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"Dave Seaman" <dseaman@xxxxxxxxxxxx> wrote in message
news:en6koa$as1$1@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> On Sun, 31 Dec 2006 03:58:57 +0900, ooo wrote:
>
> > "Dave Seaman" <dseaman@xxxxxxxxxxxx> wrote in message
> > news:en5rr7$vb7$1@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> >> On Sat, 30 Dec 2006 14:42:52 +0900, ooo wrote:
>
> >> > "Dave Seaman" <dseaman@xxxxxxxxxxxx> wrote in message
> >> > news:en104u$oe7$1@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>
> >> >> Here's a paradox: The reals are a complete ordered field. If you
> > extend
> >> >> the reals by including infinitesimals, the resulting extension field
is
> > no
> >> >> longer complete. By adding things to a complete ordered field, we
get
> > an
> >> >> incomplete ordered field.
>
> >> > I see. Thank your kind explanation. Cannot we extend that to complete
> > field?
>
> >> No. There is only one complete ordered field, up to isomorphism.
>
> > Yes.
> >> >> > A difinition of infinity is ,therefor,ambiguous that a set of
natural
> >> > number
> >> >> > contains only finit size of number ,but entire member of it is
> > infinite.
>
> >> >> How is that "ambiguous"? Is the color blue ambiguous because we can
> > fill
> >> > a
> >> >> blue basket with red objects?
>
> >> > And What do you label to that basket ? Can you know content by
basket
> > only?
>
> >> Why do you ask?
>
> > Because I want to ask about validity to name a number of members of a
set
> > that contain only fint size of member "infinite"
>
> That was your reason for asking about baskets? I don't see the
> connection.
>
> The interval (0,1) contains infinitely many numbers (uncountably many, in
> fact), but each number is less than 1 and therefore certainly "finite".
>
> >> > We can refor to "round square " ,but it is meaningless.
> >> > That might not be good examle. But the expression of infinity using
> >> > countable number of simbols, may not be necesarily consistent.
>
> >> Really? How so?
>
> > I cannot decide.
>
> Neither can I.
>
> >> >> > I concider that entire naturals is what we cannot
comprehend.Infinity
> > is
> >> >> > beyond our reach.
>
> >> >> But we can reason about it.
>
> >> > Yes, We can reason about it as concept using sinbols. Therefore what
we
> > are
> >> > really dealing with with is countable numbers of symbols.
>
> >> But we can reason about uncountable sets. The fact that there is only
one
> >> complete ordered field, up to isomorphism, is one example of such
> > reasoning.
>
> > Real field is complete and its cardinality is c(2^aleph0). Then as for
> > 2^aleph0^aleph0, 2^aleph0^aleph0^aleph0... ?
>
> What about them? Are you concerned about the fact that the string
> "2^aleph_0" represents an uncountable cardinality by using a finite
> number of symbols?
>
Sorry, what I wrote was hard to understand, and has little connection with
what you had written above.
I agree that we can reason about uncountability, and its result is good.
I didn't mean symbol 2^aleph_0 itself, but as for this question I shall
write at another place. I cannot state well now.
> Do you also worry about the fact that "Chicago" doesn't contain the names
> of all the people who live there?
>
To explane inhabitants of "Chicago" correctry, We must discribe all those
who live there.
But in the case of infinite set, if above metaphor is misunderstandable, the
fact that its members are finite size not necessarily leads to idea that its
members are infinitely many.
I don't opose necessarily to standerd point of view ,because I think we
cannot discribe infinity parfectly (if word consistant is disagreeable ).
At present condition, set theory includes some parts hard to understand and
paradoxes.
> > We can reason about them. Are these doubtlessly consistent? And large
> > cardinals?
> > I don't deny a room for uncountable set. The reason is that with
rational
> > number only ,real line cannot be filled.
> > What about this question?
> > An interval is subdivided at a position of 1/2 from one end.
> > And each sub interval are subdivided in the same way.
> > The same procedures are continued.
> > In each stage of this process ,interval remain as interval,And if it is
> > enlarged,its size is the same as it was.
> > An interval remain as an interval. And without sudden change ,it has no
> > possivility to be a point.
> > Is it possible to fill void on a line filled with rationals?
>
> If I understand your question correctly, that's what Dedekind did with
> his cuts.
>
Regardless of position points added on a line space between any two points
is interval,but not point while number of points is finite.
When all rationals are added on a line ,and number of points is countably
infinite,are void(s) left on a line point or interval?
Are uncountable number of reals necessary for void coresponding to countable
number of rationals and be filled with gathering of discreet points ?
>
> --
> Dave Seaman
> U.S. Court of Appeals to review three issues
> concerning case of Mumia Abu-Jamal.
> <http://www.mumia2000.org/>
Regards OT
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