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In real number system, given a number"n" whats the number immediately after"n" lets say the number after "n" is "y". Is "y" rational or irrational??? By the completeness property of real number system "y" can be both rational and irrational.. So "y" does not exist?

Tom Lynd
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  • Right---there is no such thing as the "next" real number. – Gerry Myerson Aug 23 '13 at 13:54
  • So "y" doesn't exist? – Tom Lynd Aug 23 '13 at 13:55
  • Do you know that the order is dense? Right? So there is no number immediatly after a real number. Given $x, y \in \mathbb{R}$ there exists infinite (non - countable) numbers $z$ such that $x < z < y$ – user40276 Aug 23 '13 at 13:56
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    Tom, I prefer to say, there is no such thing as the next real number. – Gerry Myerson Aug 23 '13 at 13:58
  • yeah, so it implies that such a number does not exist.. – Tom Lynd Aug 23 '13 at 13:58
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    @Tom for something to not exist, this something must still be defined first. You haven't defined the number immediately after $n$. – Dan Shved Aug 23 '13 at 14:02
  • @Dan: That seems completely nonsensical to me. Being mathematically (well-)defined entails existence. There are certainly no non-existent mathematical entities that are (well-)definable. A viable argument (if you buy excluded middle arguments) proceeds as follows: "Suppose that there is some least real greater than $n,$ say $y.$ But since the reals are densely ordered, then we can find a real $z$ with $n<z<y.$ Contradiction. Thus, there is no least real greater than $n$." – Cameron Buie Aug 23 '13 at 15:02
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    @Gerry's point seems to be that if something doesn't exist, then it is better not to refer to it by a naming convention that suggests it does (unless using a contradiction argument such as the above, in which case there is only briefly a name assigned). I may be off base, here, though. – Cameron Buie Aug 23 '13 at 15:04
  • @CameronBuie You see, your argument is talking about "some least real greater than $n$". That's your definition of "the number immediately after $n$". So before proving non-existence, you fashioned a definition. That is all I meant: you need something more concrete than a vague intuitive adjective like "next" or "immediate" to prove non-existence. – Dan Shved Aug 23 '13 at 15:11
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    @Dan: Ah! That makes more sense. Precision is good. – Cameron Buie Aug 23 '13 at 15:13
  • @CameronBuie Sorry for being too pedantic. It's just that I've participated more than once in conversations with non-mathematicians who talked about things like "adjacent numbers" without stopping to think what it is that they mean. – Dan Shved Aug 23 '13 at 15:15

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In the real number system, there is no number immediately after $n$. Such a number would be the least number greater than $n$--that is, the least element of $(n,+\infty),$ which has no least element.

The standard ordering of the reals is a dense linear order, meaning that between any two distinct elements, there is another element. Thus, there can be no least number greater than $n$, since if $y$ is a number greater than $n,$ there is some $n<z<y$.

Cameron Buie
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