I Questions on ##\mathbb{R}##

[issacnewton]

TL;DR Summary

question on $\mathbb{R}$ as defined in Rudin's analysis

I am learning analysis from Rudin's famous book (baby rudin). I am confused about how $\mathbb{R}$ is defined in this book. In the appendix of chapter 1, he says that members of $\mathbb{R}$ will be certain subsets of $\mathbb{Q}$, called cuts. Is this definition different from the way we understand $\mathbb{R}$ in applied sciences ? If it is, then where do irrational numbers reside ?

Thanks $\smile$

 

[andrewkirk]

Yes it's different, as each cut is a set of rational numbers, while we don't think of real numbers as being sets of other numbers. We think of them (or at least I do) as pin-point, atomic objects, not collections.
The purpose of Dedekind's cut construction is to show that, from basic set theory, we can construct a set of objects, together with operations of addition and multiplication on them, that have all the functional properties we want a number to have (commutativity of addition and multiplication etc), as well as completeness: that every Cauchy sequence converges.
In a sense all he's doing is demonstrating that it's reasonable to define real numbers as having the properties we give them, since we can construct a set that has those properties.
In the language of Model Theory, the set of Dedekind cuts is a Model for the Theory of real numbers (comprising the field axioms plus ordering axioms plus completeness axiom).

 

[fresh_42]

There is an alternative way besides Dedekind cuts to define the real numbers, e.g. in Hewitt, Stromberg, Real and Abstract Analysis (GTM 25) that uses an analytical/topological approach via Cauchy sequences. They basically define the real numbers as limits of Cauchy sequences modulo their subset of all Cauchy sequences that converge to zero. The real numbers are then a set of equivalence classes. I find this more instructive but there is still work to do for the technical details (5 pages in the book) and check of the axioms.

 

[issacnewton]

Ok, so any real number can be represented as a cut (say $\alpha$), which is a subset of $\mathbb{Q}$ with the following three properties

(I) $\alpha \ne \varnothing$, and $\alpha \ne \mathbb{Q}$
(II) If $p \in \alpha$, $q \in \mathbb{Q}$, and $q < p$, then $q \in \alpha$
(III) If $p \in \alpha$, then $p < r$ for some $r \in \alpha$

This means that its guaranteed that a cut will always be non empty. And any real number is a cut, which is basically a set formed using rational numbers.

Thanks

 

[PeroK]

Ok, so any real number can be represented as a cut (say $\alpha$), which is a subset of $\mathbb{Q}$ with the following three properties

(I) $\alpha \ne \varnothing$, and $\alpha \ne \mathbb{Q}$
(II) If $p \in \alpha$, $q \in \mathbb{Q}$, and $q < p$, then $q \in \alpha$
(III) If $p \in \alpha$, then $p < r$ for some $r \in \alpha$

And any real number is a cut, which is basically a set formed using rational numbers.

Thanks

Think first of how the rational numbers are developed from the integers, as pairs of integers. That development feels natural as $q = \frac m n$ is something we are already familiar with. Mathematically, it provides a way to develop a new set from a set we already know.

One way to represent a real number is by the set of rational numbers less than or equal to it. This is not something we normally do, but if you think about it, it works quite well. If the real number is rational, then it's the set up to and including the rational number itself. If the real number if irrational, then it's the set of all rationals less than the number, with the real number being the supremum of the set.

That discussion assumes, of course, that we already have the real numbers defined formally in some way. However, that gives us the idea of how we could define the real numbers as sets of rationals in the first place. That's what the Dedekind cuts do. This gives us a formal way to define the real numbers without presuming they exist in the first place. And, since the rationals were developed from the integers, this gives us a way of formally developing the reals from the integers.

Once that's achieved and you know that this set has all the properties you want, you can start calling it $\mathbb R$ and treating its elements like the real numbers we already thought we knew all about.

Note that if you didn't do this development (or something similar), you might always be concerned that the set of real numbers may have a logical problem somewhere and be self-contradictory in some way. It was self contradictions in the original set theory that started the whole idea of developing mathematics formally in this way - to ensure that everything you do has a provably solid foundation.

I'd be suprised if Rudin doesn't stress that point somewhere in his book.

 

[issacnewton]

This makes sense.