# What is {ℝ}?

by kokolovehuh
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 P: 23 Hi, Someone I know tried to convey me the meaning of {$ℝ$}, stating it represents a set of real numbers. But using notation, {$ℝ$}, is implying that the real space is (improperly) contained in a set, and I don't think this makes any logical sense. On the other hand, we can say {$x \in S | \forall S \in ℝ$}, etc....or simply $x \in ℝ$. Another way of thinking about this is instead of putting your foot in a sock, you are putting the sock into your foot and it's disturbing. Am I right or wrong? Thanks
 Sci Advisor HW Helper PF Gold P: 3,172 It is perfectly valid to put sets inside of other sets. For example, we may consider the set of all subsets of the complex numbers ##\mathbb{C}##. This is called the power set of ##\mathbb{C}## and is sometimes given the notation ##\mathcal{P}(\mathbb{C})## or ##2^\mathbb{C}##. Any subset of ##\mathbb{C}## is an element of ##\mathcal{P}(\mathbb{C})##. For example, we have ##\mathbb{R} \subset \mathbb{C}## and ##\mathbb{R} \in \mathcal{P}(\mathbb{C})##. We can form subsets of ##\mathcal{P}(\mathbb{C})## in the usual way, by putting elements of ##\mathcal{P}(\mathbb{C})## into a set. Thus ##\{\mathbb{Z}, \mathbb{Q}, \mathbb{R}\} \subset \mathcal{P}(\mathbb{C})##, and a special case is a subset containing only one set, such as your example: ##\{\mathbb{R}\} \subset \mathcal{P}(\mathbb{C})##.
 P: 23 Thanks bjunniii, you have a good point. However, let me rephrase my doubt. We want to use a notation to represent a set of all real number, say $X$. It is immediately apparent that $x \in X \subseteq ℝ$ for some real number $x$. In this case, we are not not considering any stronger set, for instance, $P(ℂ)$ as you mentioned. Now having limited ourselves to real space, it is rather redundant to say the set is represented as {$ℝ$} because since $ℝ$ is not a proper subspace in this case. This is the reason why I said "this does not make any logical sense"; I am ridiculed by those curly brackets! Instead, we could simply write $X \in ℝ$ that give a much more direct and sensible idea of what space we are talking about. Do you agree?
 Mentor P: 18,040 What is {ℝ}? The set $\{\mathbb{R}\}$ is a set which contains only one element. Its element is $\mathbb{R}$. There is no reason why such a construction would not be allowed. It is true, however, that sets like $\{\mathbb{R}\}$ don't play a big role in mathematics.
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 Quote by kokolovehuh But using notation, {$ℝ$}, is implying that the real space is (improperly) contained in a set, and I don't think this makes any logical sense.
Unless you state otherwise, all set theory is done in ZFC, and one of its axioms is the axiom of pairing. It asserts: given any two sets A and B, there exists set C with exactly those two elements, i.e. C = {A, B}.

So it does logically exist (in this case A = B = ℝ).
 P: 23 @micromass, @pwsnafu, I see what you are saying. Overall, you have convinced me {$ℝ$} is possible. But, The notation with curly bracket is directly defining the single element in this set as the real space which is essentially another set. I was simply saying there is no necessity to put real space as a subset of a set in the first place; there are no other disjoint elements.
 Quote by kokolovehuh Hi, Someone I know tried to convey me the meaning of {$ℝ$}, stating it represents a set of real numbers