Why there is not the next real number? We can't say what is the just next real number (or rational or irrational number) of a given real number (or rational or irrational number respectively), what is the actual fundamental reason  behind it?      Is it for the property of denseness of the sets?   Please help me to sure about the answer of this question.
 A: This is a property of the order of $\Bbb R$. We say that an order is dense if whenever $x<y$ there is some $z$ such that $x<z<y$. If $x$ is a real number, then if $y$ was "the next real number", we had some $z$ such that $x<z<y$, and so $z$ would be the next real number instead, but then we can do that again, ad infinitum.
However, one of the common assumptions in modern mathematics is "the axiom of choice", from which we can prove that there exists a well-order of any set, including $\Bbb R$. The fundamental property of a well-order is that every non-empty set has a minimum. It follows that every element which is not maximal, has an immediate successor.
So if we endow the set of real numbers with a well-ordering then there is a notion of "the next real number".
Some caveats:

*

*We cannot write an explicit formula which is guaranteed to produce a well-ordering of $\Bbb R$. We can prove its existence from the aforementioned axiom of choice. And so, since there is no natural well-ordering for $\Bbb R$ there is no deep sense to saying what is the next real number after $0$ (whereas in the natural numbers there is such sense for saying that $1$ is the successor of $0$).
Using different well-orders would give us different notions for "the next real number".


*Any such well-ordering of $\Bbb R$ is guaranteed to be almost entirely incompatible with the natural order of $\Bbb R$. This is important to note, because we often expect the structure we give $\Bbb R$ would somehow respect any "naturally occurring structure" that $\Bbb R$ already carries, but this is very far from the truth.


*The above point stands in analogy to $\Bbb Q$ which is a countable set, and therefore can be written as $q_0,q_1,q_2,\ldots$, and that is in fact a well-ordering of $\Bbb Q$. Note that this indexing is very incompatible with the usual order of $\Bbb Q$. Although in the case of $\Bbb Q$ there are explicit enumerations of the set. So the analogy doesn't stretch beyond the second point.
A: You have to be precise with what does 'next' mean. It assumed some notion of order on the reals. If that notion of order is taken to be the usual one (whatever that means), then the fact that there is no 'next real' is a property of that particular order. In more detail, it is possible to construct the real numbers (in various essentially equivalent ways) from the rationals. Then one can define the ordering on the real numbers as a derived notion from the ordering of the rationals. Then one can prove that between any two real numbers there is a third, distinct one. This density property immediately implies that there is no 'next number' in the sense that if $x$ is a real number, then the set $\{y\in \mathbb R \mid x<y\}$ does not have a least element. 
However, if one believes in the axiom of choice, then there exists an ordering on the reals which is a well-ordering, meaning that every real has a uniquely 'next real'. Explicitly exhibiting such an ordering is not possible though. 
A: The reason is if you say this so ans so number is the next number then I can immediately give you another in between both of them. Say x and y are rationals then (x+y)/2 is the real in between them. Suppose instead you have irrationals then see the digit where they differ and take rational approximation. And similarly do other cases
A: Denseness is rather a relation between two sets than a propriety of a single one. That is, it has no meaning saying that "a set is dense". That between any two rational numbers there is another one distinct from both is independent from being $\mathbb Q$ a subset of $\mathbb R$. And it follows from the definition of $\mathbb Q$ and the proprieties of an ordering relation.
