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
. A set
is called if there is a natural number n such that
. Otherwise A is . "
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 . This number
is also sometimes called the cardinality of A and its denoted
. Note that according to this definition,
is finite and ."
So that will mean that we will need to choose n=0 for an empty set, so that
. Now according to the author's defnition of
, . So
which is true since for any set A, we have .
Do you think its correct understanding ?
Thanks
Oh you meant . 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.
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....