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Topic: Another AC anomaly?
Replies: 280   Last Post: Dec 27, 2009 3:53 PM

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K_h

Posts: 419
Registered: 4/12/07
Re: Another AC anomaly?
Posted: Dec 9, 2009 1:25 AM
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"Virgil" <Virgil@home.esc> wrote in message
news:Virgil-2C0D8F.18292508122009@newsfarm.iad.highwinds-media.com...
> In article
> <Hv2dnXQ7LtSxUIPWnZ2dnUVZ_hSdnZ2d@giganews.com>,
> "K_h" <KHolmes@SX729.com> wrote:
>

>> "Dik T. Winter" <Dik.Winter@cwi.nl> wrote in message
>> news:KuAGqH.FrI@cwi.nl...

>> > In article
>> > <yvCdnW28VrXBqIXWnZ2dnUVZ_s-dnZ2d@giganews.com>
>> > "K_h" <KHolmes@SX729.com> writes:
>> > ...

>> > > > > > When you mean with your statement about N:
>> > > > > > N = union{n is natural} {n}
>> > > > > > then that is not a limit. Check the
>> > > > > > definitions
>> > > > > > about
>> > > > > > it.

>> > > > >
>> > > > > It is a limit. That is independent from any
>> > > > > definition.

>> > > >
>> > > > It is not a limit. Nowhere in the definition of
>> > > > that
>> > > > union a limit is used
>> > > > or mentioned.

>> > >
>> > > Question. Isn't this simply a question of language?

>> >
>> > Not at all. When you define N as an infinite union
>> > there
>> > is no limit
>> > involved, there is even no sequence involved. N
>> > follows
>> > immediately
>> > from the axioms.

>>
>> I disagree. Please note that I am not endorsing many of
>> WM's claims. There are many equivalent ways of defining
>> N.
>> I have seen the definition that Rucker uses, in his
>> infinity
>> and mind book, in a number of books on mathematics and
>> set
>> theory: On page 240 of his book he defines:
>>
>> a_(n+1) = a_n Union {a_n}
>>
>> and then:
>>
>> a = limit a_n.
>>
>> He writes "...that is, lim a_n is obtained by taking the
>> union of all the sets a_n". The text book I have on set
>> theory defines N as the intersection of all inductive
>> subsets of any inductive set. So clearly there are many
>> equivalent approaches to defining N. In Rucker's
>> approach,
>> we could define N as a limit:
>>
>> a_0 = {} //Zeroth member is the empty set.
>>
>> a_(n+1) = a_n Union {a_n}
>>
>> and then:
>>
>> N = limit a_n.
>>
>> My text book on set theory also explicitly states that we
>> can have a limit of a set of ordinals, for example:
>> "...the
>> phase successor ordinal for an ordinal which is a
>> successor
>> and limit ordinal for an ordinal which is a limit". In
>> fact, one of the problem sets is to prove the
>> bi-conditional: If X is a limit ordinal then UX=X (U is
>> union) and if UX=X then X is a limit ordinal.
>>

>> > >
>> > > My
>> > > book on set theory defines omega, w, as follows:
>> > >
>> > > Define w to be the set N of natural numbers with
>> > > its
>> > > usual order
>> > > < (given by membership in ZF).
>> > >
>> > > Now w is a limit ordinal so the ordered set N is, in
>> > > the
>> > > ordinal sense, a limit. Of course w is not a member
>> > > of
>> > > N
>> > > becasuse then N would be a member of itself (not
>> > > allowed
>> > > by
>> > > foundation).

>> >
>> > Note here that N (the set of natural numbers) is *not*
>> > defined using a
>> > limit at all. That w is called a limit ordinal is a
>> > definition of the
>> > term "limit ordinal". It does not mean that the
>> > definition you use to
>> > define it actually uses a limit. (And if I remember
>> > right, a limit
>> > ordinal is an ordinal that has no predecessor, see,
>> > again
>> > no limit
>> > involved.)

>>
>> N can be defined as a limit or not as a limit. These are
>> really equivalent approaches.

>
> In ZF, unions are defined only for sets of sets and for
> such a set of
> sets S, the union is defined a the set of all elements of
> elements of S.


Check out:
http://planetmath.org/encyclopedia/SequenceOfSetsConvergence.html

> Unless some other definition replaces this one, it is
> impossible to form
> the union of a family of sets unless that family are
> already the
> members of a set.
>
> In which case, N cannot be defined as the limit you
> suggest, as it would
> have to exist before it exists.


Here is one way to define N as a limit.

- Given a set x, the successor of x is the set x'=xU{x}.

- A set y is inductive if x' is in y whenever x is in y.

- Given an initial set x, the inductive set comprised of the
successors of x is called a limit set of the sequence of
sets x'=xU{x}, x''=x'U{x'}, ... .

- Let N be the limit set formed from the initial set {}.

