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Originally Posted by John Henry
Clearly, then, irrational numbers do exist. Does any more general or generalisable proof exist that all non-terminating, non repeating real numbers are irrational?
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Well, if I understand the idea correctly, once you have established that a real number is non-terminating and non-repeating, then it is by definition irrational. (This can be seen by the fact that the real number set is "larger" than the set of rational numbers. This is used in the proof that the real numbers are uncountable.) The big problem is how to prove something is irrational without necessarily knowing if it is non-repeating or non-terminating. Proving Pi irrational was a big deal for example. We still don't know if Pi^e is irrational, where "e" is the exponential constant.
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n=1000
|A|=1000
A={1,2,3...1000}
The set of all integers is infinite we can set n as infinite. If we set n infinite, then we have a set of integers which contains n, which is infinite, giving us an infinite integer.
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No, you cannot simply <em>set</em> n to infinity and expect the properties of integers to carry over formally. You may see infinity being used in Complex Analysis as part of a set, like C union {infinity}, but this is just shorthand, and shouldn't imply infinity can be treated as such.
What you are saying above makes intuitive sense, but so does the following process:
1/1=1; 100/100=1; 9999/9999=1; infinity/infinity=1. But this would be wrong. raveneye said it best, two sets can have infinite number of members yet be different sizes. Everything is intuitive up until we reach infinity. I believe this is why the notion of <a href="http://mathworld.wolfram.com/CardinalNumber.html">Cardinality</a> was introduced.