With respect to proving the smoothness of the Navier Stokes equation (Clay Millennium Prize) is it sufficient to prove that perfect smoothness is not possible in geometry because it implies critical inconsistencies within geometry in Euclidean space (hence the NS eqn. is limited to solutions in a non-smooth continuum, and turbulence is a necessary part of nature? Here is my argument: Consider Figure 1, showing two line segments on a Cartesian plane, sides A and B parallel to the X and Y axis, forming sides of a right triangle. Let each line segment have unit length, so that the total length of the two line segments is 2 units. A right triangle
Bisecting A and B and rearrangement of the parts results in two triangles configuration (Figure 2). Two triangles joined at a vertex, resting in the line of the hypotenuse of the initial triangle Rearrangement cannot affect the total length, which remains 2 units. Observe that the number of points at which the shape touches the diagonal increases from 2 to 3 points. Now observe Figure 3. Further bisection an rearrangement forms 4 triangles lying in the line of the hypotenuse By symmetry with the previous construction, the total length remains two and the number of points at which the form touches the diagonal is 1 + 2n where n is the number of bisection and rearrangements of the sub-forms. Again, by symmetry, the width of the sub-form is halved with each iteration, and without the aid of a microscope (to help think about the model) it soon becomes difficult to distinguish between this form and that of the diagonal that spans between the end points of the construction. The symmetry of the system implies that there are triangles ‘all the way down’, independent of any iterative aspects. The construction converges on a line, and, in the limit, the number of places at which the form touches the diagonal is likewise 1 + 2^aleph0. According to Cantor this covers the diagonal (with a point ‘left over’). One might note that this varies from Cantor’s conjecture that 2^aleph0 is the cardinality of the reals, which rides on the back of the continuum hypothesis in that if there is no interpolate, then this follows by deduction. However this is relatively unimportant for the main thrust of this investigation. What is important is that had I chosen to break the first triangle into any multiple of prime numbers of sub triangles and continue in this fashion, the same reasoning would imply that there is upon the line 3^aleph0, 5^aleph0, 7^aleph0, 11^aleph0 …up into higher cardinalities of points on the diagonal. Because each end of the line is fixed with respect to these sets of points—each set ends at the same point—these sets of points cannot be set into one to one correspondence. Rather these are in two to three, two to five, three to five… correspondence; every possible ratio other than ratios involving the number 1. Is Cantor’s continuum hypothesis disconfirmed by this? That any one of these options covers the diagonal is not a paradox. However, that they all fill the same continuum is at least a paradox, and indeed inconsistent, for none of these points can land on the same place except at the endpoints of the line because they are based on prime numbers. The matter has nothing to do with their countability, only that all describe the same continuum, and they all cover the same diagonal. But they can’t all have the same cardinality. If this reasoning is correct, then Cantor’s continuum problem (Gödel 1983), ‘How many points are there on a straight line in Euclidean space?’ is problematic. Separately one may argue that there is a one to one relationship between each point, or line segment on the straight line, and that of the hypotenuse. In itself this is not immediately a problem until it is recognized that while the length of the diagonal is √2, by Pythagoras’s theorem, the length of the vertical and horizontal line segments remains 2. To a person inspecting the line, or conceiving the line, without prior knowledge of the internal properties of the construct, they would be in no position to distinguish between the given construction and a straight line. In fact it is questionable as to how the construction avoids further inconsistency, given that there seems to be no basis for introducing a consistent metric or scale in the limit case. Because every right-angled triangle may be similarly rearranged, there are an infinite number of examples of this kind in geometry. Because the same may be done with semicircles and every shape between the semicircle and the triangle, there is a transfinite number of problems. This implies directly, that our notion of space as being populated by points, as we suppose them to be, is flawed. It does lend credence to a treatment of space as being populated by some other kind of structure that is not a point as we think such to be. We might propose that this implies that at some level the notion of scale loses sense and that a more holistic theory would accommodate the dichotomy. But this prompts the question: at what length scale does the change take place and, given the symmetry of the first triangle with any triangle at a lower level, what could call a halt to the devolution? More particularly, what kind of principle could act selectively on a particular small triangle and not the one above or below it, and say ‘this is the smallest triangle’, which is exactly what we find evidence for in the world that we experience – minimum increments in the form of the Planck length and Planck time (Isham 1989).
I would contend that this implies that the idea of a smooth continuum is flawed at foundation, and that there exists a minimum separation between points (which implies the existence of a minimum metric). This further implies there should be a minimum length scale in physics, and a parallel abstraction in mathematics.
Does this then imply that the Navier Stokes equation also should expect inconsistencies at some length scale and that turbulence is a necessary part of the world, given sufficient time lapse, because there is a minimum scale or difference between points such that an exact solution is unsustainable when the scale is less than some particular value? If so, it seems to me that the NS equation is incomplete and should be rewritten in terms of minimum increments rather than as a differential equation.