Date: Nov 17, 2012 3:21 PM
Author: William Hughes
Subject: Re: Matheology § 152
On Nov 17, 1:23 pm, "LudovicoVan" <ju...@diegidio.name> wrote:

> "William Hughes" <wpihug...@gmail.com> wrote in message

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> news:1ec0c2cc-f926-4fd4-a413-37ba8809ea80@y8g2000yqy.googlegroups.com...

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> > On Nov 17, 9:59 am, "LudovicoVan" <ju...@diegidio.name> wrote:

> >> "William Hughes" <wpihug...@gmail.com> wrote in message

> >>news:28bff553-f679-4e23-8932-a1fb42f1b364@c17g2000yqe.googlegroups.com...

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> >> > Note that *set* limits have some important properties.

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> >> > Given a sequence of sets {B_1,B_2,B_3,...}

> >> > then the set limit always exists (it

> >> > may be the empty set).

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> >> > If we have

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> >> > A = set limit {B_1,B_2,B_3....}

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> >> > Then

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> >> > A is a set

> >> > A cannot contain an element that is not contained

> >> > in any of the B's

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> >> Williams going around, in circles:

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> >> It was already mentioned that it is wrong to use that specific definition

> >> to

> >> solve the balls and vase problem.

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> >> <http://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior#Specia...>

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> > The problem is the above applies to *any* definition of a *set* limit.

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> But those definitions are a *specific* case of these:

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> <http://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior#Sequen...>

Well, I could defend myself by pointing out that these are talking

about limits of sets (and any limit, e.g. the usual limit on real

numbers can be considered the limit of sets) and I was talking about

set limits. However, I don't think this is very convincing.

I will simply point out that the first defintion, does not apply

in this case.

A more fundemental problem is that there is no reason to

expect the cardinality of the B's to have anything to do

with the cardinality of A.

Eg. B_n = [-1/n,1/n]. Then A is {0}

More like the current situation.

B_n: {all rational numbers, q |

q can be written as k/n^2 (k an integer) AND q in [-1/n,1/n]}

Then B_n is finite. |B_n| grows without bound.

A= {0}, |A| = 1 (if you want A the empty set,the add the condition

that

q is nonzero)

So there is no reason to change the limit to make the cardinality of

the

limit equal to the limit of the cardinalities

(nor is there a problem that WM two limits are different)