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


Re: Matheology § 300
Posted:
Jul 7, 2013 2:04 PM


"Michael Klemm" <m_f_klemm@tonline.de> wrote in message news:krc11i$j27$1@solani.org... > Julio Di Egidio wrote: > >>>>> 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: > > The expression lim_{n>oo} 1/Card(N\Fison(n)) > means: > Limes of the sequence > (1/Card(N\Fison(1)), 1/Card(N\Fison(2)),1/Card(N\Fison(3)),....) = > (0,0,0,....), the value n = oo excluded. > This limes is 0.
Again, that for all n in N the value is 1/oo just does not entail that the limit is 1/oo.
> The expression > Card(N\(lim_{n>oo} Fison(n))) > means: > Limes of the sequence > ({2,3,4,...},{3,4,5,...},{4,5,6,...},...), > {oo} not contained in the sequence and > oo not contained in any of its members.
That's not an argumen against the identity I am assuming. On the contrary, you have not supported your:
lim_{n>oo} 1/Card(N\F(n)) = lim_{n>oo} 1/oo
I.e. your "preemptive" substitution of Card(N\F(n)) with Card(N) = oo seems unwarranted. That equality is true for all n in N, but it is not true in the limit, where n>oo, and F(n) = N.
Indeed, here is how I'd print the sequence and its limit:
1/Card({1, 2, 3, 4, 5, ...}) 1/Card({2, 3, 4, 5, ...}) 1/Card({3, 4, 5, ...}) ...  1/Card({})
> This limit is by definition the set of all > naturals contained in infinitively many members > of the sequence. Thus, the limit is the empty set {}. > Hence one obtains > 1/Card(N\lim_{n>oo} Fison(n)) = oo. > Therefore the error in the calculation is the first equation lim_{n>oo} > 1/Card(N\Fison(n)) = > = 1/Card(N\(lim_{n>oo} Fison(n))).
I'd insist that it's rather your
lim_{n>oo} 1/Card(N\FIS(n)) = lim_{n>oo} 1/oo
that is incorrect. In fact, where's 'n' gone?
Julio

