(February 13, 2018 at 5:54 pm)RoadRunner79 Wrote:(February 13, 2018 at 5:39 pm)polymath257 Wrote: OK, first some definitions.
Finite means it can be counted by some counting number or some 'real' number. Infinite means it is not finite.
So, a line segment of length 1 inch is 'finite' in length, but has an infinite number of points on it, so is infinite in terms of number of components. It is 'bounded', but also infinite. Similarly, anything with finite, non-zero volume will have an infinite number of components.
(This goes to show the definition of infinite in terms of boundedness is not a very good one).
The negative integers {..,-3,-2,-1} are unbounded below and bounded above. This is an infinite set because the number of negative integers cannot be counted by a finite number. That doesn't detract from its being bounded above (by 0). This is an example of a set that is bounded in one sense, but unbounded in another.
So, yes, you can have an actual infinite (like the negative integers) that is also bounded above. Not finishing isn't a criterion: the set is complete in and of itself.
Since I don't believe in any God, I can't say what you think about his/her/its infinite nature.
The water example shows that large collections can appear to be continuous when in fact they are not. So?
I'll try to comment on the other stuff later. However I've had the discussion concerning a infinite number of points on a finite line before here (and haven't gotten very good answers).
How are you defining a point, and on what basis do you state that there are an infinite number of them on a 1 inch line?
Between any two points, there is another that is halfway that is different than both ends. That alone is enough to guarantee an infinite number of points. And that is true no matter what the definition.
