Do you not see the difference between "the
nth set ends with the number n" and "
n ever [has] to end on the nth set"? Words and grammar matter, which is why I keep returning you to definitions.
The definition of S deals with each set in S
individually. The nth set listed is {1, 2, 3, ..., n}.
That set ends.
I said nothing about
n ending! Each set is defined for
one value of n. There are an infinite set of sets, one for each natural number n.
- For a given set {1, 2, 3, ..., n}, n is a fixed number; you can't talk about it ending.
- For all of S, n increases without bound; it never ends.
But you appear to be confusing these two ideas.
In order to understand mathematical statements, you must learn to
focus on exactly what is said, and not imagine relationships that are not mentioned. (You gave
one definition for that set; there is no other definition! You are the master of what that set is.)
When I talk about "the notation for S", I am talking about
what you said at the very start:
That notation
means what I have described: a set containing finite sets. That is
your definition. It determines what we are talking about.
Now you are talking about
imagining a different notation:
{{1}, {1, 2}, ... {1, 2, 3, ... n}, {1, 2, 3, ... n ...}...}
You are probably right that this explains your confusion.
This does not mean the same thing! Here you have added a rather nonsensical "set", {1, 2, 3, ... n ...}; this notation means nothing, because n appears just to be one element in the middle of an infinite set, and in itself plays no role. That set would just be {1, 2, 3, ...}, the set of
all natural numbers. More important, you then follow this with another ellipsis (...), as if there is more beyond that "set". There can't be.
To put it another way, you really can't use n in such a definition of S, because
n is not defined anywhere. It would normally be some fixed (but unknown) number; for example, if n=5, then your "set" would be {{1}, {1, 2}, ... {1, 2, 3, ... 5}, {1, 2, 3, ... 5 ...}...}, which would be just {{1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, {1, 2, 3, 4, 5},
{1, 2, 3, ... 5 ...}...}, and, again, that last part would just be one more set, the set of all integers.
So the notation you are imagining is really bad notation, and it does not mean what you wrote in the first place.
Part of the problem is that this sort of notation with ellipses is a little ambiguous. We have other notations to clarify such ideas. Your set might be defined in set-builder notation as
[imath]S = \{\{1, 2, ..., n\} | n\in\mathbb{N}\}[/imath]
or, for even more clarity (since the reader might be confused by "{1, 2, ..., 1}" when n=1, which just means {1}),
[imath]S = \{\{k\in\mathbb{N}\ |\ 1\le k\le n\}\ |\ n\in\mathbb{N}\}[/imath]
In this form, n has a defined meaning, for each element (set) in S. We aren't using ellipses to imply a pattern.
This is why I say that you must
take the details of whatever you write seriously -- because
we will take you at your word! Mathematicians are very careful to write exactly what they mean, and to read exactly what is written. If you don't really mean what you write, then you are just going to have endless arguments about things no one agrees on.
