Doron Posted November 23, 2004 Report Posted November 23, 2004 If we have a measurable center point and no measurable line or a measurable line and no measurable center point, we cannot define a circle. It means that if r=0 or r=oo we cannot use the circle’s concept, or in other words, we cannot define a circle. r is circle’s radius. s' is a dummy variable ( http://www.mathworld.wolfram.com/DummyVariable.html ) a) If r=0 then s'=|{}|=0 -> (no circle can be found) = A B) If r>0 then s'=|{r}|=1 -> (a circle can be found) = B The connection between A,B states cannot be but A_XOR_B Also s' = 0 in case (a) and s' = 1 in case (B), can be described as s'=0_XOR_s'=1. We can prove that A is the limit of B only if we can show that s'=0_AND_s'=1 -> 1 A collection of elements, which can be found on many different scales, really approaching to some given constant, only if it has finitely many elements. <u>Further explanation:</u> 0 and 1 are the cardinal values, and they are based on the set that standing in the base of each one of them. If r=0 then we use the Empty set in order to define the value of s' = |{}| = 0 (because no circle can be found) If r>0 then we use a Non-empty set in order to define the value of s' =|{r}| = 1 (because a circle can be found) A is a center of a circle iff B is any measurable arbitrary value, which is not A. B can never be A exactly as the cardinal of a non-empty set cannot be the cardinal of the empty set. So we get A XOR B states which are equivalent to s'=|{}| XOR s'=|{r}| states. In this case A cannot be the limit of B. QED. A definition cannot include in it states where the thing that it defines DOES NOT EXIST, and the standard definition of a circle ( http://mathworld.wolfram.com/Circle.html ) includes in it states (r=0,r=oo) where no circle can be found. Strictly specking, this definition cannot be considered as a rigorous logical definition.
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