From: wolfgang.mueckenheim@tha.de   
      
   Am 19.12.2025 um 11:16 schrieb Mikko:   
   > On 18/12/2025 13:57, WM wrote:   
   >> Am 18.12.2025 um 11:20 schrieb Mikko:   
   >>> On 17/12/2025 20:22, wm wrote:   
   >>>> Am 17.12.2025 um 11:10 schrieb Mikko:   
   >>>>> On 15/12/2025 19:18, wm wrote:   
   >>>>>> The actual infinity is for instance when we "consider the points   
   >>>>>> of an interval as a totality that is completely existing"   
   >>>>>> [Hilbert, 1925]. The points of a line, when completely existing,   
   >>>>>> have a first and a last one because never two or more can exist   
   >>>>>> besides each other. This cannot happen other than by dark points.   
   >>>>> As no definition of "dark point" is given it is possible that no point   
   >>>>> is dark and that every point is dark. The "proof" does not identify   
   >>>>> any feature of a point that would dfferentiate dark from non-dark.   
   >>>>>   
   >>>> Dark points are existing but cannot be identified by their co-   
   >>>> ordinates. The existence of such points is clearly proven by the   
   >>>> first and and the last point of an open interval.   
   >>>   
   >>> In the space of hatural numbers there is an interval from the point 4   
   >>> to the point 5. How does the existence of these points prove that   
   >>> there are dark numbers?   
   >>   
   >> For every point x < 5 there are points between x and 5.   
   >   
   > In the space of the natural numbers there are no points between 4   
   > and 5.   
   >   
   The interval consists of real numbers, most of which are dark. Dark   
   natural numbers are larger and not so easy to understand. Therefore I   
   recommend that you start with a real interval as Hilbert discussed it above.   
      
   Regards, WM   
      
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