Difference between revisions of "Infinity and beyond"
(New page: What is the nature of infinity? It is not the largest possible number because you can always add 1 to a number so it is not a number at all. Look at infinity as a point beyond our reach of...) |
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What is the nature of infinity? It is not the largest possible number because you can always add 1 to a number so it is not a number at all. Look at infinity as a point beyond our reach of counting because we can never reach it by counting. | What is the nature of infinity? It is not the largest possible number because you can always add 1 to a number so it is not a number at all. Look at infinity as a point beyond our reach of counting because we can never reach it by counting. | ||
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So how do we reach beyond Infinity? | So how do we reach beyond Infinity? | ||
| − | == | + | ==Cantor's diagonal slash== |
| − | Line up N integers with N reals | + | Line up N [[Integer|integers]] with N reals |
1 - '''1'''.111 | 1 - '''1'''.111 | ||
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Take one digit from each line of the reals going diagonally | Take one digit from each line of the reals going diagonally | ||
| − | 1.234 | + | 1.234 |
Change every digit of the number to something else (randomly will do) | Change every digit of the number to something else (randomly will do) | ||
| − | 2.345 | + | 2.345 |
The resulting number can't be a member of the original list of reals because it differs by at least one digit from every number on the list. | The resulting number can't be a member of the original list of reals because it differs by at least one digit from every number on the list. | ||
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Since there are more real numbers than integer numbers then there must be more than one level of infinity. | Since there are more real numbers than integer numbers then there must be more than one level of infinity. | ||
| − | [[Category: | + | [[Category:General information]] |
Latest revision as of 14:38, 23 March 2008
What is the nature of infinity? It is not the largest possible number because you can always add 1 to a number so it is not a number at all. Look at infinity as a point beyond our reach of counting because we can never reach it by counting.
So how do we reach beyond Infinity?
Cantor's diagonal slash
Line up N integers with N reals
1 - 1.111 2 - 2.222 3 - 3.333 4 - 4.444
Take one digit from each line of the reals going diagonally
1.234
Change every digit of the number to something else (randomly will do)
2.345
The resulting number can't be a member of the original list of reals because it differs by at least one digit from every number on the list.
This holds for any size of lists and proves that there are more reals than there are integers because for all possible lists of integers you can create at least one more real than there are integers.
Since there are more real numbers than integer numbers then there must be more than one level of infinity.
