Limit?

[Doron]

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.

You can prove that A is the limit of B only if you can show that s'=0_AND_s'=1 -> 1

A collaction of elements, wich can be found on many different scales, really approaching to some given constant, only if it has finitely many elements.

Please tell me what do you think, thank you.[

[Bo]

basicly what you say is this: (please correct me if i'm wrong!)
If r=0, then A=true, B=false. Else A=false, B=true.
for all r: A XOR B = true.

you then claim that the limit of r->0 means that A=true AND B=true, and here i disagree. my main problem is that the definition of A and B is Boolean; and so non-continuous, and i'm quite convinced the limiting case isn't well defined... (you can't limit True to False, which is i think what your claim {s'=0_AND_s'=1 -> 1 } basicly means). If you can find somewhere a continuous case of boolean algebra, that would be very interesting.

A collaction of elements, wich can be found on many different scales, really approaching to some given constant, only if it has finitely many elements.

i dont see where the notion of finite elements comes from here.

Bo

[Doron]

Dear Bo, you missed it a little.

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&gt;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.

[paultrr]

We define a sphere of which a circle would be an element of that sphere when it comes to cosmology and physics irrespective of the fact that from what we can observe the universe has no visual center. That defining of a sphere, while observationally based, also relies upon geometry. However, as far as cosmology goes while there is no center to that expanding sphere there is a visual limit imposed by light itself. That limit is generally considered as some 300,000 years after the BB point. The outer edge of that sphere is also imposed from the restriction of the lightcones of the original BB event. So not all cases known require an actual physical center to define a sphere or a circle.

In theory, even given a known radius of any circle if one divides the area in half and keeps on doing such the solution in each case while approaching zero or that center point will never actually reach zero. There is an infinite decreasing volume between the edge of that original circle and that center which means in essence that center is not really known or defined except as a singularity from a geometric point of view. Given that we really never do have a solid definition of the center of any circle.

Bit of math/logic based thought to throw into this discussion. But it does have bearing upon the discussion itself.