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 ?
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
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....