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Originally Posted by John Henry
Since the set of integers is infinite and passes through every sequence of digits, there can be no sequence of digits which is not included in it, even those which are non-repeating and non terminating. Irrational numbers between 1 and 0 are expressed as a non-terminating, non-repeating sequence of digits after the decimal place. So why can we not extend the pattern above infinitely?
If this pattern cannot be continued infinitely, what is its limit?
If it can, then does that not mean that all the non-repeating, non-terminating decimals that make up irrational numbers can be expressed as a ratio of two integers and as such are simply infinitely long rational numbers and not truly irrational?
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Because long yet finite sequences, and infinite sequences, are qualitatively different beasts.
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Originally Posted by asaris
In any case, it can be proven that the set of real numbers is larger than the set of rational numbers. So there must be irrational numbers.
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The proof in question relies on contradiction and is non-constructive.
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Originally Posted by John Henry
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.
...don't we?
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You used proof by example.
3 is prime
5 is prime
7 is prime
thus, all odd numbers are prime.
Proof by example isn't a proof.
There is difficulty in ruling out infinite integers from the set of integers. But the infinite integers you cannot rule out behave more like finite integers than the ones you want to use.
(Godel's incompleteness theorem can be interprited as the impossiblity of restricting the set of integers to only numbers which can be written out finitely)
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Originally Posted by raveneye
Yep, the set of integers is infinite. It is countably infinite (as is the set of all rational numbers). The irrationals and reals are not countable, although they are also infinite. So the set of reals is larger than the set of rational numbers, even though both are infinite.
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Once again, this is a non-constructive statement, and requires proof by contradiction to be valid in order for it to be true. There is a logic of mathematics that includes all demonstrateable mathematical truths in which you cannot prove that there are more reals than rationals.
For example, if you insist that all real numbers can be described algorithmically, then you end up with a real number line that behaves a hell of a lot like the real number line you play with normally, yet it contains countably many numbers.
This is a slightly obtuse branch of mathematics, and I'm still learning about it myself. I just thought I'd point out that 'truth' is in the eye of the axiom system.