Cantor diagonalization argument

In summary, the conversation discusses Cantor's diagonal argument and its implications for finding a 1-1 correspondence between real numbers and integers. The argument states that if you have a countable list of real numbers, you can always find one that is not on the list. This shows that there is no 1-1 correspondence between the two sets.
  • #1
arshavin
21
0
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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  • #2
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.
 
  • #3
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.
 

FAQ: Cantor diagonalization argument

What is the Cantor diagonalization argument?

The Cantor diagonalization argument is a mathematical proof that was developed by Georg Cantor in the late 19th century. It is used to show that the real numbers are uncountable, meaning that they cannot be put into a one-to-one correspondence with the natural numbers.

How does the Cantor diagonalization argument work?

The argument works by assuming that there is a list of all the real numbers, and then constructing a new number that is not on the list. This new number is created by taking the diagonal elements of the list and changing each digit to a different number. Since this new number is not on the list, it is not accounted for in the one-to-one correspondence with the natural numbers.

Why is the Cantor diagonalization argument important?

The Cantor diagonalization argument has significant implications in mathematics and philosophy. It shows that there are different levels of infinity, and that some infinities are larger than others. This challenges the traditional idea that infinity is a singular concept, and has led to further developments in set theory and the foundations of mathematics.

Is the Cantor diagonalization argument a proof of the existence of uncountable sets?

No, the Cantor diagonalization argument is not a proof of the existence of uncountable sets. It is a proof that the real numbers are uncountable, but it does not prove the existence of other uncountable sets. However, it does provide a framework for understanding the concept of uncountability and can be applied to other sets as well.

Are there any criticisms of the Cantor diagonalization argument?

Yes, there have been some criticisms of the Cantor diagonalization argument. One of the main criticisms is that it relies on the assumption that a list of all the real numbers can be created, which some argue is not possible. Additionally, some have raised concerns about the validity of the one-to-one correspondence used in the argument. Despite these criticisms, the Cantor diagonalization argument remains a widely accepted proof in mathematics.

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