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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 8, 2009 5:03 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply


"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.

k




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