What is ℝ? Understanding Real Numbers

[kokolovehuh]

What is {ℝ}?

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

 

[jbunniii]

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: $\{\mathbb{R}\} \subset \mathcal{P}(\mathbb{C})$.

 

[kokolovehuh]

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?

 

[micromass]

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.

 

[pwsnafu]

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).

 

[kokolovehuh]

@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.

 

[pwsnafu]

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.

I was about to write "what's your point?" until I re-read the original post:

Hi,
Someone I know tried to convey me the meaning of {$ℝ$}, stating it represents a set of real numbers

Your friend is wrong. The set of real numbers is ℝ. (I bold "the" because there is only one.)

But $\{\mathbb{R}\} \neq \mathbb{R}$. The left hand side is "a set with one element, and that element is ℝ". They are not the same thing.

 

[kokolovehuh]

Aha, there we go. Thanks a lot pwsnafu for the verification! as well as others for your time generous reply.