Another question I have is about isomorphisms of enumerations.Cool Hand wrote:Perhaps you could expound on your dissatisfaction with Cantor's diagonal method and others could critique it.

It's obvious that if there's a bijection between set S and the counting numbers, then the cardinality of set S is aleph_0.

Is the converse true? If set S has cardinality of aleph_0, then is it always the case that it can be enumerated in the form used in Cantor's diagonal argument?

How about the inverse? If there is no bijection between S and the counting numbers, does that mean S cannot have a cardinality of aleph_0?

Moving on . . .

On page 421 of Douglas Hofstadter's book,

**, he gives an example of Cantor's diagonal argument. An example I believe to be flawed. I'm going to leave out some of Hofstadter's description of Cantor's argument, assuming that you already know it.**

*Godel, Escher, Bach*He goes on to explain why that number is not in the original list, using Cantor's diagonal argument. But there is a fatal flaw in his choice of the digits for the numberHofstadter wrote:Let us consider just real numbers between 0 and 1. ... Since real numbers are given by infinite decimals, we can imagine that the beginning of the table might look as follows:

r(1): .14 1 3 9 2 6 5 3 . . .

r(2): . 333 3 3 3 3 3 3 . . .

r(3): . 7 182 8 1 8 2 8 . . .

r(4): . 4 1 421 3 5 6 2 . . .

r(5): . 5 0 0 000 0 0 0 . . .

The digits that run down the diagonal are in boldface: 1, 3, 8, 2, 0, ... Now those diagonal digits are going to be used in making a special real numberd, which is betwen 0 and 1 but which we will see, is not in the list. ... Suppose, for example, that we subtract 1 from the diagonal digits (with the convention that 1 taken from 0 is 9). Then our numberdwill be:

. 0 2 7 1 9 . . .

*d*, which should be immediately obvious to those who have been following your thread about 0.999... = 1.

Hofstadter says that

*d*cannot be equal to r(5) because even if all the other digits of

*d*matched up with r(5), the fifth digit does not. Except Hofstadter overlooks that r(5) can also be expressed as .49999... in which the fifth digit IS the same as

*d*. So he has not proven that

*d*is not in the list.

However, this flaw is a result of the way he chooses the digits for

*d*. Instead of subtracting 1 from each digit of the diagonal to make

*d*, suppose we subtract 3, and the flaw seems to be fixed. Or is it?

Can you see why I have an uneasy feeling about Cantor's diagonal argument?