"Virgil" <Virgil@home.esc> wrote in message news:Virgil-1E8B09.email@example.com... > In article > <c8idnc6JG6hv34LWnZ2dnUVZ_q-dnZ2d@giganews.com>, > "K_h" <KHolmes@SX729.com> wrote: > >> It should be pointed out that N is a limit set even if N >> is >> initially given by a definition that doesn't involve the >> notion of a limit. > > > The issue between Dik and WM is whether the limit of a > sequence of sets > according to Dik's definition of such limits is > necessarily the same as > the limit of the sequence of cardinalities for those sets. > > And Dik quire successfully gave an example in which the > limits differ.
I suspect those definitions are not valid. The definition I used is the one on wikipedia and is generally `standard' -- as I've seen it in numerous places, including books and websites. That definition is below. Using this definition it is possible to prove that N is a limit and I presented a proof of this in my previous post.
Let /\ = Intersection Let \/ = union Define infimum and supremum as follows: liminf X_n=\/(n=0, oo)[/\(m=n, oo) X_m] (n-->oo) limsup X_n=/\(n=0, oo)[\/(m=n, oo) X_m] (n-->oo)
> Here is another way to see that N is a >> limit even if you consider it bad taste to define it in >> those terms: >> >> Let \/ = union >> >> Let /\ = Intersection >> >> Define infimum and supremum as follows: >> >> liminf X_n=\/(n=0, oo)[/\(m=n, oo) X_m] >> (n-->oo) >> >> limsup X_n=/\(n=0, oo)[\/(m=n, oo) X_m] >> (n-->oo) >> >> If these two are the same then the limit exists and is >> both >> of them. > > The issue is not whether the naturals are such a limit but > whether for > every so defined limit the cardinality of the limit equals > the limit > cardinality of their cardinalities, which is different > sort of limit.
It doesn't sound like a valid limit. With the definitions I provided it is possible to prove, in your words, that "the cardinality of the limit equals the cardinality of their cardinalities". It is actually intuitively obvious: the set of finite cardinals 0,1,2,3,... and the set of finite ordinals 0,1,2,3,... both have cardinality ALEPH_0.