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Topic: Another AC anomaly?
Replies: 43   Last Post: Dec 21, 2009 8:08 AM

 Messages: [ Previous | Next ]
 K_h Posts: 419 Registered: 4/12/07
Re: Another AC anomaly?
Posted: Dec 13, 2009 9:03 PM

"Jesse F. Hughes" <jesse@phiwumbda.org> wrote in message
news:878wd7lczh.fsf@phiwumbda.org...
> "K_h" <KHolmes@SX729.com> writes:
>

>> "Jesse F. Hughes" <jesse@phiwumbda.org> wrote in message
>> news:87vdgcksy2.fsf@phiwumbda.org...

>>> "K_h" <KHolmes@SX729.com> writes:
>>>

>>
>> So, you're claiming that he is not using { and } just to
>> bracket the argument (i.e. the X_n to be limited) but
>> {X_n}
>> refers to a set containing the one set X_n.

>
> Er, yes. Though, something seems to be wrong with your
> notation. At
> issue is the set X_n = {n}, not the set {X_n}.
>
> In summary:
>
> |lim X_n| = |lim {n}| = |{}| = 0.
>
> lim |X_n| = lim |{n}| = lim 1 = 1.
>

>> Then it seems like the meaning has changed because in
>> previous posts
>> he writes that lim|S_n|=/=|limS_n| follows from a
>> wikipedia
>> definition applied to sequences of natural numbers n --
>> not to the
>> non-naturals {n}. For example:
>>

>> > > I have explicitly defined the limit of a sequence of
>> sets. With that
>> > > definition (and the common definition of limits of
>> sequences of natural
>> > > numbers) I found that the cardinality of the limit
>> is
>> not necessarily

>> > > equal to the limit of the cardinalities.
>
> And that's absolutely correct, as we see above.

Only if the sequences were of the non-naturals {n} not
sequences of the naturals n.

>> Okay, if {X_n} refers to a set containing the single set
>> X_n
>> then lim(n-->oo){n} is not a limit of the natural numbers
>> since the naturals are not the sets {n} but the sets n.

>
> Er, yes. Of course.
>

>> In this case my proof shows that lim(n-->oo)n=N.
>> Applying the
>> wikipedia definitions to n is sensible but applying them
>> to {n}
>> makes a mockery of the notion of a limit.

>
> You have some very odd notions yourself. It's a simple
> application of
> a perfectly sensible definition of limit.

It violates the spirit of what a limit is in some cases.
So, although it is sensible, it is not perfectly sensible.

>> The basic idea behind a limit is that things in one state
>> tend to
>> some final state and a good definition and application of
>> a limit
>> should embody that. In looking at the sequence {n}, with
>> 0={} and
>> 1={0}, saying that it tends to 0={} is a betrayal of the
>> core idea
>> behind a limit:
>>
>> 1, {{0}}, {{0,1}}, {{0,1,2}}, {{0,1,2,3}}, ... --> 0
>>
>> The basic idea of what a limit is suggests that an
>> appropriate definition for lim(n-->oo){n} should yield
>> lim(n-->oo){n}={N}:
>>
>> {}, {{0}}, {{0,1}}, {{0,1,2}}, {{0,1,2,3}}, ... -->
>> {{0,1,2,3,4,...}}
>>
>> In other words, applying the wikipedia definitions to {n}
>> is
>> an abuse of those definitions. The definition that is
>> used
>> for a limit should make sense for the kind of object it
>> is
>> applied to.

>
> You're welcome to your own cockamamie opinions about
> whether a
> particular definition is sensible or not, but they're
> utterly
> irrelevant to the issue at hand. The fact is that with
> this
> *perfectly standard* definition of limits, we see that
>
> lim |X_n| != |lim X_n|.
>
> That's all there was at issue.

The sensibility of a definition is the real issue. Applying
the so-called standard definitions to {n} leads to a
cockamamie limit which is at odds with the general notion of
a limit. For a better definition, first choose one of the
wikipedia definitions. If a sequence of sets, A_n, cannot
be expressed as {X_n}, for some sequence of sets X_n, then
lim(n-->oo)A_n is defined by the wikipedia limit. Otherwise
let L=lim(n-->oo)X_n be the specified wikipedia limit for
X_n. If L exists then:

lim(n-->oo)A_n = lim(n-->oo){X_n} = {L}

otherwise lim(n-->oo)A_n = lim(n-->oo){X_n} does not exist.
Under this definition lim(n-->oo){n}={N} and
|lim(n-->oo){n}|=lim(n-->oo)|{n}|=1. Even this definition
can be improved. In the spirit of what a good definition of
a limit should be, we should require that, for example,
lim(n-->oo){n,n,n}={N,N,N}. This can be done by a simple
generalization: if a sequence of sets A_n cannot be
expressed as {X_n,Y_n,Z_n,...}, for one or more arguments,
then lim(n-->oo)A_n is defined by the specified wikipedia
limit. Otherwise let L=lim(n-->oo)X_n, K=lim(n-->oo)Y_n,
J=lim(n-->oo)Z_n, ... be the specified wikipedia limits for
X_n, Y_n, Z_n, ... . If L, K, J, .. all exist then:

lim(n-->oo)A_n = lim(n-->oo){X_n,Y_n,Z_n,...} = {L,K,J,...}

otherwise lim(n-->oo)A_n does not exist. Viewers of this
thread may want to see if there is a way to generalize
and/or improve this definition further. A good definition
should always seek to capture the essence of the notion it
is defining.

k

Date Subject Author
12/12/09 Jesse F. Hughes
12/13/09 K_h
12/14/09 Dik T. Winter
12/14/09 K_h
12/15/09 Dik T. Winter
12/15/09 K_h
12/16/09 Dik T. Winter
12/16/09 K_h
12/17/09 Dik T. Winter
12/18/09 K_h
12/18/09 Dik T. Winter
12/18/09 K_h
12/21/09 Dik T. Winter
12/14/09 Dik T. Winter
12/14/09 K_h
12/15/09 Dik T. Winter
12/15/09 Dik T. Winter
12/15/09 K_h
12/16/09 Dik T. Winter
12/17/09 K_h
12/17/09 Dik T. Winter
12/18/09 K_h
12/18/09 K_h
12/18/09 Dik T. Winter
12/18/09 K_h
12/18/09 Dik T. Winter
12/18/09 K_h
12/15/09 K_h
12/16/09 Jesse F. Hughes
12/17/09 Dik T. Winter
12/17/09 Jesse F. Hughes
12/16/09 Dik T. Winter
12/15/09 ross.finlayson@gmail.com
12/13/09 K_h
12/13/09 Jesse F. Hughes
12/13/09 Jesse F. Hughes
12/14/09 Ilmari Karonen
12/14/09 Jesse F. Hughes
12/15/09 Chas Brown