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

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.

--
"[I want to] stand at the pinnacle of human achievement with no one
else in all of history even close, no human being having faced what I
have--and survived. Because when all is said and done, make no
mistake, the simple truth is, I am better." --James S. Harris
.

Relevant Pages

  • Re: -- Wrong limits do not commute
    ... Han de Bruijn wrote: ... The idea of limit that Han is entertaining is ... only educate us about the notation but about limits in higher dimensions ... complex analysis for example, there are things like circles with radius ...
    (sci.math)
  • Re: Binomial coefficient
    ... Han de Bruijn a écrit: ... As well as the OP, I've never seen this notation, indeed .. ... f^is tne n-th derivative of f, ...
    (sci.math)
  • Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
    ... fed this fish to the penguins: ... was just a silly way of writing lim S_nso let's ... Here is another reason that the bracket notation isn't so good. ...
    (sci.math)
  • Re: Binomial coefficient
    ... Denis Feldmann wrote: ... As well as the OP, I've never seen this notation, indeed .. ... Han de Bruijn ...
    (sci.math)
  • Re: Binomial coefficient
    ... Denis Feldmann wrote: ... Han de Bruijn ...
    (sci.math)