The Cardinality of the Ether
By Robert J. Swartz

The ether transmits light waves at the speed of 300000 kps.  This is relative to its inertial reference frame.  Is this reference frame due to some kind of 3D particle lattice, or is it the case that the ether is continuous?  If the ether is an integer lattice, then its cardinality approaches aleph-0 as a whole, and the cardinality of a finite region of ether is finite.  However, there is the possibility that the ether is continuous, meaning that its cardinality is greater than aleph-0.  There is the intermediate possibility that the ether is a rational space in the sense that its points are 1-1 with the rational numbers cubed (Q^3).  I will prove that the cardinality of the ether is greater than aleph-0 by using a motion argument.

Theorem:  The cardinality of a finite region of ether is greater than aleph-0.

Proof:  The Earth orbits the sun with a velocity of 30 kps.  This proves that motion is possible through the stationary cosmic ether.  The Earth also accelerates through the ether because its orbit is circular.  The integral of the Earth's acceleration from time t0 to t1 is a positive real number.  This number is the velocity.  The fact that this integral exists and is non-zero proves that the measure of the ether is non-zero.  This is because the integral of a function over a space of measure zero equals zero.

Set theory says that any set of cardinality aleph-0 has measure zero.  Therefore, a finite region of ether cannot have cardinality aleph-0.  So, therefore, its cardinality is greater than aleph-0.

The set Q^3 has cardinality aleph-0.  Therefore, the ether is not in one to one correspondence with Q^3.  Therefore, transcendental coordinates exist in the ether.