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In Rudin's Principles of Mathematical Analysis, it is said that real numbers will be certain subsets of rational numbers called cut. I am having trouble accepting this fact, because if we consider real numbers as those having members that are subsets of rationals, then how do one account for gaps left by rationals? I.e. irrationals.
Please help me understand the actual idea being conveyed here. Thank you.
What you're overlooking is that there aren't any "gaps." The rationals form a "dense" subset of the reals, which means that any open set in the reals contains at least one rational number.
The usual way in which Dedekind cuts are used to define the reals is to define a Dedekind cut as $\{R \subset \Bbb Q ~ \vert ~ x \in R \Rightarrow \forall y \lt x ~ (y \in R) \}, \text{ where } R \neq \emptyset, \Bbb Q$. We think of a real number as simply all rationals (strictly) less than that number. So, for example, in this construction we think of $\pi$ as $\{x \in \Bbb Q~ \vert ~ x \lt \pi ~ \}$.
Sets are not the same things as numbers and the sets actually "fit" in the "gaps".
And as a cut is a set that has no maximum element it need match up with any rational number. (That's actually the entire point.)
Consider this set: $A=\{q\in \mathbb Q|q < \frac 23\} $. This set is a "cut" (it fits all the criteria). And it sits directly "on top of" The rational number $\frac 23$ as clearly $\sup A=\frac 23$. In this way set $A$ is linked to and represents the real (and rational) number $\frac 23$.
Now consider the set $B=\{q\in \mathbb Q|q <0$ or $q^2 <2\} $. This set is also a cut. But it does not "sit on" any rational number as there is no rational number that is $\sup B $. This set falls into a "gap" between rational numbers. Set $B $ is linked to and represents the real (and irrational) number $\sqrt 2$.
Intuitively the idea behind the Dedeking cuts is the following. Assuming that you "know" what the real numbers are, then you can associate to each real number $a$ the set $$( - \infty ,a ) \cap \mathbb Q=: C_a$$
Now, a set $C_a$ satisfies the four axioms of definition of a Dedekind cut, and moreover, each Dedekind cut is exactly one and only one of the sets $C_a$. Indeed, if $C$ is a Dedekind cut, and $a= \sup(C)$ then $C=C_a$.
Because of this, the correspondence $a \leftrightarrow C_a$ is a bijection between the real numbers and the set of Dedekind cuts.
Now, the issue is what to do if you don't know the real numbers, but you want to construct them. Well, you can construct the Dedekind cuts without assuming the existence of a real number. This way we can give an axiomatic construction for the reals.
And always remember, when we think about Dedekind cuts as "real numbers", what we mean is that, once we formally define $\mathbb R$, for each cut $C$ there exists an unique real number $a$ such that the cut $C$ is exactly $$C= (- \infty, a) \cap \mathbb Q$$
