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Does the symbol $\mathbb{R}$ refers to the set of the real numbers or to the field of the real numbers?

In other words, when you say "$x \in \mathbb{R}$", are you just sayng that $x$ is a real number? Or, on the other hand, that $x$ is a real number and obbeys the real numbers field laws (AKA properties): associative, additive identity, etc.?

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    $\begingroup$ It means both depending on what you want to do $\endgroup$ – Guy Fsone Sep 29 '17 at 10:46
  • $\begingroup$ Usually any collection of mathematical objects is a set and a field is a set with certain structures, depending on the context. $\endgroup$ – Vim Sep 29 '17 at 10:46
  • $\begingroup$ Where is the "why not both?-meme when you need it?... Btw, there are also other things that $\mathbb{R}$ might stand for, e.g. a quotient of certain rational sequences. $\endgroup$ – Dirk Sep 29 '17 at 10:47
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    $\begingroup$ That's like asking “Is Donald Trump a human being or the President of the United States?” $\endgroup$ – José Carlos Santos Sep 29 '17 at 11:01
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    $\begingroup$ I'm not sure he's a human being… $\endgroup$ – Bernard Sep 29 '17 at 11:11
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Sometimes in model theory, when this distinction is important, you will see something like $$ \mathfrak R = \langle\mathbb R, +, \times, 0, 1, < \rangle $$ But in the rest of mathematics we write $\mathbb R$ for whatever we want. When someone asks about "open sets" in $\mathbb R$, for example, then of course we assume the usual topology. When they ask about metric properties, we assume the usual metric. When they ask about uniform properties, we assume the usual uniformity. And so on.

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In the formal context, $\mathbb{R}$ refers to the set of all real number, whereas $(\mathbb{R}, + , *)$ refers to the field of real number with the usual addition and multiplication.

Note that, a field is the $3$-tuple consisting of a set, and two binary operation, which satisfying some conditions.

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$\mathbb{R}$ is merely just a regular old set.

But once you equip it with some sort of additional structure such as operators +, *. And if those 3 guys $(\mathbb{R}, +, *)$ behave as a field should (satisfy field axioms), then you can call it a field.

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