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I am reading Baby Rudin and worked through the Appendix of Chapter 1, on the construction of the real numbers with Dedekind cuts. I understand the construction in terms of the cuts, and describing them as the elements of an ordered field $\mathbb{R}$.

Now what I think I don't understand is what $\mathbb{R}$ itself is. At least as far as I understand, $\mathbb{R}$ is constructed as a set of particular sets called cuts. But are the elements of $\mathbb{R}$ actually cuts? I ask this because the book says if, say, $\alpha$ is a cut, then $\alpha \in \mathbb{R}$.

But aren't the elements of $\mathbb{R}$ just numbers? I know each cut corresponds to a number, like for example the cut: $$r^*=\{x: x<r\}, r \in \mathbb{Q}$$

corresponds to the rational $r$. But the cut $r^*$ is a set of numbers, not the number r itself. I am just really confused with this since I just started learning real analysis. Hope this question makes sense, and thanks in advance.

PinkyWay
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    This might sound stupid, but the elements of $\mathbb{R}$ are what you define them to be. Put differently, you ask whether the elements of $\mathbb{R}$ aren't just numbers, but what is a number to begin with? – Ben Steffan Feb 10 '20 at 16:54
  • But are the elements of R actually cuts? Yes they are. and the real number is the set of all cuts on rational numbers. and yes a r* is a set of rational numbers. I did not read the book but the following might help. some text books define natural numbers as $$0= {\emptyset}, 1 = {\emptyset,0}, 2 = {\emptyset,0,{0}}, ...$$, hope you see the similarity – Aven Desta Feb 10 '20 at 17:00
  • Also remember how you defined the rational numbers. a positive rational number is ordered pair of a natural number, as @Ben said, numbers are what you define them to be. the most important thing is that the definition corresponds to the way we use them and each property we already know are included there. So each real number is a cut, or a set of rational numbers satisfying a certain property. – Aven Desta Feb 10 '20 at 17:06
  • @BrianSteffan I knew the definition of numbers would cause trouble. Let me say a rational number is an element of Q. The cuts are defined as sets of these elements of Q. We also Q is a subset of R, thus every element in Q is in R. So say r is in Q. Then how can r be in R if r itself is not in R? It is just the r* that corresponds to it, but clearly r=/=r* –  Feb 10 '20 at 17:08
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    @NicolásMaílloGómez, cuts define real numbers, that is rational and irrational numbers. Now we have a different definition of rational numbers. take for example r*, it is a rational number iff r is a rational number in our previous definition of rational numbers. This is the same with integers-rational numbers case. After defining rational numbers we redefine integers as a set of all ordered pairs (a,b) where b = 1 – Aven Desta Feb 10 '20 at 17:13
  • @NicolásMaílloGómez You say that $\Bbb Q$ is a subset of $\Bbb R$. Well... certainly something that behaves exactly the same as $\Bbb Q$ is a subset of $\Bbb R$, so much so to the point where we might call them the same thing in many common settings. You are touching upon type theory and the idea that the $1$ of the natural numbers is in fact technically a different mathematical object than the $1$ of the integers or the $1$ of the rationals or the $1$ of the reals and so on... – JMoravitz Feb 10 '20 at 17:27
  • Most people will choose to interpret the number $1$ as whatever the most complicated type it is that is required of the problem they are currently trying to solve. This is analogous to how "Smart Typecasting" works in programming languages like Python or Javascript. The question is "$a+0i$" in every way equal to just "a" gives some more information and there is a great paper linked there titled When is one thing equal to some other thing?. – JMoravitz Feb 10 '20 at 17:32
  • @JMoravitz alright, I will be sure to check that out, thanks! –  Feb 10 '20 at 17:33
  • Also highly relevant: Is it an abuse of language to say "the integers," "the rational numbers," or "the real numbers," etc.? – JMoravitz Feb 10 '20 at 17:36

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If you define the real numbers as cuts, then yes: each cut is a set and $\mathbb R$ is a set of sets. But this is not the only way to construct the real numbers. It does not really matter which definition you use, as long as your definition can be proven to be equivalent to my definition. That is, there is a one-to-one mapping $\Phi$ from your real numbers to my real numbers, that preserves all relevant operations: e.g. $\Phi(x+y) = \Phi(x)+\Phi(y)$, where the first $+$ is your definition of addition of real numbers and the second $+$ is my definition of addition.

We don't ever care about what something "really is", just what operations can be defined on it and what properties those have.

Robert Israel
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  • I think the book kind of mentions it, but it talks about two distinct sets Q and Q, where the elements of Q are rational numbers and the elements of Q are the cuts corresponding to those numbers. It is also said that Q=/=Q*, but that they are isomorphic (I think this is a concept in abstract algebra and I really don't know much about it) –  Feb 10 '20 at 17:35
  • But the thing I want to take is: how should I think about the set R? –  Feb 10 '20 at 17:36
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    You should think about it as the set of real numbers. No mathematician ever actually thinks of the reals as Dedekind cuts unless forced to do so. – Robert Israel Feb 10 '20 at 21:29
  • OK, thank you very much. I still think this topic is really confusing though. –  Feb 10 '20 at 21:31