Is there a real number which comes just after a particular real number?

This is like when we say that the integer which comes just after $2$ is $3$.

Is there a real number which comes just after a particular real number?

For example: Is there a number which comes just after $0.5$?

If any one didn't get me, read Asaf's answer because that explains my question better.

Solution 1:

"Just after 0.5", well, in what context?

In the natural numbers the number that comes "just after $2$" is indeed $3$. But in the rational numbers, or the real numbers, what is the next real number after $2$? Is it $3$? Or is it $2.5$? Or is it $2.25$? Or so on and so forth.

Not every ordering comes with a well-defined notion of "just after", like the integers and the natural numbers have. And as luck would have it, the rational numbers and the real numbers are both examples for ordered sets which do not posses this property.

So as a real number, or even as a rational number, with the standard ordering, there is no "just after" any number.

If one no longer desires to use the standard order, then it is not hard to come up with all manner of alternative orders, where the notion of "next real number" makes a lot of sense.

For example, we can prove that there is a bijection between $\Bbb R$ and $\Bbb{R\times Z}$. Then any structure we can give the set $\Bbb{R\times Z}$ can be translated to a structure on $\Bbb R$.

In particular the lexicographic order, $(r,k)\preceq (s,m)$ if and only if $r<s$ or, $r=s$ and $k\leq m$. Namely, we replace each real number with a copy of $\Bbb Z$. Now given any point on this order, $(r,k)$ it has a unique "next number" which is $(r,k+1)$.

Of course the translation from $\Bbb{R\times Z}$ to $\Bbb R$ is not in any way canonical or unique. It just exists, and so there's no way to just say what is the next real number after $0.5$ because that would greatly depend on this translation.

Solution 2:

If you are talking about a rational number or a real number after $\frac{1}{2}$, it is not possible. For suppose $x$ were such a number. Then $\frac{\frac{1}{2}+x}{2}$ is a number after $\frac{1}{2}$, but closer than $x$.

Solution 3:

This is a great question! You stumbled upon an important problem:

"I know how to tell if a real number is bigger than another one, but apparently (even if there is an "order" between them) I cannot tell which one comes after $\frac 12$"

And you are right; there is no way to meaningfully define the "next" real number.

To understand why, let's think about natural numbers; they have a lot of properties, and among them, is the fact that you can order them. First you have $1$, then $2$, and so on.

So, which sets can be ordered? Of course any finite set can. What about infinite ones? Can $\mathbb Q$ be ordered?

The idea is that if you can find a bijection between your set $X$ and $\mathbb N$, then you can "copy" the order of $\mathbb N$ (because $1$ would correspond to an element of your set $X$, $2$ to another one and so on)

All the set that can be put in a such a relationship with $\mathbb N$ are called countable and all countable set can be ordered. (Note however that this does not ensure that the "next" element of $X$ is bigger; in fact the way you define bigger in $X$ may have nothing to do with the order in which its elements are arranged. )

For example, $\mathbb Q$ is (very un-intuitively) countable, so you can order the fractions and you can tell which (rational) number comes after $\frac 12$.

On the other hand, $\mathbb R$ is not countable, so you can't "automatically" do that. There is, as @Asaf Karagila points out in the comments, the concept of "well ordered" but I admit I know nothing about that. Refer to Asaf's comment for more info! :-)

The proof of both these fact were given by Cantor; look here, for example