Definition of Infinitesimal

Infinitesimal is defined as an infinite recursive subset of positive integers. The definition visualizes the unit interval as a set of infinitesimals and the unreachable infinite universe as a set of black-wholes.

1. Definition of Infinitesimal and visualization of the universe by Kannan Nambiar Voltaire’s view of infinitesimal: it is a thing whose existence cannot be conceived My vision of infinitesimal: inconceivable, but reachable infinity: conceivable, but unreachable FrontCover

2. Definition of Infinitesimal as a Confluence a possible solution for the millennia conundrum by Kannan Nambiar verS

3. ... it is a thing whose existence cannot be conceived. ___ Voltaire ... ghosts of departed quantities. ___ George Berkeley ... If infinitesimals were ever accepted, the Jesuits feared, the entire world would be plunged into chaos. ___ Amir Alexander (in his book "Infinitesimal") What is an infinitesimal?

4. In binary notation any number in the interval (0,1] can be written as a nonterminating infinite sequence of 0s and 1s, and from this we infer that an infinite subset of positive integers can also represent a number in the interval (0,1]. As an example, the number 2/3 can be represented by the recursive binary sequence .10101010 ... and hence also by the set of odd integers (corresponding to the positions at which the 1s occur). Here, by recursive sequence is meant a sequence which can be generated by a computer. The rationale behind this is explained below. Recognizing these facts gives us a simple definition for the infinitesimal. Definition of Infinitesimal: An infinite recursive subset of positive integers. A definition for the infinitesimal

5. In set theory there are two kinds of numbers that are important: natural numbers and real numbers. These we claim can be represented by two graphs, "natural" graph and "infinitesimal" graph , respectively: |------------|------------|------------|-----------|-----------|-----------|-----------|-------- 0 1 2 3 4 5 6 7 The natural graph shown above is infinite and the nodes are represented by vertical lines representing the mathematical concept "cuts". In the graph each cut represents a natural number. Natural graph

6. Infinitesimal graph The infinitesimal graph for the real numbers is shown below, but only for a unit interval . 0 4 2 6 1 5 3 7 |------|------|------|------|------|------|------|------| 0 1 We have shown only 7 cuts, but in theory we should proceed indefinitely with 15, 31, 63, ... cuts. Like infinite graphs, of course, we cannot totally complete an infinitesimal graph.

7. Recursive sequences as labels Just as we label the cuts (nodes) of the natural graph with integers, we can label the cuts of the infinitesimal graph also with natural numbers and eventually with recursive sequences: For example, to get the label corresponding to the cut 3/8, write its binary sequence .011 in the reverse order as 110 = 6. If we follow this procedure, it should be clear that we will have a one-to-one correspondence between the infinitesimals in the unit interval and the recursive sequences .

8. Cardinality aleph_0 As said earlier, an infinite nonterminating sequence of 0s and 1s, and the corresponding infinite set, can represent a number in the interval (0,1]. Mathematicians have a name for those sequences which can be calculated by a computer, they call them recursive sequences and the corresponding sets recursive sets . Since computer programs are written using a finite alphabet, it follows that infinite recursive sets can be enumerated. In other words, the cardinality of the set of infinite recursive sets is aleph_0 .

9. Confluence Definition of confluence: Consider an infinite sequence of binary strings of even length starting with a binary point, as given below. .x1 .xx11 .xxx111 ... ... ... . xxxxxxxxxxx ... ... 1111111111 In the last sequence, if the initial string of x's of infinite length represent a recursive sequence, we call the entire sequence a 1-confluence . For the 0-confluence . xxxxxxxxxx ... ... 0000000000 a similar definition holds good.

10. Recap Here is the upshot of our discussion: 1. 1-confluence represents an infinitesimal edge and its right node. 2. 0-confluence represents an infinitesimal edge and its left node. 3. The cardinality of the set of infinitesimal edges which constitute the real line is aleph_0 . 4. If we can call the sequence 0.xxxxxxx ... an infinitesimal, there is no reason why we cannot call the symmetric sequence ... xxxxxxxx.0 a supernatural number or black-whole . 5. We can assume that all problems of interest for the scientists can be worked out using the concept of infinitesimal defined here.

11. Conclusion A significant result we have from our discussion is that the real line can legitimately be imagined as a set of infinitesimals of cardinality aleph_0 . Since there is a one-to-one correspondence between infinitesimals and black-wholes, we can conclude that the cardinality of the set of black-wholes is also aleph_0 . For more details and applications, see the YouTube video “Enhanced Set Theory”.

12. BackCover If we can accept an infinite sequence of digits going towards the right from a decimal point as a legitimate mathematical concept, why can’t we similarly accept an infinite sequence going towards the left and call it a supernatural number or black-whole ? Of course, supernatural numbers are no more supernatural than transcendental numbers are transcendental. If we extend the idea of the infinitesimal to three-dimension, corresponding to every infinitesimal sphere within a sphere of unit radius, we can visualize a spherical black-whole of infinite radius. Our description of natural graph makes it clear that it can be visualized as an aleph_0 fold magnification of the infinitesimal graph . Let us not forget that it is Euclid’s geometry that led us to mathematical logic. Here are some visualizations that are possible with the infinitesimal: If we can accept a unidirectional infinite sequence as a number , why can’t we accept the idea of a bidirectional infinite confluence and call it an infinitesimal ?