Cantor's diagonal proof?
Discussion
Is this actually a proof, or more a paradox?
It states there are more decimals 0>1 than real numbers.
Cantor said
1 Write all the real numbers one below the other.
2 Next to them write all the decimals,one below the other,in any order. Each decimal can be represented by an infinite string of digits, and there are infinitely many of them.
So far so good, the two columns have a 1:1 correlation which shows that although they are infinite, they are equal.
Now for the fun bit.
3 Go down the list of decimals, and for the first one, write down the first digit +1. For the second one, write down the second digit +1 next to the digit you just wrote, and so on all the way down the list.
Cantor claims:
The number you end up with Cannot Be On The List, because it differs from every number on the list by at least one digit, so there are more decimals 0>1 than reals. qed
But I think this can't be correct, because if it were so, then you haven't written all the decimals!
Obviously it is not possible to write down All the numbers or decimals in practice, it's all about 1:1 correlation, but people far cleverer than me consider Cantor's diagonal proof valid.
It states there are more decimals 0>1 than real numbers.
Cantor said
1 Write all the real numbers one below the other.
2 Next to them write all the decimals,one below the other,in any order. Each decimal can be represented by an infinite string of digits, and there are infinitely many of them.
So far so good, the two columns have a 1:1 correlation which shows that although they are infinite, they are equal.
Now for the fun bit.
3 Go down the list of decimals, and for the first one, write down the first digit +1. For the second one, write down the second digit +1 next to the digit you just wrote, and so on all the way down the list.
Cantor claims:
The number you end up with Cannot Be On The List, because it differs from every number on the list by at least one digit, so there are more decimals 0>1 than reals. qed
But I think this can't be correct, because if it were so, then you haven't written all the decimals!
Obviously it is not possible to write down All the numbers or decimals in practice, it's all about 1:1 correlation, but people far cleverer than me consider Cantor's diagonal proof valid.
Edited by Super Sonic on Friday 19th January 19:36
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