netzweltler
Posts:
342
From:
Germany
Registered:
8/6/10


Re: Countably Infinite Sets
Posted:
Dec 29, 2012 4:19 AM


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?

