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

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 Virgil Posts: 870 Registered: 7/27/09
Re: Another AC anomaly?
Posted: Dec 9, 2009 2:35 AM

In article <c8idnc6JG6hv34LWnZ2dnUVZ_q-dnZ2d@giganews.com>,
"K_h" <KHolmes@SX729.com> wrote:

> "Virgil" <Virgil@home.esc> wrote in message

> > 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
> > 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'}, ... .

It is the set of all of them, if that is what you mean by union
>
> - 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

However, in the discussion between Dik and WM, Dik gave SPECIFIC
definition of what HE meant by the limit of a seqeunce of sets which
differed from that in your citation.
>
> 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.

The issue between Dik and WM is whether the limit of a sequence of sets
according to Dik's definition of such limits is necessarily the same as
the limit of the sequence of cardinalities for those sets.

And Dik quire successfully gave an example in which the limits differ.

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.

The issue is not whether the naturals are such a limit but whether for
every so defined limit the cardinality of the limit equals the limit
cardinality of their cardinalities, which is different sort of limit.

Date Subject Author
11/23/09 Jesse F. Hughes
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12/1/09 Dik T. Winter
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