I recently watched a YouTube video titled "Infinity is bigger than you think"...
In the video he brings up the idea of countable and uncountable sets (mentioning that he prefers to call them listable rather than countable, because you can't literally count an infinite sequence).
He starts with by pointing out that the set of whole numbers iscountable listable, because if you keep adding the next number in the sequence to the list forever, you end up listing all whole numbers.
Then he explains how all possible fractions (using whole numbers) are also countable/listable.
Then he explains that real numbers are not countable/listable.
But this is where I have a problem. Thinking of how he made the fractions countable/listable, it occurs to me that there might be a way to do this with real numbers as well.
So I'll give my process for listing all possible real numbers, and then everyone else can explain to me exactly why I'm wrong.
(For simplicity I'll ignore negative numbers, because you can easily add them in afterward by having every positive number in the sequence followed its equivalent negative.)
Listing all the whole numbers is easy:
Example 1 - List of all whole numbers
But to list all the real numbers, we'll need to be able to list everything between the whole numbers as well. We need some method of listing every possible sequence of digits that can follow the decimal place.
But since the infinite sequence in Example 1 lists every possible sequence of digits, all we need to do is flip the numbers around so that all the digits are on the opposite side of the decimal place (while keeping all the digits the same relative distance from the decimal place as they were in Example 1).
Example 2 - List of all partial numbers
All that remains to do is to merge the two lists together to get the list of all whole numbers.
The obvious problem is that you can't just take the first whole number and combine it with every possible partial number before moving onto the second whole number, because you'll never get to the second number even if you kept on listing it forever.
But that's exactly the same problem as with the fractions. He solved that in the video by putting them on a grid, an then listed the diagonals of the grid. We can apply the same solution here, listing whole values across the X-axis and partial values down the Y-axis...
Now we just have to list the diagonals...
Example 3 - List of all real numbers
Given that a list of all real numbers seems so easy to generate, can someone please explain why they're "uncountable" when the set of all possible fractions is "countable"?
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| I AGREE |
In the video he brings up the idea of countable and uncountable sets (mentioning that he prefers to call them listable rather than countable, because you can't literally count an infinite sequence).
He starts with by pointing out that the set of whole numbers is
Then he explains how all possible fractions (using whole numbers) are also countable/listable.
Then he explains that real numbers are not countable/listable.
But this is where I have a problem. Thinking of how he made the fractions countable/listable, it occurs to me that there might be a way to do this with real numbers as well.
So I'll give my process for listing all possible real numbers, and then everyone else can explain to me exactly why I'm wrong.
(For simplicity I'll ignore negative numbers, because you can easily add them in afterward by having every positive number in the sequence followed its equivalent negative.)
Listing all the whole numbers is easy:
Example 1 - List of all whole numbers
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, .... (And so on)
But to list all the real numbers, we'll need to be able to list everything between the whole numbers as well. We need some method of listing every possible sequence of digits that can follow the decimal place.
But since the infinite sequence in Example 1 lists every possible sequence of digits, all we need to do is flip the numbers around so that all the digits are on the opposite side of the decimal place (while keeping all the digits the same relative distance from the decimal place as they were in Example 1).
Example 2 - List of all partial numbers
.0, .1, .2, .3, .4, .5, .6, .7, .8, .9, .01, .11, .21, .31, .41, .51, .61, .71, .81, .91, .02, .12, .22, .32, .42, .52, .62, .72, .82, .92, .03, .13, .23, .33, .43, .53, .63, .73, .83, .93, .04, .14, .24, .34, .44, .52, .64, .74, .84, .94, .05, ... (And so on)
All that remains to do is to merge the two lists together to get the list of all whole numbers.
The obvious problem is that you can't just take the first whole number and combine it with every possible partial number before moving onto the second whole number, because you'll never get to the second number even if you kept on listing it forever.
But that's exactly the same problem as with the fractions. He solved that in the video by putting them on a grid, an then listed the diagonals of the grid. We can apply the same solution here, listing whole values across the X-axis and partial values down the Y-axis...
| 0.0 | 1.0 | 2.0 | 3.0 | 4.0 | 5.0 | 6.0 | 7.0 | 8.0 | 9.0 | 10.0 | 11.0 | ... |
| 0.1 | 1.1 | 2.1 | 3.1 | 4.1 | 5.1 | 6.1 | 7.1 | 8.1 | 9.1 | 10.1 | 11.1 | ... |
| 0.2 | 1.2 | 2.2 | 3.2 | 4.2 | 5.2 | 6.2 | 7.2 | 8.2 | 9.2 | 10.2 | 11.2 | ... |
| 0.3 | 1.3 | 2.3 | 3.3 | 4.3 | 5.3 | 6.3 | 7.3 | 8.3 | 9.3 | 10.3 | 11.3 | ... |
| 0.4 | 1.4 | 2.4 | 3.4 | 4.4 | 5.4 | 6.4 | 7.4 | 8.4 | 9.4 | 10.4 | 11.4 | ... |
| 0.5 | 1.5 | 2.5 | 3.5 | 4.5 | 5.5 | 6.5 | 7.5 | 8.5 | 9.5 | 10.5 | 11.5 | ... |
| 0.6 | 1.6 | 2.6 | 3.6 | 4.6 | 5.6 | 6.6 | 7.6 | 8.6 | 9.6 | 10.6 | 11.6 | ... |
| 0.7 | 1.7 | 2.7 | 3.7 | 4.7 | 5.7 | 6.7 | 7.7 | 8.7 | 9.7 | 10.7 | 11.7 | ... |
| 0.8 | 1.8 | 2.8 | 3.8 | 4.8 | 5.8 | 6.8 | 7.8 | 8.8 | 9.8 | 10.8 | 11.8 | ... |
| 0.9 | 1.9 | 2.9 | 3.9 | 4.9 | 5.9 | 6.9 | 7.9 | 8.9 | 9.9 | 10.9 | 11.9 | ... |
| 0.01 | 1.01 | 2.01 | 3.01 | 4.01 | 5.01 | 6.01 | 7.01 | 8.01 | 9.01 | 10.01 | 11.01 | ... |
| 0.11 | 1.11 | 2.11 | 3.11 | 4.11 | 5.11 | 6.11 | 7.11 | 8.11 | 9.11 | 10.11 | 11.11 | ... |
| 0.21 | 1.21 | 2.21 | 3.21 | 4.21 | 5.21 | 6.21 | 7.21 | 8.21 | 9.21 | 10.21 | 11.21 | ... |
| ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... |
Now we just have to list the diagonals...
Example 3 - List of all real numbers
0.0, 0.1, 1.0, 0.2, 1.1, 2.0, 0.3, 1.2, 2.1, 3.0, 0.4, 1.3, 2.2, 3.1, 4.0, 0.5, 1.4, 2.3, 3.2, 4.1, 5.0, 0.6, 1.5, 2.4, 3.3, 4.2, 5.1, 6.0, 0.7, 1.6, 2.5, 3.4, 4.3, 5.2, 6.1, 7.0, 0.8, 1.7, 2.6, 3.5, 4.4, 5.3, 6.2, 7.1, 8.0, 0.9, 1.8, 2.7, 3,6, 4.5, 5.4, 6.3, 7.2, 8.1, 9.0, ... (And so on)
Given that a list of all real numbers seems so easy to generate, can someone please explain why they're "uncountable" when the set of all possible fractions is "countable"?
via JREF Forum http://forums.randi.org/showthread.php?t=264483&goto=newpost
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