Cantor diagonalization argument

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SUMMARY

Cantor's diagonal argument demonstrates that for any countable list of real numbers, it is possible to construct a real number that does not belong to that list. This establishes that there cannot be a one-to-one correspondence between the set of real numbers and the set of integers, as the list of real numbers is uncountable. The critical distinction lies in the term "countable," emphasizing that Cantor's argument applies to infinite sets rather than finite ones.

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  • Understanding of set theory concepts, particularly countability
  • Familiarity with Cantor's diagonalization method
  • Basic knowledge of real numbers and integers
  • Comprehension of one-to-one correspondence in mathematics
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  • Study the implications of Cantor's diagonal argument on set theory
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arshavin
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sorry for starting yet another one of these threads :p

As far as I know, cantor's diagonal argument merely says-

if you have a list of n real numbers, then you can always find a real number not belonging to the list.

But this just means that you can't set up a 1-1 between the reals, and any finite set.

How does this show there is no 1-1 between reals, and the integers?
 
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arshavin said:
As far as I know, cantor's diagonal argument merely says-

if you have a list of n real numbers, then you can always find a real number not belonging to the list.

No. It says that if you have a countable list of real numbers, you can form one that isn't anywhere on the list.
 
CRGreathouse said:
No. It says that if you have a countable list of real numbers, you can form one that isn't anywhere on the list.
Crucial point being "countable" not "n". So there is "no 1-1 between reals, and the integers". Cantor's "list" is not finite.
 

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