It is infinite because even if you reach a diameter infinitesimally thin you can just half it. Remember, we are not physically creating diameters, just theoretically. Lines of symmetry take up no space, they are superimposed on the image by the mind. In any case it is possible to get a 'line' that has no width. It is a non-traditional line, but it still counts. A cut through the circle still is a line. Just don't start referring to it as two semi circles, you can still call it a circle. And then you continually cut it into sectors. Infinite lines of symmetry.

Points take up no space in a 2D plane or 3D space and neither do lines. Numbers become infinitesimally small while approaching zero. A circle has an infinite number of lines of symmetry. That's basic math.

Wrong a line has no width to begin with, any line that has width is not a line, but a 2d shape. So lines have no width, circles have an infinite number of lines of symmetry, and this does not change as it changes size, by definition a circle has an infinite number.

Wouldn't a circle have zero? Maybe I'm misunderstanding the question but lines of symmetry refer to lines, correct? Lines are defined by length? Circles are made up of an infinite amount of points equidistant from the one directly across from it? So where are the lines?

well it all depends on the size of the circle and the width of the lines, but it is a very high number if it does stop, its like trying to count all the stars, it will take forever, and you will prob fail, or if you do count em all you have a sad life

I understand that this is irrelevant, but; is there a difference between that definition and the one that indicates a line is a figure that is straight, infinitely long and infinitely thin? Each definition produces the same figure, right?

I am aware. If you'd like I'll rephrase it - lines are defined by length. Don't they have the ability to turn since one of the things a line can be is a trace of a moving point? Lines at their simplest are just infinite lengths with no width or depth. The graph of x^2 is a line - right?