Re: Olber
This principle can easily be demonstrated at home. As, I believe, it was by one of the Greek mathematicians.
Take a small circle.
From the edge draw a line to the centre.
next to that line, draw another line from the edge to the centre.
Leave no gap at the edge.
continue doing this until the circle is filled.
Note, no gaps.
Now, draw another larger circle with the same centre.
Extend the first set of lines out from the first circle to the new one.
There are now gaps between the lines at the edge of the outer circle.
You can carry on filing the gaps with lines back to the centre and adding more lines and more circles forever.
It is a simple, if rather tedious, demonstration of the principle of infinity.
INT21
I suppose that what the Greek mathematician called a principle, is that you can do no consistent mathematical work on infinity. Greeks were unsettled by the infinite and its apparently paradoxical nature, and this demonstration was for them another tool, in addition to Zeno's paradox, to prove that it could not be a mathematical object.
But there is another way to see things, as detailled in the video posted by David Plankton*, and which apprehends the infinite in a perfectly consistent way. What happens if you can perform the operations described by the Greek mathematician, what would you do with the gaps between the two circles ?
Answer : if you extend this first set of lines, leaving no gaps, out from the first circle to the new one, you will have no gaps between the lines at the edge of the outer circle, it will be already fully filled.
As we can easily demonstrate : take a point on the inner circle, only one line can be drawn from this point to the center of the circle ; as anyone will admit.
Also, that the circle is filled means that every point is already included in a line.
Now, if you draw a line on the outer circle, it will necessarily cross the inner circle at one point, in fact at one of those points, as there are no other ones.
So if you repeat the same operation of filling entirely the outer circle with straight lines from the centre, as each of the lines has to cross the inner circle, and to cross one of those points necessarily already included in the first set of lines, and
only one of them, each of them will be aligned in one of the straight line with one point and only one on the outer circle.
So that in fact, these 'new' lines are the same ones that were already extended from the first circle to the new one, leaving no room for other lines.
Or, if you put it in modern mathematical terminology, you have a perfect bijection drawn between the points on the inner circle and the points of the outer circle, a one-to-one relation, every point on the inner circle being tied with one and only one point on the outer circle, and vice versa. This is true whether the diameter of the outer circle is 1.001 times larger than the inner one, 2 times, 1000 times, one million times,one bliion times,a goggle times, a goggleplex times etc... So that the number, the amount, the cardinality of the points on any circle, how large or small, is always exactely the same.
The same principle that applies when you see that there are 'as many' even numbers, or odd numbers, than there are whole numbers, and that was frightening ancient Greeks. This apparently paradoxical nature can be handled mathematically, once you admit that the fact that common sense is, indeed, offended.
*
Succeeds to encompass even such esoteric notions as grand cardinals...