white hole and black whole
Here, white hole is the terminology we use when we cram a set of points of cardinality 2^\aleph_0 into an interval of infinitesimal length and black whole is the terminology for the entire space beyond the finite space.
We will consider only the real line in its entirety, infinite on both sides, with the understanding that we can have a visualization of the three-dimensional space, if we have a clear mental picture of the one-dimensional line.
White hole. First of all, let us note that corresponding to every real number it is possible to visualize a white hole attached to it. We will illustrate this with an example. Consider the number 2/3 written as an infinite sequence 0.101010... and its finite terminations 0.1, 0.101, 0.10101, ... which can be used to represent the intervals (1/2,2/3), (5/8,2/3), (21/32, 2/3), ... respectively. Note that the length of the interval decreases monotonically when the length of the termination increases and the cardinality of the set of points inside these intervals remain constant at 2^\aleph_0. Simply stated, we can say that a white hole is what we get when we visualize the interval corresponding to the entire nonterminating sequence, and this infinitely small interval contains 2^\aleph_0 points in it. From this description it should be clear that the term white hole does introduce us to the notion of an infinitesimal.
It is easy to see that any point in a unit interval can be represented by an infinite subset of natural numbers. For example, the representation 0.101010... shows that 2/3 can be represented by the set of odd integers, the places where the 1's occur in the sequence. Recognizing this, we can define the white hole or infinitesimal corresponding to 2/3 as the set of all those subsets of \aleph_0 which contain the odd integers. We will take it for granted that the cardinality of this set is 2^\aleph_0.
Black Whole. When we flip the real number xxx...xxx.xxxxx... around the binary point, the resulting ...xxxxx.xxx...xxx we define as a supernatural number and from symmetry considerations we assert that a transfinite stretch comes attached with every supernatural number. Using the notion of point at infinity of complex analysis, a transfinite stretch may also be called a black hole, without violating the conventional sense of the word. If we define the black whole as the set of all black holes and the white whole as the set of all white holes, we will have the universe neatly divided into the two halves, white whole and black whole.
Epilogue. According to our visualization, the infinitesimal corresponding to 2/3 is a dedekind edge between the rational numbers less than 2/3 and greater than 2/3. We do not call it a Dedekind Cut because of its nonzero length.
If we call the elements of a white hole, figments, and consider it as axiomatic that not even the axiom of choice can pick up a figment from a white hole, scientists will have the pleasant situation where they will not have to deal with sets which are not Lebesgue measurable. This follows from the fact that the axiom of choice is crucial for the creation of nonmeasurable sets. As a corollary, we can also claim that we will never be
able to walk along a transfinite stretch.
White Hole and Black Whole: Visualizing the Physical Universe
Tuesday, April 10, 2007
