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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 8, 2009 8:29 PM
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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.

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.


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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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