Practice question · Multiple choice
Cantor showed the reals cannot be listed even in principle, while the rationals can. Both are infinite. What does the diagonal argument establish?
Hints
- Given a list, build a number differing from the nth entry in its nth decimal place. Where is it on the list?
- The same construction fails for the rationals. Ask why.
Show the answer
D. That infinities differ in size, since no list of reals is complete
Why
Change the nth digit of the nth entry and you have a real that differs from every listed number somewhere, so the list was incomplete, whatever it was. The rationals survive the same challenge because they can be enumerated by a zig-zag, which is why 'infinite' turned out to name more than one size.
Practise Number Systems
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