Originally posted by @deepthought
The definition of a circle is the set of points equidistant from its centre. Since the distance from the centre to the points is infinite any shape all of whose points are an infinite distance from its centre, such as an infinite square, has this property as there is only one infinity defined in the extended real line. So we can't distinguish betwee ...[text shortened]... ent amounts to a statement that because a circle is not a square, infinite numbers do not exist.
I find your above argument very interesting and it has given me an idea for an alternative argument;
For any said infinitely large shape (meaning its diameter in any direction along a plain/volume it is defined in is of infinite length) that has an infinite length from its center to all points on its surface/circumference, because there is mathematically only one positive infinity, the would be no distinguish between such a said infinitely large shape from any other infinitely large shape of 'different' shape because each (and all of them) would have that exactly same positive infinity from their center to all their points on their surface/circumference.
So if there can be said to be, for example, an infinitely large circle, that infinitely large circle is also an infinitely large square and an infinitely large triangle and an infinitely large ellipse and so on i.e. it is also all the infinite number of different 2D shapes!
But that makes it completely meaningless to say it is specifically an "infinitely large circle" because then why then call it a "circle" and not, say, a "square" i.e. why not call it and say it is, say, an "infinitely large square" instead?
And that also makes it meaningless to say it is specifically an "infinitely large sphere" because then why then call it a "sphere" and not, say, a "cube" i.e. why not call it and say it is an "infinitely large cube" instead?
And if there is nothing to distinguish between different shapes of such 'infinitely large' size, that makes it calling it a 'shape' meaningless i.e. it is meaningless as a 'shape' thus there isn't such thing as an 'infinitely large' shape thus there isn't such thing as an 'infinitely large' circle.
Can you explain any error in my above argument? Because not sure if my above argument is really sound.
And can you make a better argument/counterargument?