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Non-computables, hyperreals, surreal, and even more obscure numbers exist between even just, e.g., 0 and 1. Does that mean they are all real numbers? Is a number real as long as it is on the number line, even how obscure, specific, unreal tiny it is?

If not, what is and isn't a real number? And does "all numbers on the number line" have a name?


marked as duplicate by Lord Shark the Unknown, max_zorn, mrtaurho, Shailesh, Kemono Chen Mar 5 at 9:23

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    $\begingroup$ if it's on a number line, it's a real number $\endgroup$ – Vasya Mar 4 at 22:43
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    $\begingroup$ Hyperreals and surreals are, in general, not reals. These are number systems strictly larger than the reals. $\endgroup$ – GEdgar Mar 4 at 22:50
  • $\begingroup$ @Vasya What about, for example, the number line that forms the imaginary axis of the complex plane? $\endgroup$ – Théophile Mar 4 at 22:50
  • $\begingroup$ @Théophile Obviously the real "number line" was assumed. $\endgroup$ – user Mar 4 at 22:59
  • $\begingroup$ @user I don't think that one can make such an "obvious" assumption. Just as there's a world of a difference between "a number" and "a real number", there's a difference between "a number line" and "the real number line". $\endgroup$ – Théophile Mar 4 at 23:00

These are very different things... Computability (or non-computability) is a property of the real numbers. However, the hyperreals and the surreals are both extensions of the reals, which means they contain the real numbers by definition, but they contain other things as well. So a general hyperreal number may not be real.

The definition of the real numbers is a complicated thing in itself. I like to think of the set of real numbers as the completion of the set of rational numbers, but there are many different, equivalent ways of constructing the reals. Take a look at the wikipedia page; it's interesting stuff!


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