I understand that $\mathbb{R}$ has no supremum, because the real numbers go on forever. My question is that if I was to write $\sup\mathbb{R}$, would I say it doesn't exist or would I say that $\sup\mathbb{R}=\infty$? It seems more likely that it would be the former, as infinity isn't actually a number, but when I searched on Google, a Wikipedia article said that $\sup\mathbb{Z}=\infty$, so I thought I'd just make sure. (The Wikipedia link)
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The supremum of a set $X$ depends not only on $X$ itself but also on the ambient linear ordering (which is usually understood from context.) If you take the supremum of the set $\mathbb{R}$ (or the set $\mathbb{Z}$) in the extended real line then you get $\infty$. However, in the ordinary real line $\mathbb{R}$ these sets ($\mathbb{R}$ and $\mathbb{Z}$) do not have suprema.
Trevor Wilson
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+1. I also want to emphasize that the extended real line is not some esoteric trick: using it is the "right" way to do calculus and analysis. (although introductory calculus usually treats it in an ad-hoc fashion, such as giving special definitions for what $\lim f = +\infty$ means in a way that references only ordinary real numbers) – Sep 28 '13 at 05:58
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@Hurkyl Agreed. But I suppose $\infty$ will always be a second-class citizen of sorts because it doesn't fit into the field structure of $\mathbb{R}$. – Trevor Wilson Sep 28 '13 at 06:02