Jain Journal 1990 07

Author(s): Jain Bhawan Publication
Publisher: Jain Bhawan Publication

Page 17

JAIN JOURNAL
number implying that even a is a transfinite number. This would be a conclusion if asamkhyāta is regarded as an actual infinity in a physical sense and if such a physical actual infinity is used in number calculations it would be a transfinite number.
A much stronger reason to regard asamkhyāta as mathematical actual infinity or a transfinite number is in the nature of mathematical operation which Jains developed to handle various categories of numbers. One such example is the following set of formulas given in Dhavala.31
(1)
J2/(J+J/n)=J—J/(n+1) where J=total number of souls, and n=countable (samkhyāta) number
(2) J/(J+J/us)=J_J/jpA
us and jpA from Table I
(3) J2/(+J/asamkhyāta)=J—J/(asamkhyāta)
(4)
J2/(J+J/uAA)=J-J/jpa uAA and jpa from Table I
(5) J2/(J+J/ananta)=J--J/ananta
Last four relation have obviously been constituted by analogy with relation (1) which is valid for all natural numbers. Besides, some more operations on various kinds of asamkhyāta and ananta have been assumed. These corresponding to the five formulas are :
(1) nn+1 if n is finite
(2) jpA=us +1
(3) asamkhyāta=asamkhyāta+1, where asamkhyāta is neither
minimum nor maximum asamkhyāta number
(4) jpa=uAA+1
(5) ananta=ananta+1, where ananta stands for neither minimum nor maximum ananta number
1
Dhavala (III, pp. 45-46) gives these five formulas.