LudovicoVan
Posts:
4,165
From:
London
Registered:
2/8/08


Re: Matheology § 300
Posted:
Jul 7, 2013 9:52 AM


"Michael Klemm" <m_f_klemm@tonline.de> wrote in message news:krbng1$i4i$1@solani.org... > Julio Di Egidio wrote: > >> Pardon the basic question, but I do not understand your result. Isn't: >> >> lim_{n>oo} 1/Card(N\Fison(n)) = >> = 1/Card(N\(lim_{n>oo} Fison(n))) = >> = 1/Card(N\N)= >> = 1/Card({}) = >> = 1/0 >> = oo > >> What am I doing wrong? > > The first lim_{n>oo} refers to a sequence of naturals
But that is not a sequence of naturals: N is a set, Fison(n) is a set, N\Fison(n) is a set, Card(N\Fison(n)) is a cardinal (in fact, equal to aleph_0 for all n in N), and 1/Card(N\Fison(n)) is again not a natural (in fact, it is equal to 1/aleph_0, or just to 1/oo, for all n in N).
> and the second to a sequence of sets giving a set as > its result. Without a check it is not allowed to draw the limit under > changement of its meaning inside the expression.
I'd rather not see what other meaning we could give to that limit...
In other words, relative to what you originally wrote:
lim_{n>oo} 1/Card(N\FIS(n)) = lim_{n>oo} 1/oo
note that FIS(n) depends on n, and I'd rather do not see a justification for your step there. More specifically, I'm thinking that:
for all n in N: 1/Card(N\FIS(n)) = 1/oo = 0
does *not* entail:
lim_{n>oo} 1/Card(N\FIS(n)) = lim_{n>oo} 0
Julio

