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Topic: Countably Infinite Sets
Replies: 7   Last Post: Dec 29, 2012 12:45 PM

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netzweltler

Posts: 293
From: Germany
Registered: 8/6/10
Re: Countably Infinite Sets
Posted: Dec 29, 2012 4:19 AM
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On 29 Dez., 00:58, James Waldby <n...@valid.invalid> wrote:
> On Fri, 28 Dec 2012 14:16:14 -0800, netzweltler wrote:
> > On 28 Dez., 21:53, James Waldby  wrote:
> >> On Fri, 28 Dec 2012 12:51:29 -0600, fasnsto wrote:
> >> > "netzweltler" <reinhard_fisc...@arcor.de> wrote ...
> >> >> Does the set {{1}, {1,2}, {1,2,3}, ...} contain all natural numbers?
>
> >> > no, it would contain the set of all natural numbers.
>
> >> Let S = {{1}, {1,2}, {1,2,3}, ...}.  The infinite union of the members
> >> of S would be a set containing all natural numbers, but no member of S
> >> is itself a set containing all natural numbers.  Of course, as noted in
> >> some earlier replies, S is isomorphic to the naturals.

> > I understand, that the union of the members of S contains all natural
> > numbers. But, did I really write the union here?
> > {{1}, {1,2}, {1,2,3}, ...}

>
> Here in math.sci, one expects (or at least hopes for) accurate use of
> terminology.  {{1}, {1,2}, {1,2,3}, ...} is not a union as such.  This
> set is a set of sets.  That is, this set is a set where each member is
> a set.  This set is not a list of sets or a list of rows because a list
> is not a set and a set is not a row.
>

> > Isn't it a list of infinitely many rows like
> > 1
> > 1,2
> > 1,2,3
> > ...

>
> > Does the list contain all natural numbers? Is this list an union?
>
> A list is not a union.  Regarding the list of infinitely many rows
> indicated just above, if we call the display of all the rows a
> "tableau", we could say that the tableau contains all natural numbers.
> We could also say that the list contains the natural number 1, and
> for each natural number contains a list that ends with that natural
> number.  But in my opinion it is wrong to assert that every natural
> number appears in this list of lists, because each thing (except 1)
> appearing in this list is in turn a list, not a number.
>
> --
> jiw


I guess what I am heading for is the ultimate abuse of set notation
then:

1
1,2
1,2,3
...

Since I can see the set of natural numbers infinitely many times
represented in the diagonals of this list I would reorder the elements
of this list and write

{{1}, {1,2}, {1,2,3}, ...} = {{1,2,3,...}, {1,2,3,...},
{1,2,3,...}, ...}

in set notation then. Doesn't make sense at all, does it?




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