In this case N is a convergent set:

http://planetmath.org/encyclopedia/SequenceOfSetsConvergence.html

It should be pointed out that N is a limit set even if N is
initially given by a definition that doesn't involve the
notion of a limit. Here is another way to see that N is a
limit even if you consider it bad taste to define it in
those terms:

Let \/ = union

Let /\ = Intersection

Define infimum and supremum as follows:

liminf X_n=\/(n=0, oo)[/\(m=n, oo) X_m]
(n-->oo)

limsup X_n=/\(n=0, oo)[\/(m=n, oo) X_m]
(n-->oo)

If these two are the same then the limit exists and is both
of them. So, let's consider the naturals:

X_0 = 0 = {}
X_1 = 1 = {0}
X_2 = 2 = {0,1}
X_3 = 3 = {0,1,2}
X_4 = 4 = {0,1,2,3}
X_5 = 5 = {0,1,2,3,4}
...

Evaluate the liminf case:

/\(m=0, oo) X_m = X_0 /\ X_1 /\ X_2 ...

= {} /\ {0} /\ {0,1} /\ {0,1,2} /\ ...
= {}

/\(m=1, oo) X_m = X_1 /\ X_2 /\ X_3 ...

= {0} /\ {0,1} /\ {0,1,2} /\ ...
= {0}

/\(m=2, oo) X_m = X_2 /\ X_3 /\ X_4 ...

= {0,1} /\ {0,1,2} /\ {0,1,2,3} /\ ...
= {0,1}
...

etc. Now, we \/(n=0, oo) to union all these together and we
get N:

\/(n=0, oo) = {} \/ {0} \/ {0,1} \/ ...
\/(n=0, oo) = {0,1,2,3,...}
\/(n=0, oo) = N


Next, evaluate the limsup case:

\/(m=0, oo) X_m = X_0 \/ X_1 \/ X_2 ...

= {} \/ {0} \/ {0,1} \/ {0,1,2} \/ ...
= {0,1,2,3,...}
= N

\/(m=1, oo) X_m = X_1 \/ X_2 \/ X_3 ...

= {0} \/ {0,1} \/ {0,1,2} \/ ...
= {0,1,2,3,...}
= N

\/(m=2, oo) X_m = X_2 \/ X_3 \/ X_4 ...

= {0,1} \/ {0,1,2} \/ {0,1,2,3} ...
= {0,1,2,3,...}
= N
...

etc. Now, we /\(n=0, oo) intersect all of these to get:

/\(n=0, oo) = N /\ N /\ N /\ ...
/\(n=0, oo) = N

Limsup and liminf both give N and so the limit is N.

k





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Read Re: Another AC anomaly?
Virgil
12/18/09
Read Re: Another AC anomaly?
Dik T. Winter
12/19/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/19/09
Read Re: Another AC anomaly?
Virgil
12/19/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/19/09
Read Re: Another AC anomaly?
Marshall
12/19/09
Read Re: Another AC anomaly?
Virgil
12/19/09
Read Re: Another AC anomaly?
Virgil
12/19/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
12/19/09
Read Re: Another AC anomaly?
Virgil
12/22/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
12/22/09
Read Re: Another AC anomaly?
Marshall
12/27/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
12/21/09
Read Re: Another AC anomaly?
Dik T. Winter
12/21/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/21/09
Read Re: Another AC anomaly?
Virgil
12/22/09
Read Re: Another AC anomaly?
Dik T. Winter
12/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/27/09
Read Re: Another AC anomaly?
Marshall
12/27/09
Read Re: Another AC anomaly?
Virgil
12/27/09
Read Re: Another AC anomaly?
Virgil
12/27/09
Read Re: Another AC anomaly?
Ralf Bader
12/27/09
Read Re: Another AC anomaly?
Virgil
12/13/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/13/09
Read Re: Another AC anomaly?
Virgil
12/4/09
Read Re: Another AC anomaly?
K_h
12/4/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/4/09
Read Re: Another AC anomaly?
Virgil
12/4/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/4/09
Read Re: Another AC anomaly?
Virgil
12/5/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/5/09
Read Re: Another AC anomaly?
Virgil
12/5/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/6/09
Read Re: Another AC anomaly?
Virgil
12/5/09
Read Re: Another AC anomaly?
Carsten Schultz
12/2/09
Read Re: Another AC anomaly?
Virgil
12/1/09
Read Re: Another AC anomaly?
george
12/1/09
Read Re: Another AC anomaly?
Virgil
11/27/09
Read Re: Another AC anomaly?
Virgil
11/27/09
Read Re: Another AC anomaly?
anonymous.rubbertube@yahoo.com
11/30/09
Read Re: Another AC anomaly?
William Hughes
11/27/09
Read Re: Another AC anomaly?
William Hughes
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
William Hughes
11/27/09
Read Re: Another AC anomaly?
anonymous.rubbertube@yahoo.com
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
Virgil
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
Virgil
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
William Hughes
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
Virgil

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