I don't think there is such thing and by reasoning goes as follows;
There isn't such thing as an infinitely large circle because and infinite large circle would have any finite length of its circumference a straight line thus as you follow it around it will never curve around to form a circle even if you followed it around forever!
But is this above a sound explanation?
And, assuming you agree there isn't such thing as an infinitely large circle, can you explain why better than above?
Originally posted by @humyWhat do you mean by infinitely large?
I don't think there is such thing and by reasoning goes as follows;
There isn't such thing as an infinitely large circle because and infinite large circle would have any finite length of its circumference a straight line thus as you follow it around it will never curve around to form a circle even if you followed it around forever!
But is this above a so ...[text shortened]... ree there isn't such thing as an infinitely large circle, can you explain why better than above?
We have a mapping from the set of positive real numbers R+ to the set of circles C by assigning the radius of the circle to the real number. So there is no largest circle (or smallest one). If one extends the set of real numbers to include infinity as a point to the right of all the reals then one could argue for a circle corresponding circle which is larger than all the circles mapped to from the ordinary reals. So as a limit point of the set of circles I don't see why not. It depends on whether you accept the reality of infinite quantities.
I don't think your argument quite works as you've defined a circle in terms of its curvature, when curvature is defined in terms of the radii of circles. 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 between different shapes when they are infinitely big. Your argument amounts to a statement that because a circle is not a square, infinite numbers do not exist.
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Originally posted by @humyThis sounds like a question for the philosophy forum.
I don't think there is such thing and by reasoning goes as follows;
There isn't such thing as an infinitely large circle because and infinite large circle would have any finite length of its circumference a straight line thus as you follow it around it will never curve around to form a circle even if you followed it around forever!
But is this above a so ...[text shortened]... ree there isn't such thing as an infinitely large circle, can you explain why better than above?
Originally posted by @wolfe63I neglected to mention the context which prompted me to ask this question which was somebody suggesting to me that the whole universe is of a shape of an infinitely large sphere, which I immediately sensed may be an erroneous concept.
This sounds like a question for the philosophy forum.
Originally posted by @deepthoughtI find your above argument very interesting and it has given me an idea for an alternative argument;
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.
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.
ANYONE;
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?
I had thought I just spotted a possible subtle flaw in my above argument but then I think I worked out how an "infinitely large circle" is still meaningless anyway;
I said there would be nothing to distinguish between such said infinitely large shapes from each other because all points on their outer surfaces are the SAME positive infinite length from the shape's center. That argument appears to work well for explaining how you cannot distinguish between, say, infinitely large circles and ellipses because neither has corners but could you still claim that, despite all points on their outer surfaces being that SAME positive initinite length away from the center, you can still distinguish between, say, a circle and a square, because even an 'infinity' large square has 4 corners by definition on the finite scale? But then I thought how would you distinguish between an infinitely large square and an infinitely large rectangle? You couldn't. And, for each and every shape with a given number of corners (and also edges if it is a 3D shape), there is still an infinite set of DIFFERENT shapes that have that same number of corners thus you cannot distinguish between shapes within that infinite set of shapes (which doesn't include all shapes) thus my claim that there isn't such things as an infinitely large circle is still valid BUT by argument still needs modifying to make the necessary corrections.
So in fact the discussion is about infinity.
Infinity has a Symbol, but it is not a jnumber. So a circle with infinite Diameter, radius or circumference is not within the rule of the game and thus meaningless.
We had that at School, when trying to get a circle's area without it's circumfernce. Somebody cam up with the fromula pi times (R-1/infinte)2...which is no valid mathematical formula.
I have just spotted a subtle flaw in my argument that there was a subtle flaw in my original argument which means I now think my original argument may have been entirely correct after all!
Suppose you had an 'infinitely' large square (assuming that makes sense). That doesn't logically entail that it has 4 corners on a finite scale! That is because lets suppose on a finite scale those 'corners' aren't 'true' corners but rather they are rounded off so that they are actually gentle curves with each of the 4 doing a 90 degree turn. As you zoom out to ever larger scales and towards infinity, those curved corners will look more and more like sharp corners so that viewed from 'infinitely far away' (assuming 'infinitely far away' makes sense which I say doesn't) they should be infinitely sharp corners thus 'true' corners.
BUT at 'infinitely far away' you are no longer viewing the corners on a finite scale but an infinite scale thus every point on the square line would be the same length away from the square's center because that length would be positive infinity and there is only one positive infinity! And that would mean they STILL cannot be 'true' corners because how can they be 'true' corners of a square when they are no closer the the square's center than any point along its edges! And that means on both the infinite scale and the finite scale, for different reasons, that 'infinitely' large square has no corners and therefore it is wrong to say that 'infinitely' large square must have 4 corners because I have just shown how this may not be the case!
Originally posted by @kazetnagorraBut is it still meaningfully a 'circle' after radius R becomes infinite?
This describes a circle with radius R. An infinitely large circle is obtained after taking the limit R -> \infty.
Originally posted by @humyThe circle in the limit R -> \infty is as "meaningful" as a real number R in the limit R -> \infty.
But is it still meaningfully a 'circle' after radius R becomes infinite?
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The post that was quoted here has been removedHere are my observations. You are a fraud that goes around regurgitating math proofs from text books or the internet. I know this because you have never once provided a single iota of original mathematics to solve a posters problem. There have been plenty of opportunities for you to "give your two cents" , and you never show up...I find this extraordinarily odd given your "love" of mathematics, and the fact that there is hardly a thread on the site that isn't tainted by your pretentious babblings. You are a repugnant human being, minus the human being part.
Originally posted by @ponderableYes, but. On the ordinary real line there is no "point at infinity", there are just larger and larger numbers. This is why Kazet expressed it in terms of limits. However, it is possible to extend the real line by adding a point to the right of all the points in the line (+ infinity) and a point to the left of all the numbers (- infinity). This is known as the extended real line [1].
So in fact the discussion is about infinity.
Infinity has a Symbol, but it is not a jnumber. So a circle with infinite Diameter, radius or circumference is not within the rule of the game and thus meaningless.
We had that at School, when trying to get a circle's area without it's circumfernce. Somebody cam up with the fromula pi times (R-1/infinte)2...which is no valid mathematical formula.
There is a bijection (jargon for a rule that associates each element in one set with an element in another set so that every element in each set has a partner in the other set) from the positive real line R+ = {x > 0 | x in R} to the set of circles C. If we extend the real line then this mapping ceases to be a bijection because there is nothing to map the point at infinity to. So we invent the infinite circle, which is just the circle whose radius is the point at infinity on the positive extended real number line.
So we can just change the rules of the game.
[1] https://en.wikipedia.org/wiki/Extended_real_number_line