Two questions about infinity and finiteness

[Alfred001]

#1 If you have a sphere with an axis through it around which it rotates, is the number of possible axis it can have finite or infinite?

#2 This is maybe the same question in a different way, but what's got me confused is this:

Is the number of possible positions (for a point) on a finite line finite or infinite? Because the way I'm thinking about it, you can take a point and put it a certain distance from one end of the line and then you half the distance and get another position and you half it again and again... you get the idea, you can half it an infinite number of times, which seems to give an infinite number of positions on a finite line.

So how can there be an infinite number of positions on a finite line?

[timo]

You indeed have an infinite number of points on a finite line segment. The length of a line segment determines if it is finite or not, not the number of points contained (which is always infinite for line segments with non-zero length - and therefore a pretty useless measure).

[Strange]

4 hours ago, Alfred001 said:

So how can there be an infinite number of positions on a finite line?

You probably want to read up on the difference between countable and uncountable infinities.

For example, if you had an infinitely long line with regular divisions marked on it (every inch or every centimetre, depending what part of the world you are in) then there would be an infinite number of those marks. But between each mark, there would be an infinite number of points. And the infinity of points between each mark is larger (infinitely larger) than the number of marks.

Cantor came up with a clever proof of this.

[taeto]

The mathematical idea of how a line, or more generally a curve, is thought of in physics and mathematics is something we got from the work of Euclid, a greek mathematician who lived about 2300 years ago. Accordingly a line is formally just the same as the collection of points that lie on it. In particular where you seem to think of "point" and "position" as two different things, they are actually just the same thing, in the usual understanding. We have other words like "distance" and "length" to talk about where points are located with respect to each other.

Other than that, your reasoning is fairly spot on. If we take a line segment, and we consider any finite collection of different points on it, then between any two points \(x\) and \(y\) there is at least one additional point, no matter how close \(x\) and \(y\) are to each other. The `midpoint' between them is one such point that one is typically taught in basic geometry to construct with a ruler and compass. Therefore any finite collection cannot contain all points of the line segment, and this is what it means to say that the set of points is infinite.

Euclid didn't really consider concepts such as countability of points, which is a topic that came later. His arguments basically apply in the same way to both the `rational line' on which all points can be described by coordinates that are rational numbers, and to the `real line', with more points, including some that have irrational coordinates, like \(\sqrt{2}\) and \(\pi\) and others. And the `algebraic line' which is inbetween the two and contains \(\sqrt{2}\) but not \(\pi\). The rational and algebraic lines have countably many points, the real line uncountably many. Almost always when people speak of a line without qualification they mean the real line.