0

In set theory cardinality of empty set is 0 but How the cardinality of $|\{\emptyset\}| $ is $1$ ? Well one can say $\emptyset$ is an element here but $\emptyset$ set has no elements in it so when it enclosed in a curly bracket how the new set has an element now as the enclosed set has no element?

Arnaud D.
  • 20,884
Swapnil Dash
  • 11
  • There are two commands for the empty set: \emptyset and \varemptyset – ajotatxe Oct 28 '19 at 14:37
  • 1
    Why would it matter that $\varnothing$ has no members? You're counting the members of ${\varnothing}$, not the members of its members. – Malice Vidrine Oct 28 '19 at 14:40
  • 1
    ${\emptyset}$ is the set that has one element in it - the empty set. It is a set of sets, just like $|{{1,2}}|=1,$ not $2$. – Adrian Keister Oct 28 '19 at 14:40
  • @ajotatxe : You mean \varnothing not \varemptyset – Toby Bartels Oct 28 '19 at 14:41
  • You are mixing up mereological sums and sets of ZFC. It is interesting. – Slup Oct 28 '19 at 14:43
  • Do not forget that the "depth" here matters. $\emptyset\neq {\emptyset}$ and is further different than ${{\emptyset}}$. These are all different objects. For some reason early students might think that you can remove some of these outer braces and not affect things but that is wholly untrue, it affects it very much. – JMoravitz Oct 28 '19 at 14:43
  • See also the post What is an empty set ? – Mauro ALLEGRANZA Oct 28 '19 at 14:45
  • @JMoravitz. Thanks for the details . I was thinking that. You said if we remove some of these outer braces it will affect the set but I still don't understand how ? – Swapnil Dash Oct 28 '19 at 14:52
  • Consider an analogy of sports teams. The Astros and the Nationals are playing in the world series. Now, the Astros happen to have several players like Justin Verlander, Zack Greinke and so on... We could have thought of it as Astros = {Justin, Zack, ... } and similarly Nationals as {Raudy, Victor, ...}. Now, I want to talk about the teams of who is playing in the world series right now. That would be ${\text{Astros}, \text{Nationals}}$. There are only two teams who are playing. I am not interested right now in talking about the number of players, I am only interested in teams – JMoravitz Oct 28 '19 at 15:00
  • Sure, if I wanted to, I could talk about the number of players, but that is an entirely different question. Now, if your complaint is about the set containing the emptyset as not having any "content" deep inside of it, however deep, that is irrelevant to the point. Consider another analogy of Christmas decorations. Some people put empty wrapped boxes underneath of the tree so that it looks pretty. That doesn't mean that the boxes aren't there. If we ask how many boxes are under the tree, even the empty ones should count. – JMoravitz Oct 28 '19 at 15:03

1 Answers1

3

Think of $\emptyset$ as an empty box.

Think of $\{\emptyset\}$ as a box with an empty box inside. The outer box has one thing.

ajotatxe
  • 65,084