantor's diagonal argument (CDA). > Specifically it's use in proving the uncountability of the reals from > 0 to 1. > > I don't understand why you can't apply CDA to the natural numbers > themselves. If we list the natural numbers padding to the left with > zeros like so: > > ...000 > ...001 > ...002 > ...003 > . > . > . > > and apply CDA by adding one to the nth digit (from the right) of n and > constructing our new number from these digits. > > We then have a number (consisting of an infinite series of ones)
What kind of integer has an infinite series of ones? Integers must have a finite number of digits.
> To put it another way, what is wrong with mapping of the natural > numbers to the reals by reversing the digits of the natural numbers > and placing them after a decimal point like so: > > 0 > 0 > 1 > 0.1 > 2 > 0.2 > ... > 10 > 0.01 > 11 > 0.11 > ... > 123456 > 0.654321 > ... > > This is driving me mad. Can someone point out what's wrong with this > argument because I can't think of a real that can not be generated by > the above mapping and I can't see why applying CDA to the natural > numbers does not lead to a contradiction.
You've only included decimal numbers that terminate (that is, they only contain a finite number of digits after the decimal point). Many real numbers (for example 1/3 = .333333....) require an infinite number of digits to represent them.
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