# Is this a good argument against Cantor's diagonal argument?

#### Mates

##### New member
This will attempt to show that the size of the set of the naturals are one-to-one with the size of the set of the reals.

I will try to match every natural number with every real number between 0 and 1.

Here are all possible combinations of real numbers from 0 to 1,

0.1, 0.11, 0.12, 0.13, 0.14 ...
0.2, 0.21, 0.22, ...
0.3, 0.31, 0.32, ...
0.4, 0.41, 0.42, ...
0.5, 0.51, 0.52, ...
0.6, 0.61, 0.62, ...
0.7, 0.71, 0.72, ...
0.8, 0.81, 0.82, ...
0.9 , 0.91, 0.92, ...

I don't know of a function off the top of my head that will match each natural to each real, but this is how we can imagine them matching. We start with matching 1 to 0.1, then 2 to 0.2, then 3 to 0.3 etc. and go down the column until we get to 9 and 0.9. When we get to 10 we match it with 0.11 and then start all over again. Each natural number can easily match to each real number as the matching goes down the columns.

I would think that this has been thought of before, so where am I going wrong?

#### Dr.Peterson

##### Elite Member
This will attempt to show that the size of the set of the naturals are one-to-one with the size of the set of the reals.

I will try to match every natural number with every real number between 0 and 1.

Here are all possible combinations of real numbers from 0 to 1,

0.1, 0.11, 0.12, 0.13, 0.14 ...
0.2, 0.21, 0.22, ...
0.3, 0.31, 0.32, ...
0.4, 0.41, 0.42, ...
0.5, 0.51, 0.52, ...
0.6, 0.61, 0.62, ...
0.7, 0.71, 0.72, ...
0.8, 0.81, 0.82, ...
0.9 , 0.91, 0.92, ...

I don't know of a function off the top of my head that will match each natural to each real, but this is how we can imagine them matching. We start with matching 1 to 0.1, then 2 to 0.2, then 3 to 0.3 etc. and go down the column until we get to 9 and 0.9. When we get to 10 we match it with 0.11 and then start all over again. Each natural number can easily match to each real number as the matching goes down the columns.

I would think that this has been thought of before, so where am I going wrong?
You are enumerating only the terminating decimals. You will not include numbers like 1/3 = 0.33333... , which can't be written with a finite number of digits, and therefore are not in any of your columns.

#### Mates

##### New member
You are enumerating only the terminating decimals. You will not include numbers like 1/3 = 0.33333... , which can't be written with a finite number of digits, and therefore are not in any of your columns.
But don't I have an infinite number of digits? My matching goes on forever just like 0.3333... does.

#### Dr.Peterson

##### Elite Member
But don't I have an infinite number of digits? My matching goes on forever just like 0.3333... does.
Not at all. Every number you enumerate has a finite number of digits! As I said, 0.333... is not in your table at all. (There is no infinitieth column!)

The same is true, of course, of many other numbers. I fact, your table omits more numbers than it includes!

#### lev888

##### Elite Member
But don't I have an infinite number of digits? My matching goes on forever just like 0.3333... does.
Continue this row:
0.3, 0.31, 0.32, 0.33, 0.34...
As you can see, 0.3333... is skipped.

#### Jomo

##### Elite Member
You wrote 0.1, 0.11, 0.12, 0.13, 0.14 ...
Where does this end?
Sure it includes 0.1, 0.11, 0.12, 0.13, 0.14, 0.15, 0.16, 0.17, 0.18, 0.19. Now what is the next number? You can use "..." whenever you want, but it must be extremely clear what the pattern is. Personally I would assume that the next numbers would be 0.20, 0.21, 0.22, ...0.29, but this the 2nd row. That first row ends!

#### pka

##### Elite Member
This will attempt to show that the size of the set of the naturals are one-to-one with the size of the set of the reals.
To Mates; Do you even know much less understand what the diagonal argument says?