The Shāntī Pāth & Cantor’s Infinities

The ‘lazy eight’ symbol ‘’ has long been used to denote the infinite since this shape can be traversed endlessly.

Till the mid 19th century, the only concept of infinity, in the mathematical world, was linked to ‘countable numbers’ or integers or natural numbers – as they are commonly known. As we count, we go :

1, 2, 3, 4,…..n…….. ∞

Here ‘n’ stands for any countable integer. At the end of this seemingly endless sequence, the number becomes too large to count and is denoted by ‘’.

The earliest limitation of this way of thinking was probably expressed by Euclid (330 – 275 BC) when he tried to answer the question – “How many points are there in a ‘line segment’ ?” Now a line is a ‘line’; it is continuous and how can we talk about it as ‘discrete’ points ? But then, that is what mathematicians are all about; they question what seems obvious and keep building a pyramid of well-grounded higher knowledge. In fact Euclid’s axioms and theorems are even today part of elementary school Geometry. But he could not answer this one!Galileo Galilei (1564 – 1642) published in 1638 Dialogue of the Two New Sciences gives some excerpts of his arguments on this). In this work, he outlined a paradox relating to ‘natural numbers’ and their ‘squares’. He derived a geometric equivalent by arranging two concentric circles as shown in Fig. 1. and raised the question– “Are the number of points on the inner, smaller circle the same as those on the outer, bigger one ?” For every point A or B on the inner circle, there is a corresponding ray OA or OB respectively and a corresponding A’ or B’ point on the outer circle and so on. Thus, infinite rays emanate from ‘one’ center O. These diverge to an infinite number of points A,B etc. on the smaller circle and each in turn connects to a corresponding A’, B’ etc. on the outer one. The paradox explicitly is – “how can the rays emanating from the center link an identical number of infinite points on the inner and outer circle ?” The answer to our question is not obvious and it did trouble intellectuals for almost three centuries till Cantor resolved it rigorously.