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

 Messages: [ Previous | Next ]
 Jesse F. Hughes Posts: 9,776 Registered: 12/6/04
Re: Another AC anomaly?
Posted: Dec 12, 2009 11:22 PM

"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:
>>

>>> The only way lim(n ->oo){n}={} is if limsup and liminf
>>> both
>>> equal {}. If limsup and liminf are different then the
>>> limit
>>> does not exist and cannot equal an existing set like {}.
>>> Since the empty set exists, and since you are claiming
>>> that
>>> lim(n ->oo){n}={}, you need to show that limsup and
>>> liminf
>>> are both {} by the definition you are using.

>>
>> Yes, he needs to show that, but it is utterly trivial and
>> obvious from
>> the definition of limsup and liminf given here. I'm not
>> sure why you
>> think it's not obvious, but here's the proof.
>>
>> Now, let X_n = {n}. Thus, n is in X_k <-> n = k.
>>
>> n in lim sup X_k iff n is in infinitely many X_k, but we
>> see from the
>> above that n is in only one X_k. Thus, lim sup X_k = {}.
>>
>> n in lim inf X_k iff there are only finitely many X_k such
>> that n not
>> in X_k, but again, we see that this is false for every n.
>> Hence
>> lim inf X_k = {}.

>
> 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.

> 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.

> 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.

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.

--
"Sorry, wakeup to the real world. You're on your own dependent on me
as your guide. Luckily for you, I'm self-correcting to a large extent,
so if the proof were wrong, I'd tell you. It's not wrong."
--- James Harris confirms that his proof is correct.

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