What is the size of a set of sets of the empty set {{}, {{}}, {{{}}}}?

What is the size of a set of sets of the empty set {{}, {{}}, {{{}}}}?

I am not sure if the empty set, in this case, can be considered as a set and make the size 3 or if it is 2 or 0.

thanks

• one $)$ is meant to be $\}$ I believe. There are three elements in the set. – Alvin Lepik Sep 20 at 14:19
• That's not an empty set. – Lord Shark the Unknown Sep 20 at 14:22
• As you are probably aware, $\{\}$ represents the empty set. $\{\{\}\}$ represents the set containing the empty set, and so on. Your set has in it the empty set, the set containing the empty set, and the set containing the set that contains the empty set. How many distinct elements do you have? – Cameron Williams Sep 20 at 14:22
• The empty set is a set. It just has nothing in it. This set has three things in it. On thing is the empty set, the second thing is a set containing the empty set. The third thing is a set containing a set containing the empty set. That's three things. The things may be nothing more than an empty bag, a bag with an empty bag in it, and a bag containing a bag with an empty bag; so the may have nothing of substance, but they are still things. – fleablood Sep 20 at 15:34
• It would be easier to read if you use \,and \; to add a little space, as $\{\;\{\},\,\{\{\}\},\,\{\{\{\}\}\}\;\}$ – DanielWainfleet Sep 21 at 2:24

Your set can be written as $$\{a,b,c\}$$ where $$a$$ happens to be the emptyset, $$b$$ happens to be the set containing the empty set, and $$c$$ happens to be the set containing the set containing the emptyset.

You should not have any trouble seeing that $$\{a,b,c\}$$ is of size three so long as they are all distinct. The only possible source of confusion in this is in recognizing that each of $$\emptyset, \{\emptyset\}$$ and $$\{\{\emptyset\}\}$$ are not only valid possible elements of a set, but are distinct and different than one another. Indeed, they are all valid possible elements of sets and are all different than one another. Remember that the "depth" of each is relevant and is part of what makes these distinct.

• but {a,b} can be writen as { ∅, a,b} and its size is 2 – Takobell Sep 20 at 15:21
• @TarekHich yes, well, I did include pointing out that $a\neq b$, $a\neq c$ and $b\neq c$ that $a,b,c$ are distinct, and any set $\{a,b,c\}$ where $a,b,c$ are distinct is of size three. – JMoravitz Sep 20 at 15:22
• In the event that $x$ and $y$ are distinct and neither are equal to the emptyset, then $\{\emptyset,x,y\}$ is still of size three. The only reason why $\{\emptyset,a,b\}$ was of size two in your example in the comment was because $\emptyset$ and $a$ were in reality the same element. – JMoravitz Sep 20 at 15:24
• TarekHich: "but {a,b} can be writen as { ∅, a,b} and its size is 2" and JMoravitz: "yes, well". Me: Um, no..... $\{a,b\}$ can not be written as $\{\emptyset, a, b\}$. The empty set is a thing and it is not in $\{a,b\}$ and it is in $\{\emptyset, a,b\}$. Maybe you meant say we can write $\{a,b\}$ as $\{\ \ \ \ , a, b\}$ where "$\ \ \ \$" is nothing. But $\emptyset = \{\ \ \ \}$ is not nothing. It is something. "$\ \ \ \ \$" isn't anything at all. $\emptyset$ doesn't HAVE anything at all but it IS something. – fleablood Sep 20 at 15:42
• @fleablood perhaps you missed where I said $a$ is the empty set. Are you implying {x,y} can't be written in the case that x=y or is somehow different than {x}? Remember we are dealing with sets and not multisets – JMoravitz Sep 20 at 15:59

As you are only interested in the number of elements in the set and not in the elements themself, you basically just have to count the commas, so your set has three elements, so three is its size.

• I have mixed feelings about this answer. On the one hand it directly gets that a set has a list of things-- that the content of the things doesn't matter. But on the other hand it begs and avoids the questions. Also consider $\{apple, \{orange, banana\},banana, \{peach, cherry\}\}$. That has has five commas but only four items. And $\{a,a\}$ has a comma indicating two items but it actually has only one. – fleablood Sep 20 at 19:33
1. The empty set is indeed an element of this set.