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.