How many elements are there in {1,1,1}?
Thanks for answer and reference.
However, suppose I rephrase question as follows:
What is 1? 1 is an integer. How many integers are there in {1,1,1}?
It seems that no matter how I answer the question "what is 1?" (integers, apples, etc) the answer has to be three. Your reference calls three the cardinality of the set.
But if 1 designates an individual (Joe), {1,1,1} doesn't exist (is impossible), so given that {1,1,1} exists it must have three members. Neat, I like that.
My train of thought can be a pain at times- it's the way I think. I really don't mean to be argumentative. I'm trying to learn and understand and hope the back and forth is useful to others.
EDIT: I was going to edit out my reply above, which was starting to look like blah blah- but I'll leave it and try another approach.
How many elements (members) in {1,1,1,3}: Four, three of which are equal.
(I'm trying to feel comfortable with countability by playing with it. Should I be playing with it on a Forum? Not sure, really for others to decide.)
Damn! Still not there. What does equal mean? The same, or has the same value?
Because $\displaystyle \{1,1,1\}=\{1\}$ there is only one element in that set. The reason is the agreed upon rules of set theory.
As for the question “What is 1?” that is a different question.
Here is one of many ways to answer that.
For any set $\displaystyle X$ define $\displaystyle \mathcal{S}(X)=X\cup \{X\}$.
Now for the definition of Natural Numbers.
$\displaystyle \begin{gathered} 0 = \emptyset \hfill \\ 1 = \mathcal{S} (0) \hfill \\ 2 = \mathcal{S} (1) \hfill \\ 3 = \mathcal{S} (2) \hfill \\ \quad \vdots \hfill \\ \end{gathered} $.
So you see that $\displaystyle 1=\mathcal{S}(0)=\emptyset\cup \{\emptyset\} $.
Thus the answer to “How many integers are there in” $\displaystyle \{1,2,2,3,3,3\}$ is three, we understand that answer in purely about sets. It not about counting but about set membership.
On the other hand if we were to ask “how many members in the string ‘122333’ then the answer is six.
So context is everything.
Thanks. That's the answer I was looking for: "The reason is the agreed upon rules of set theory."
I'm really not interested in the meaning of "1." I just took it as an arbitrary symbol.
Still a little confused though. Given the set {a1,a2,a3} how many members when a1=a2=a3=5? Do we go from 3 to 1 depending on the value? I think the answer has to be, it depends on whether a1,a2,a3 are symbols or values of the symbols.
EDIT: You guys are right {a,a,a} is the same thing as symbol or value. So the answer is 1. You learn something every day.
An interesting perspective on above discussion comes from Rudin '64 theorem 2.14: A countably infinite union of countably infinite sets is a countably infinite set.
In the course of the proof Rudin says, in effect:
{x1,x2,x3,x4,x5} is equivalent to {1,2,3,4,5}
If x2=x3=x4 ("are elements in common") then {x1,x2,x3,x4,x5} is equivalent to {1,2,3},
which is what emakorov and Plato said, equal (common) elements are counted once (assigned the same integer).