On 1/4/2007 1:53 PM, Albrecht wrote: > > On 27 Dez. 2006, 03:19, "ooo" <fara...@gmail.com> wrote: >> I am biginer in English and mathrmatics. >> If real line is filled with points and each point is >> distinguished,then each point has difference from every other points. >> Therfore real line has void.
In order to make sure that I understood ooo correctly I will try and "translate" it into what I believe to read:
>> If real line is filled with points and each point is >> distinguished from the other ones, then each point has >> to be separated by a difference from every other point. >> Therfore the piece of real line in between has to be void.
> What means: "something is filled with points"? > I see two possibilities concerning the relation of /totality of real > numbers/ <-> /totality of points on a straight line/:
In the subject, ooo got more specific: "completely filled" not just sufficiently. "Sufficiently" was used by Fraenkel in 1923.
In so far, Albrecht is justified when he refers to totalities. However, the line cannot be resolved into single points, and a continuum of real numbers cannot be resolved either.
> 1) The straight line is build up out of points
> 2) We can found points on the straight line but it isn't build up out > of them > > The 1) leads to the paradoxon that points don't have extent but the > straight line have. So there arise the question how points are able to > build up extent without having extent by themself.
The required qualitative step corresponds to fictitious transition from the realm of single points and discrete (rational) numbers to continuous mere potentialities of location and real numbers.
> The next question > arise how there could be different extents as lines of different length > but all are build up of the same "amount" of points.
I see such misconceptions related to Cantorian naivity. Refer to Galilei's clarity, instead: There is no amount of elements inside any piece of continuum.
With 1) math is unable to explain expansion, extent and measure.
Only as long as it follows Dedekind, Cantor, and other trolls.
> 2) is consistent to our experience that we can found as many points on > a line as we want. But than we must consider that lines consist of > lines, and nothing more. Points are properties of lines but not parts. > Infinitely many points denotes the incapability to have them all. In > this view there is no actual infinity.
Be not stupoid, follow Leibniz. Accept infinity and the reals like valuable fictions. Calculate as if they were rationals if admissible.
> > The set theory is based on the view 1).
No. Even worse, set theory is based on schizophrenia in re. Cantor's definition of a set claimed to allow both options at a time. Therefore its torso has beem mumified into ZFC axioms.
