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

On Fri, 8 Feb 2008 13:08:54 -0800 (PST), Han.deBruijn@xxxxxxxxxxxxxx
wrote:

WARNING: It seems to me that you are lacking some prerequisites for this
discussion. Hence the following might not be clear at all. Now YOU have
to be patient.


s[A] := {s(x) : x e A}   (A c N)

"the image of A under s"

Am I allowed to hook in here, with one of my small but certain
steps ?

Sure. After all you are running the show in this thread! :-)


Let's repeat the definition of S(). I hope this is the latest version
and that it has been found correct by you and Jesse:

y is in S(x) iff there is some n e N such that for all m e N ,
with m > n , y is in S_m(x) .

Right, in symbols:

y e S(x) <-> En e N Am e N: m > n -> y e S_m(x).


Here, S_m(x) = s o s o ... o s(x), where there are exactly
m compositions. (m e N)

For the sake of the argument I'll use a slightly different definition
for S_n(x) than Jesse did - _but that's in no way essential_.

S_n(x) := s o s o ... o s(x) (n e N)
`------.------´
n-times s

(Though I'd prefer a precise definition via recursion here.)

This means

S_0(x) = x
S_1(x) = s(x)
S_2(x) = s(s(x))
:


I have a minor problem with this definition. It seems to me that n is
an unused quantity ...

No, it's not. See the symbolic (re-)formulation. (But that's just part
of the answer...)


...and that it could be left out. The definition then would read as
follows:

y is in S(x) iff for all m e N, y is in S_m(x).

Is this correct? And if not, why not?


:-)

Again, there's no "correct" here... The question is if the (original)
definition is reasonable/sensible - it is. Of course one would have to
EXPLAIN that. [Well...]

To make a long story short: (for an arbitrary x)

(S_n(x)) is a sequence of sets.

Now you know (by now) that we cannot define a function consisting of
"infinitely many compositions of s" in ZFC. BUT we can consider the
limit of the sequence (S_n(x)) instead!

So just define (for an arbitrary x)

S(x) = lim S_n(x)
n

Then one might consider

S(x)

to be a reasonable/sensible "realization" of "applying the successor an
infinte number of times" in ZFC.

But now the question arises: How is the notion of "limit" defined for
sequence of sets? --- Actually, we DO have a reasonable definition for
that notion. [Details skipped.]

Now IF that limit (defined in this way) for (S_m(x)) exists, then the
following will hold:

y e S(x) <-> En e N Am e N: m > n -> y e S_m(x).

That amounts exactly to Jesse's "definition" (or criteria).

Anyway, now let's take a fixed x = 0, to simplify the discussion.

S_0(0) = 0 = {}
S_1(0) = s(0) = {0}
S_2(0) = s(s(0)) = {0, 1}
S_3(0) = s(s(s(0))) = {0, 1, 2}
: :

Now it's (intuitively) reasonable to assume that

lim S_n(0) = N
n

if such a limit (for (S_n(x)) exists. No?

(Isn't that what one would expect from "applying the
successor an infinite number of times" starting with
the empty set?)

And actually, exactly that's what we get using the definition for
"limit" mentioned above. So let's define

S(0) := lim S_n(0)
n

Then (as already mentioned)

y e S(0) <-> En e N Am e N: m > n -> y e S_m(0).

And from the latter we can get S(0) = N, of course.

Now let's consider your "alternative approach" (with x = 0):

y e S(0) <-> Am e N: y is in S_m(0).

But then S(0) would (have to) be {}. Since none of 0, 1, 2, ... would be
in S_0(0). Hence none of 0, 1, 2, ... would be in S_m(0) for ALL m e N
(since it would not be in S_m(0) for m = 0).

Now imho

lim S_n(0) = {}
n

doesn't look very reasonable as a limit for the sequence (of sets)

S_0(2) = {}
S_1(2) = {0}
S_2(2) = {0, 1}
S_3(2) = {0, 1, 2}
:

Won't you think so?


K. H.

--

E-mail: info<at>simple-line<dot>de
.

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