Re: 1-1/2+1/3-1/4+1/5-1/6+1/7

On Mon, 11 Feb 2008 09:13:50 -0500, "Jesse F. Hughes"
<jesse@xxxxxxxxxxxxx> fed this fish to the penguins:

Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> writes:

Jesse F. Hughes wrote:
S[x] = { S(z) | z in x }
= { y | (E z in x)(E n in N)(A m in N)(m > n -> y in S_m(z)) }
I do not see why that should be the same set as
T(x) = { y | (E n in N)(A m in N)(m > n -> (E z in x)(y in S_m(z))) }
So, you have to show that (E z in x)(E n in N)(A m in N)(m > n
-> y in S_m(z)) iff (E n in N)(A m in N)(m
n -> (E z in x)(y in S_m(z)))
I don't believe it's so.

In the lucid notation by G. Frege:

lim S_n[N] = {N} . Hence S[N] = {N} . But also:
n -> oo

How do you figure that? The claim lim S_n[N] = {N} comes from
nowhere, as does the claim that S[N] = lim S_n[N].

Again, if we chase through the definitions, we find:

S[x] = { S(z) | z in x }
= { y | (E z in x)(E n in N)(A m in N)(m > n ->
y in S_m(z)) }

lim S_n[x] = { y | (E n in N)(A m in N)(m > n ->
(E z in x)(y = S_m(z))) }

(Of course, T(x) was just a silly way of writing lim S_n[x] so let's
use G. Frege's notation.)

Thus, if you want to show that these two sets are equal, you need to
show that

(E z in x)(E n in N)(A m in N)(m > n -> y in S_m(z))
iff
(E n in N)(A m in N)(m > n -> (E z in x)(y = S_m(z)))

at least in the special case when x = N.

lim S_n[N] = lim N \ {0,1,2,3 .. ,n} = N \ lim {0,1,2,3 .. ,n} = N \ N
n -> oo n -> oo n -> oo

Right.

Hence: S[N] = {} . No ?

No.

Is there a problem while interchanging a set difference with our
limit ?

Not sure what set difference you're talking about. The problem is
that you think S[N] = lim S_n[N], but if you unravel the definitions
you'll see that this just isn't so.

Here is another reason that the bracket notation isn't so good. It's
true that

S(x) = lim S_n(x)

but it is not true that

S[N] = lim S_n[N].

In terms of that more categorical notation, we'd say something like:
S = lim S_n, but not PS = lim PS_n, i.e., that the powerset functor
doesn't commute with colimits. Something like that, anyway, but not
*quite* that, since s isn't really a function in the category Set.
But since category theory probably won't clear up your issues, let's
leave the details alone.

Is the limit notion involved here the one about lim sup and lim inf of
sequences of sets the G. Frege introduced with some quotes from
Halmos? If yes, then the answer to Han's question (if I understood it
correctly) is yes.

Regards,
G. Rodrigues
.

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