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On Mon, 1 Jan 2007 03:13:05 +0900, ooo wrote:
> "Dave Seaman" <dseaman@xxxxxxxxxxxx> wrote in message
> news:en6koa$as1$1@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>> > 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 ?
The irrationals do not contain any intervals but they do account for the
full measure of the line. That is, the rationals are a set of measure 0.
The irrationals in any interval [a,b] have the same measure as the
interval itself, namely its length.
--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
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