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Before you read, Yes I have seen the other posts on stack exchange about this topic, but I have something new.
I'm an elementary student that just started sets so please bear with me :)
I've seen online that the set of all real numbers is simply defined as: $$ \Bbb R = \{x \mid x \in\Bbb R \}, $$ which doesn't seem right. I randomly came up with this notation while studying sets $$ \Bbb R = \{x \mid a \le x \le b \text{ and } a, b \in \Bbb{Q} \}, $$ where $\mathbb{Q}$ is the set of all rational numbers.
So a number like $\pi$ would be greater than $3$ and less than $3.15$ but you couldn't say an imaginary number like $i$ is greater or less than any rational or irrational number.

What is wrong with this notation? (if there is anything wrong?)

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    $\begingroup$ Your notation is certainly broken, but maybe Dedekind cuts is what you have in mind? $\endgroup$
    – Dmitry
    Dec 12, 2023 at 23:36
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    $\begingroup$ You are certainly correct that defining the real numbers as the set of all real numbers is circular, at best. But defining the real numbers properly is very hard, as you will have seen from the other posts you looked at. Really, there aren't any shortcuts...if there were, that's the way people would present it. $\endgroup$
    – lulu
    Dec 12, 2023 at 23:43
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    $\begingroup$ What is wrong is that there is always a rational number between two distinct rationals (take the average). You have to find a precise way of defining a number that is not rational. Dedekind cuts do that (eg GH Hardy's Pure Mathematics gutenberg.org/files/38769/38769-pdf.pdf). $\endgroup$ Dec 12, 2023 at 23:45
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    $\begingroup$ btw your intuition is very good - not many would have got as far as you by themselves $\endgroup$ Dec 12, 2023 at 23:47
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    $\begingroup$ @MarkBennet What if we defined it as "$R$ is the set of $x$ such that a≤x≤b for some pair of integers $a$ and $b$"? $\endgroup$ Dec 13, 2023 at 4:05

2 Answers 2

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While the existing answer mentions some good things, I believe that its explanation is a bit backwards, and it misses the weakest point in your definition: namely, attempting to use $\le$ leads to circular dependency.

Wikipedia provides a good description of $\le$. Namely, for a given set $P$, it's a binary relation on $P$ which satisfies some properties. While theoretically you can define it however you want (as long as it satisfies the conditions), there are standard choices of $\le$, e.g. for rational and real numbers.

Note that for different sets, the relations are different: $\le$ on $\mathbb Q$ is not the same as $\le$ on $\mathbb R$, simply because they have different domains. To avoid confusion, let's denote them as $\le_{\mathbb Q}$ and $\le_{\mathbb R}$ respectively. Now the questions is: which $\le$ do you use in your definition?

  • If it's $\le_{\mathbb Q}$, then $x$ must be rational (otherwise, $a \le_{\mathbb Q} x$ is not defined). So your resulting set must only have rational numbers.

  • On the other hand, if you want to use $\le_{\mathbb R}$, then things seemingly work out. However, note that $\le_{\mathbb R}$ is a binary relation on $\mathbb R$. So, to define $\mathbb R$, you use $\le_{\mathbb R}$, which in turn requires $\mathbb R$ to be defined. So you have a circular dependency.

To emphasize the latter point, let's be a little more pedantic (which is a good practice when defining basic algebraic structures). For the sake of argument, assume that $\mathbb R$ and $\le_{\mathbb R}$ are already defined. Then you might want to say that $a \le_{\mathbb R} x$ is defined when $a \in \mathbb Q$ and $x \in \mathbb R$. However, strictly speaking, this is not true, since, strictly speaking, $\mathbb Q$ is not a subset of $\mathbb R$: if you look at any common construction of $\mathbb R$, then $\mathbb Q$ clearly doesn't belong to this construction; e.g. in Cauchy construction, a rational number is not a sequence of rational numbers.

Instead, there exists a very natural mapping $id\colon \mathbb Q \to \mathbb R$, which maps a rational number to the "equivalent" real number: e.g., in Cauchy construction, you can map $q \in \mathbb Q$ to a sequence $q, q, q, \ldots$. This mapping satisfies all of the properties you might possibly expect, so it's convenient to equalize $q \in \mathbb Q$ with its image $id(q) \in \mathbb R$. Hence, with a little abuse of notation, people write $\mathbb Q \subseteq \mathbb R$. (Similarly, $\mathbb N \subseteq \mathbb Z \subseteq \mathbb Q \subseteq \mathbb R \subseteq \mathbb C$ are all abuses of notation, and another answer describes the mapping $\mathbb R \to \mathbb C$.)

If we are pedantic, then for $a \in \mathbb Q$ and $x \in \mathbb R$, instead of $a \le_{\mathbb R} x$ we must write $id(a) \le_{\mathbb R} x$. However, to define $id$, you must define $\mathbb R$ first. So we again have another circular dependency.

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The main problem with the approach you're taking is that it assumes that you're already in a certain context, and usually that context doesn't exist until you've actually constructed the real numbers to begin with. If we look at

$$\{ x : a \leq x \leq b; a, b \in \mathbb{Q}\}$$

This could be a set of rational numbers, in which case it's just the same as $\mathbb{Q}$. But it could also be a set of something like the surreal numbers, in which case you're going to capture a bunch of things with infinitesimal components along the way.

Now if you say something like

$$\{ x \in \mathbb{C} : a \leq x \leq b; a, b \in \mathbb{Q}\}$$

then you do indeed describe the real numbers, but that's only because the relation $\leq$ is generally not defined for complex numbers, and you're also abusing the fact that we typically associate the real part of the complex numbers $\{x + 0i\}$ with the plain real numbers $\{x\}$ in a way that usually doesn't matter, but it kind of matters when you're using structure that is only really defined for the actual real numbers.

So instead, when we want to construct the real numbers, what we actually do is construct a set of objects that captures some idea of "finding the gaps in the rational numbers" (e.g. Cauchy sequences, or Dedekind cuts) and show that it has all the properties we want, and then associate that with the symbols we typically use to represent real numbers.

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    $\begingroup$ I'd like to add that I'm pretty freaked out by the notation $\{ x \in \mathbb{C} : a \leq x \leq b; a, b \in \mathbb{Q}\}$. If I read that in a book I'd assume that either the author made a mistake, or they've defined a special meaning of $\leq$ in the context of their book. That is, $3 \leq 4$ is true; $4 \leq 3$ is false; but $1+i \leq 3$, I wouldn't call "false", I would call "nonsensical". $\endgroup$
    – Stef
    Dec 13, 2023 at 13:10
  • $\begingroup$ That could be simplified(?) to $\mathbb R=\{\,x\in\mathbb C\mid x\le 0\lor 0\le x\,\}$. In this form, it is crucial that $\mathbb C$ does not contain any infinities. $\endgroup$ Dec 13, 2023 at 21:39

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