Originally posted by @joe-shmoIt was Zahanzi, not vivify. The post that initiated the exchange was the first one on page 17 of thread Thread 177012 My reply is fourth from bottom on the same page. Duchess's response is top of the next page.
I don't buy it. I know nothing of the continuum hypothesis, yet If I were reading your post about it and you mentioned it by name as simple google search of "continuum hyptho..." and google literally suggest "continuum hypothesis undecidable". from which I can skim through the text and pull the statement from Wolfram
"Together, Gödel's and Cohen's re ...[text shortened]... to tear the argument apart or verify it as long as the content was recognizable as mathematics.
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Originally posted by @deepthoughtAs I suspected, its a short quip that reads like a line from Wikipedia. I don't see any deep mathematical analysis there, do you? You were talking of pure mathematical truth and she knew of a loophole. In fact, it seems like a very trivial piece of information for any admirer of mathematics to have. I personally have a range of mathematical familiarity that far exceeds my ability to competently perform it, and it was obtained by casually reading some math books.
It was Zahanzi, not vivify. The post that initiated the exchange was the first one on page 17 of thread Thread 177012 My reply is fourth from bottom on the same page. Duchess's response is top of the next page.
Probably something that Duchess is good at, reading a regurgitating information. In my opinion I don't see anything to change my mind about her in that post. I think she is a fraud.
Here I will start with the assumption that an infinitely large perfect circle (one with radius = +infinity) makes sense in pure geometry to then see if I can then make it continue to make sense or if I find it makes no sense.
An infinitely large perfect circle (one with radius = +infinity) would, on its infinite scale, have a curved line and not straight lines for its circumference (else it wouldn't be an infinitely large circle by definition).
But if you keep zooming in on any circle's circumference, the ever smaller segment of the circumference would appear to be more and more like a straight line.
So if you could then somehow 'infinitely zoom in' to see just one finite segment (out of an infinite number of such finite segments) of that circumference of an infinitely large circle but now on the finite scale, it should be a perfect straight line i.e. it should be a straight line on the finite scale (but still not on the infinite scale).
But if you extend a finite straight line to infinity while keeping each and every finite part of it perfectly straight on the finite scales, surely that line would stay straight and thus wouldn't curve around or bend on an infinite scale? And even if it somehow did curve around on an infinite scale, apart from the question why would it, why would it curve evenly on an infinite scale to form an infinitely large perfect circle and why would it ever curve around to eventually meet the same point to form an infinitely large circle rather than, for example, just keep randomly change direction to form a random scribble on the infinite scale?
Any flaw with my above reasoning?
Originally posted by @humyI do not agree that the definition of a circle involves curvature. A circle is an object composed of points equidistant from a centre. In set builder notation, on the Euclidean plane we have the following:
Here I will start with the assumption that an infinitely large perfect circle (one with radius = +infinity) makes sense in pure geometry to then see if I can then make it continue to make sense or if I find it makes no sense.
An infinitely large perfect circle (one with radius = +infinity) would, on its infinite scale, have a curved line and not straight li ...[text shortened]... direction to form a random scribble on the infinite scale?
Any flaw with my above reasoning?
C₁ = {P ∊ ℝ² | x(P)² + y(P)² = 1, x,y ∊ ℝ}
where C₁ is the set of points contained in the unit circle. P represents a point and x(P) and y(P) are the coordinates of point P. The set of circles is then:
C = {{P ∊ ℝ² | x(P)² + y(P)² = r², x,y ∊ ℝ} | r ∊ ℝ}
This set contains one limit point at r = 0. If we extend the real numbers to include points at infinity we can define a new set of circles which has an infinite circle:
C* = {{P ∊ ℝ*² | x(P)² + y(P)² = r², x,y ∊ ℝ*} | r ∊ ℝ*}
Where ℝ* is the extended real number line and C* is the corresponding set of circles.
The curvature at a point on a curve K = 1/r where r is the radius of the circle whose centre is at the point at which two lines, normal to the curve, and arbitrarily close together meet. For a straight line the corresponding circle does not exist in C; but is in C*, being the infinite circle.
In non-Euclidean geometry the set of points equidistant from a centre may not be what you normally think of as a circle. A portion of the corresponding curve may well be straight. Basically you are begging the question by defining an object in terms of its curvature (or absence of curvature) and then noting the absence of curvature.
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Originally posted by @humyCould it be that the shape of infinity is determined by gravity?
Here I will start with the assumption that an infinitely large perfect circle (one with radius = +infinity) makes sense in pure geometry to then see if I can then make it continue to make sense or if I find it makes no sense.
An infinitely large perfect circle (one with radius = +infinity) would, on its infinite scale, have a curved line and not straight li ...[text shortened]... direction to form a random scribble on the infinite scale?
Any flaw with my above reasoning?
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The post that was quoted here has been removedNo....I think you came up with every bit of that elegant little morsel...all on your very own.
"Gravity has no influence in the realm of mathematics."
Until the "Unified Field Theory" is proven. Steven Hawking believed it one day would be.
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The post that was quoted here has been removedThe lovable, likable and ever-popular Duchess64 knows when to throw complimentary bones of knowledge to the wee knaves of Red Hot Pawn.
Fortunately, her genius at winning friends and influencing people is exclusive solely to her...and cannot be transferred to plebes.
Originally posted by @deepthoughtI think I am finally getting what your are saying.
I do not agree that the definition of a circle involves curvature.
....
The curvature at a point on a curve K = 1/r where r is the radius of the circle whose centre is at the point at which two lines, normal to the curve, and arbitrarily close together meet. For a straight line the corresponding circle does not exist in C; but is in C*, being the infinite circle.
If I now understanding you correctly, you imply that you can define without any contradictions an infinitely large circle (meaning radius = +infinity) and its circumference would be an infinity long straight line and it would be meaningless to say you could "zoom infinitely out" to be able to see it from infinitely far away to see the whole of that infinite circle as a whole on its infinite scale and so see it as you would normally see a circle i.e. see it with a curved circumference.
So there doesn't exist any "infinitely large scale" from which you could view its shape as a whole and thus at NO scale, infinite or finite, is its circumference curved.
Thus its an error to imply you could rationally symbolize an infinitely large circle as a whole on the infinite scale as you would normally picture a finite circle as a whole on the finite scale.
Before I more on, have I got that all exactly correct?