question about cardinality of empty set

**issacnewton** · December 5th 2011, 01:48 AM

Hi

I am doing the chapter "Equinemerous sets" from Velleman's "How to prove it" and
I have some doubts. At one point he says that "for each natural number n , let
$I_n=\{i \in \mathbb{Z^+}\lvert i \le n \}$ . A set $A$
is called $\mathit{finite}$ if there is a natural number n such that
$I_n \sim A$ . Otherwise A is $\mathit{infinite}$ . "

Further down he says that "it makes sense to define the number of elements of
a finite set A to be the unique n such that $I_n \sim A$ . This number
is also sometimes called the cardinality of A and its denoted
$\lvert A \rvert$ . Note that according to this definition,
$\varnothing$ is finite and $\lvert \varnothing \rvert =0$ ."

So that will mean that we will need to choose n=0 for an empty set, so that
$I_0 \sim \varnothing$ . Now according to the author's defnition of
$I_n$ , $I_0=\varnothing$ . So

$I_0 \sim \varnothing \Rightarrow \varnothing \sim \varnothing$

which is true since for any set A, we have $A \sim A$ .

Do you think its correct understanding ?

Thanks

**Plato** · December 5th 2011, 02:43 AM

Originally Posted by issacnewton

At one point he says that "for each natural number n , let
$I_n=\{i \in \mathbb{Z^+}\lvert i \le n \}$ . A set $A$
is called $\mathit{finite}$ if there is a natural number n such that
$I_n \sim A$ . Otherwise A is $\mathit{infinite}$ . "
Further down he says that "it makes sense to define the number of elements of
a finite set A to be the unique n such that $I_n \sim A$ . This number
is also sometimes called the cardinality of A and its denoted
$\lvert A \rvert$ . Note that according to this definition,
$\varnothing$ is finite and $\lvert \varnothing \rvert =0$ ."

So that will mean that we will need to choose n=0 for an empty set, so that
$I_0 \sim \varnothing$ . Now according to the author's definition of
$I_n$ , $I_0=\varnothing$ . So $I_0 \sim \varnothing \Rightarrow \varnothing \sim \varnothing$ which is true since for any set A, we have $A \sim A$ .
Do you think its correct understanding ?

That is consistent with his definition because $S_0$ must be $\emptyset$ .

**issacnewton** · December 5th 2011, 04:58 PM

What is $S_0$ ?

**Plato** · December 5th 2011, 05:12 PM

Originally Posted by issacnewton

What is $S_0$ ?

Well look at Velleman's own definition: $S_n=\{i\in\mathbb{Z}^+:i\le n\}$ .
Now what positive integer is less than or equal to zero?

So answer your own question.

Note that Velleman did not specify the nature of n in that definition.
So $S_{0.5}=S_0$

**issacnewton** · December 5th 2011, 05:22 PM

Oh you meant $I_0$ . Ya I get that. Does your book have a different symbol there ? I think you have first edition. anyway...

Velleman's definitions are too formal. Even wikipedia doesn't give such definitions and I have not seen other maths books giving
the definitions so formally. I think he is a set theorist/logician, that's why he is too formal. Or since he is teaching "how to prove it" so
it makes sense to give definitions as rigorous as possible

**Plato** · December 5th 2011, 05:44 PM

Originally Posted by issacnewton

Oh you meant $I_0$ . Ya I get that. Does your book have a different symbol there ? I think you have first edition. anyway...
Velleman's definitions are too formal. Even wikipedia doesn't give such definitions and I have not seen other maths books giving the definitions so formally. I think he is a set theorist/logician, that's why he is too formal. Or since he is teaching "how to prove it" so
it makes sense to give definitions as rigorous as possible

This may surprise you but I have never seen that textbook.
But if you look at Velleman's pedigree you see that one of his advisers was Mary Ellen Rudin (Walter Rudin's wife) and an R L Moore's PhD student. That should at once tell you that her students should be a stickler for precise definitions. There you have some of the best mathematicians of the last century . So if I were you I would rethink that remark.

**issacnewton** · December 5th 2011, 06:26 PM

Well, thanks for the information. Not being from US, didn't know that.

Its strange that even the real analysis books I have seen, don't use such precise
definitions. So that style is not followed everywhere. I was checking Moore's wikipedia page and there is mention of Moore's method. May be this style comes from there....

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