Infinitesimal Graph: Visualizing the Unit Interval

Briefly stated, infinitesimal graph is what we get when we consider an infinitesimal in the unit interval as a dedekind edge separating rational numbers above and below a number. The dedekind cut itself we imagine to be on the right side of the edge and call it a dedekind node.

Infinitesimal Graph. In representing the unit interval as a graph, we consider only the recursive numbers, that is, the numbers that are programmable on a computer. Since the programs in a computer can be listed, it follows that the cardinality of recursive numbers in a unit interval is \aleph_0. Thus the cardinality of the set of edges in the infinitesimal graph corresponding to a unit interval is \aleph_0.

Just as an infinite graph cannot be drawn in its entirety, we cannot draw a completed infinitesimal graph also. But we can imagine a limit process which will ultimately provide the infinitesimal graph in the limit, as described below.

Write the numbers 1,2,3,4,... in binary in the reverse order as 1,01,11,001,... and put a binary point in the beginning of each resulting in the infinite sequence .1,.01,.11,.001,... representing fractions in the unit interval. Mark the dedekind cut corresponding to these points and consider them as nodes. The graph we get as a limit is the infinitesimal graph for the unit interval (0,1].

The cardinality of the set of points in a dedekind edge can easily seen to be 2^\aleph_0. First of all, note that any fraction in the unit interval can be written as an infinite binary sequence. For example, we can write the sequence for 2/3 as 0.10101... , which we can use to have an infinite set representation for 2/3. If we take the positions at which the sequence takes the value 1, we get the set representation for 2/3 as the set of odd integers. Further, we can claim that the dedekind edge for 2/3 contains all those sets of positive integers which contain the odd integers. The cardinality of this set we can easily recognize as 2^\aleph_0.

The description above allows us to state the following facts. The unit interval is a set of dedekind edges, each edge representing a recursive number, and the cardinality of the set is \aleph_0. A dedekind node corresponding to a number cuts the unit interval into two pieces, each piece representing reals below and above the number. Each dedekind edge contains 2^\aleph_0 elements.

Real Cardinality. We may call an element of an edge a figment and the edge itself a relement (real element), and claim that not even the axiom of choice can pick a figment from a relement. We define real cardinality as the cardinality of a set when only relements in it are considered.

Skolem Paradox. These definitions allow us to resolve some of the vexing problems of set theory. Cantor's theorem asserts that every model of Zermelo-Fraenkel set theory (ZF) has to have cardinality greater than \aleph_0. On the other hand, Lowenheim-Skolem theorem (LS) says that there is a model of ZF theory, whose cardinality is \aleph_0. These two statements together is called Skolem Paradox. The notion of relements gives us a reasonable way to circumvent the paradox. We merely take the LS theorem as stating that the real cardinality of a model of set theory need not be greater than \aleph_0. Cantor's theorem is satisfied when we consider figments also as elements.

Nonmeasurable Sets. With the introduction of the concept of relements, we are in a position to dispense with the notion of sets which are not Lebesgue measurable. This is because a nonmeasurable set cannot be constructed without using the axiom of choice.

Further reading: See the paper http://www.e-atheneum.net/science/ist_visualization.pdf for some more details. Note that the axioms given in this paper are dated, even though nothing has been found to be wrong with them.