I can't remember many of the details of the following post from 2007. But I can visualize what I meant, which is that if the reals are mapped onto any finite closed loop, this set of reals has 2N permutations, whereby each point is 0 distance from two "adjacent" points.
Kryptograff contains Paul Conant's thoughts on scientific and mathematical matters. Conant is a journalist who holds no scientific degrees. This blog was set up after problems at the previous address: http://kryptograff.blogspot.com. Please check there for previous posts.
Draft 3
Assertion
The Jordan curve theorem holds for a set of non-visualizable, "wild" curves that can be posited in accord with the Well-Ordering Principle.
Prefatory remarks
An intuitive idea of the well-ordering of the reals can be had by considering the following:
We have that any infinite digit string of 0s and 1s may represent a real. Now a string of length n digits may be ordered from least to greatest in 2n ways, with the awareness that any digit after n is a 0.
This denumerable ordering holds for any finite n. However, denumerable ordering does not hold for the entire set 2N of course. But the Well-Ordering Principle can be interpreted to mean that the set of string permutations is precisely ordered at 2N.
We can obtain the set of all curves, including wild non-differentiable and non-fractal-like curves thus:
Orienting ourselves on the x-y grid, using Quad I for convenience, we can arbitrarily assign any y height to any x value in the interval, say, [1,2]. By this, two neighboring x points can have wildly different heights, though there would still exist a slope for them. But, by the Well-Ordering Principle, there must exist two points that are precisely ordered and that have 0 distance between them. These two points will have no slope and constitute a wild, non-visualizable curve "section." That is, we have a situation where there is a y height for every real but no slope for the curve anywhere, even though y does not equal 0.
Though the area under such a curve must be less than 1*ymax, we may find it difficult to evaluate this integral or even give a good approximation by numerical methods.
To complete the set of all planar curves, we mention fractal and fractal-like curves, which are discussed briefly below.
Proof
We form what I call a molecule, or bubble, with the understanding that such an entity may not be visualizable with a drawing.
We may define an origin-based molecule as r = cosx where, with a well-ordering of the set of radians, r is any finite length. Additionally, r(x) = cosx is a relation with 2n-1 values, accounting for "fingers" -- any of which may be infinitely short -- and "interior molecules" beyond the neighborhood of origin. An interior bubble is defined in the same way as a basis bubble, except that its relation r' = cosx requires that for every value of x, r'(x) is less than r(x) and falls between origin and r(x). (Note: an interior bubble is defined by the relation and is not considered an a priori figure here.)
This will suffice to describe any n-loop; i.e., simple loop or pretzel, though we must shortly consider some sets.
[To help one's mental picture, we can proceed thus after forming a basis molecule whereby there are no fingers or holes. We form a second molecule, which may or may not be congruent, and map it onto the first by arranging a translation of coordinates, the effect of which is to intersect the two bubbles such that they share at least two points. All points other than the join points are then erased. The join points are those which do not intersect only a bubble's points.
[We can orient a bubble any way we like and add bubbles to bubbles to our heart's content. We may get a simple loop or an n-loop pretzel.
[A pretzel may appear when two or more bubbles intersect and the intersection set is construed to be "empty" or "background." If a pretzel hole appears, then the intersection obeys the relation requirements of a molecule. A single-hole pretzel is defined by the relation s(1) and s(2) each have one value and there is a subset for which there are exactly four distinct values of s(x).]
Now the interval [r(x)low,r(x)high] is (ignoring fractals), a finite line segment. So we regard those two values as the end points of the line segment. We then require a Dedekind cut between such an end point and the end point of the corresponding, coinciding half-line. In the case of fingers and pretzel holes, we have rather than a half-line, another line segment, of course.
Now the set of such Dedekind cuts maps as a continuous curve about a finite area with no end points. Suppose the boundary curve had two end points. In that case the relation r would have 0 or 2n values, a contradiction.
So, with respect to a molecule, we have that a point on the plane is either an element of the figure's Dedekind boundary set, an element of a line segment r = cos(x) such that there are 2n-1 values of r(x) (not including origin), or neither. So then, the Jordan curve theorem is proved for molecules -- n-loops -- with continuous or wild curves. (We have not bothered with the matter of nested sets of interior loops, which clearly follows from the preceding.)
In the matter of fractal, or fractal-like, curves, it is plain that a fractal construction at any finite step n is composed of a set of self-similar bubbles of diminishing size. Clearly our proof holds for any such step. By the way, we can see immediately that we can apply, at least notionally, a wild curve to a fractal, giving us a whole new set of fractals.
In the case of a fractal, the non-trivial fractal "slopes," though non-differentiable, take on infinitesimal form in the infinite limit, but what form does a wild fractal curve take? The wild part has 0 slope. So whether the curve can be said to exist in the algorithmic limit must be determined from consideration of axioms. At this point, I am content to point out such a situation.
The Jordan curve theorem also applies to any curve of infinite length that is found at, above or below the x axis. We have the relation r(x) which may have 2n-1 distinct values. We then follow the arguments above and have that a point is found in a Dedekind boundary, between a Dedekind boundary point and r(a)2n-1 or not in the Dedekind boundary but above r(a)max or below r(a)min.
So that implies that there must be "wild" curves where x < y < z does not necessarily hold. Since the set of permutations covers all possibilities, we have the subset of all possible continuous loops, including an infinitude of unvisualizable ones.
