Is there such thing as an infinitely large circle?

Originally posted by @humy
I like that.
So perhaps the strategy here should be;

(1) Make a definition of a square that is correct for finite squares and is not broken in the infinite case.

(2) Make a definition of a circle that is correct for finite circles and is not broken in the infinite case.

(3) Now compare the two definitions for the infinite case and see if they are i ...[text shortened]... both the answers 'yes' and 'no' legitimate depending on which arbitrary definitions you choose.

I've been thinking about this. I'm worried of my use of the word "straight line". One could specify a construction where there are four vertices joined by curves. Lines of construction joining opposing vertices (preferred points on the perimeter of the shape) meet at some point we call the centre, at that point we draw another two lines which bisect the first two, think of a plus sign drawn over an X. All points on the vertical line of construction should be equidistant from the two vertical curves joining vertices in the original figure and at the same horizontal distance all the way to the top of the figure, ensuring the edges are straight lines, similarly with the horizontal line. This ensures that the figure has straight edges in the finite case and isn't broken in the infinite case, specifically we don't need a way of specifying what a straight line at infinity is.

The definition is required to be unique in the finite cases (I need symmetry operations to rule out rectangles in the above). Not make reference to any concept that might be difficult to specify in the infinite case (straight, curved, distance along perimeter). I don't know how much of a problem that is likely to be.

Originally posted by @humy
I've just had another thought;

Lets say we accept as valid the definition of an infinitely large circle as being the set of all points in an infinitely large 2D plain that is at infinity distance from an arbitrary chosen center point.
There would obviously be an infinite number of such points that would make up the 'circumference' of such an infinite circl ...[text shortened]... oint or if they are all an infinite distance from each other, whatever that's supposed to mean!?

Not sure I agree with this conclusion. I agree that measure is a dicey concept for the set of points at infinity. However, all we need to do is avoid reference to words like straight, curved and distance (along the perimeter). I think my above definition of a square (using lines of construction to establish that the edges fulfill rules that would produce straight lines in a finite case) avoids that problem.