Do Dedekind cuts fill the gaps in the rational number line?

[FallenApple]

So I'm reading the first page of Rudin. This is right after proving that there is no rational solution to x^2=2.

How does this show that the rational number system has gaps? All it shows is that A has no upper bound and B has no lower bound. Is it really necessary to have two sets A&B perfectly glued together? It seems to show that the real number line has gaps(if unfilled), but not the rational number line.

Also, proving there is no rational number x such that x^2=2 is different from saying that there exists a number x
such that I can take the square of it and get 2.

 

[Delta2]

Well the book is trying to say that since there is no rational x such that $x^2=2$ there is a gap in the rational line at the point $x=\sqrt{2}$. The discussion you post just says that you can approach $\sqrt{2}$ as close as you want with a rational number sequence $p_{n+1}=\frac{2p_n+2}{p_n+2}$ (and this approach can be either from the upper or from the lower side , chose $p_0=1$ for the lower and $p_0=3$ for the upper approach) but you can never get to it using just rational numbers. (The limit of sequence $p_n$ is $\sqrt{2}$ but there is no k such that $p_k=\sqrt{2}$).

 

[Logical Dog]

the real number line has no gaps, only the rational smf other subsets. i believe the property is called completeness. i don't know how to say it rigourously but the definition I got is that if a number set fills out a number line completely, without any gaps then it is complete. This is different from density which states between any two elements there exists a third in between them, which also requires another property called order he refers to density in remark 1.2 the limit of a sequence need not be an element of the sequence.

ROOT 2 is NOT IN the set { a/b | a, b in integers and b not equal to zero}

the irrational numbers or the root of 2 exist, but they are carrying infinite non repeating decimal components and hence would take forever and ever to list them. this is what I understand. Because of this all irrational numbers are denoted by the set R\Q = R - Q. the difference set between Reals and rationals, it leaves you with points like root 2, pi and other famous irrationals.

Is it right to say Set A in the example is the interval [0, root 2) and B is (root 2, infinity)?

 

[cpsinkule]

"Gaps" may not have been the best wording, but he is trying to convey that the rational numbers are lacking "something". In this example, you see that you can get very close to root 2, indeed arbitrarily close to root 2, but we both know root 2 is not rational, so the rationals are lacking something. The property it lacks is called completeness, it will be discussed in the section on sequences. A metric space is complete if every cauchy sequence converges in that space, but that is also equivalent to saying any bounded subset of that space has a supremum, which is what i think this section covers. In this example, your subset is all rationals whose square is less than or equal to 2, and you see this set has no supremum. This missing supremum is the "gap" that we call root 2, all of the gaps would be the set of irrational numbers. When you join the irrationals to the rationals you remove all of the "gaps" and you have a nice complete space where all of your sequences which should converge, do converge.

 

[FallenApple]

Well the book is trying to say that since there is no rational x such that $x^2=2$ there is a gap in the rational line at the point $x=\sqrt{2}$. The discussion you post just says that you can approach $\sqrt{2}$ as close as you want with a rational number sequence $p_{n+1}=\frac{2p_n+2}{p_n+2}$ (and this approach can be either from the upper or from the lower side , chose $p_0=1$ for the lower and $p_0=3$ for the upper approach) but you can never get to it using just rational numbers. (The limit of sequence $p_n$ is $\sqrt{2}$ but there is no k such that $p_k=\sqrt{2}$).

Makes sense. Without R, one cannot go from one set to another. If I start in B and try to go to A, I would just keep transversing lower and lower values of B. There needs to be a number between B to get to A.

 

[Svein]

Rudin talks about Dedekind cuts (https://en.wikipedia.org/wiki/Dedekind_cut) which is a way to construct real numbers.