As there is no way for a curve or line to pierce the continuous curve, there is a set of points which that segment does pierce and another set which cannot be intersected. That latter set is called the "inside" of the loop and the other set is outside the loop. (The case where outer points come arbitrarily close to the boundary is covered.)
The article below was intended to cover not only single loops, but pretzels, but from this vantage point I don't find it all that clear.
Kryptograff contains Paul Conant's thoughts on scientific and mathematical matters. Conant is a journalist who holds no scientific degrees. This blog was set up after problems at the previous address: http://kryptograff.blogspot.com. Please check there for previous posts.
The Jordan curve theorem holds for wild curves
Draft 3
Assertion
The Jordan curve theorem holds for a set of non-visualizable, "wild" curves that can be posited in accord with the Well-Ordering Principle.
Prefatory remarks
An intuitive idea of the well-ordering of the reals can be had by considering the following:
We have that any infinite digit string of 0s and 1s may represent a real. Now a string of length n digits may be ordered from least to greatest in 2n ways, with the awareness that any digit after n is a 0.
This denumerable ordering holds for any finite n. However, denumerable ordering does not hold for the entire set 2N of course. But the Well-Ordering Principle can be interpreted to mean that the set of string permutations is precisely ordered at 2N.
We can obtain the set of all curves, including wild non-differentiable and non-fractal-like curves thus:
Orienting ourselves on the x-y grid, using Quad I for convenience, we can arbitrarily assign any y height to any x value in the interval, say, [1,2]. By this, two neighboring x points can have wildly different heights, though there would still exist a slope for them. But, by the Well-Ordering Principle, there must exist two points that are precisely ordered and that have 0 distance between them. These two points will have no slope and constitute a wild, non-visualizable curve "section." That is, we have a situation where there is a y height for every real but no slope for the curve anywhere, even though y does not equal 0.
Though the area under such a curve must be less than 1*ymax, we may find it difficult to evaluate this integral or even give a good approximation by numerical methods.
To complete the set of all planar curves, we mention fractal and fractal-like curves, which are discussed briefly below.
Proof
We form what I call a molecule, or bubble, with the understanding that such an entity may not be visualizable with a drawing.
We may define an origin-based molecule as r = cosx where, with a well-ordering of the set of radians, r is any finite length. Additionally, r(x) = cosx is a relation with 2n-1 values, accounting for "fingers" -- any of which may be infinitely short -- and "interior molecules" beyond the neighborhood of origin. An interior bubble is defined in the same way as a basis bubble, except that its relation r' = cosx requires that for every value of x, r'(x) is less than r(x) and falls between origin and r(x). (Note: an interior bubble is defined by the relation and is not considered an a priori figure here.)
This will suffice to describe any n-loop; i.e., simple loop or pretzel, though we must shortly consider some sets.
[To help one's mental picture, we can proceed thus after forming a basis molecule whereby there are no fingers or holes. We form a second molecule, which may or may not be congruent, and map it onto the first by arranging a translation of coordinates, the effect of which is to intersect the two bubbles such that they share at least two points. All points other than the join points are then erased. The join points are those which do not intersect only a bubble's points.
[We can orient a bubble any way we like and add bubbles to bubbles to our heart's content. We may get a simple loop or an n-loop pretzel.
[A pretzel may appear when two or more bubbles intersect and the intersection set is construed to be "empty" or "background." If a pretzel hole appears, then the intersection obeys the relation requirements of a molecule. A single-hole pretzel is defined by the relation s(1) and s(2) each have one value and there is a subset for which there are exactly four distinct values of s(x).]
Now the interval [r(x)low,r(x)high] is (ignoring fractals), a finite line segment. So we regard those two values as the end points of the line segment. We then require a Dedekind cut between such an end point and the end point of the corresponding, coinciding half-line. In the case of fingers and pretzel holes, we have rather than a half-line, another line segment, of course.
Now the set of such Dedekind cuts maps as a continuous curve about a finite area with no end points. Suppose the boundary curve had two end points. In that case the relation r would have 0 or 2n values, a contradiction.
So, with respect to a molecule, we have that a point on the plane is either an element of the figure's Dedekind boundary set, an element of a line segment r = cos(x) such that there are 2n-1 values of r(x) (not including origin), or neither. So then, the Jordan curve theorem is proved for molecules -- n-loops -- with continuous or wild curves. (We have not bothered with the matter of nested sets of interior loops, which clearly follows from the preceding.)
In the matter of fractal, or fractal-like, curves, it is plain that a fractal construction at any finite step n is composed of a set of self-similar bubbles of diminishing size. Clearly our proof holds for any such step. By the way, we can see immediately that we can apply, at least notionally, a wild curve to a fractal, giving us a whole new set of fractals.
In the case of a fractal, the non-trivial fractal "slopes," though non-differentiable, take on infinitesimal form in the infinite limit, but what form does a wild fractal curve take? The wild part has 0 slope. So whether the curve can be said to exist in the algorithmic limit must be determined from consideration of axioms. At this point, I am content to point out such a situation.
The Jordan curve theorem also applies to any curve of infinite length that is found at, above or below the x axis. We have the relation r(x) which may have 2n-1 distinct values. We then follow the arguments above and have that a point is found in a Dedekind boundary, between a Dedekind boundary point and r(a)2n-1 or not in the Dedekind boundary but above r(a)max or below r(a)min.
