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JULY, 1990
Are asamkhyāta and ananta transfinite numbers?
It is our contention that asamkhyāta and ananta are used in the sense of actual infinity" in Jain literature dealing with cosmology. A physical notion of actual infinity need not imply a mathematical notion of actual infinity or a transfinite number. A finite size physical object might be understood as comprising of infinite material monads but it is not imperative that such an understanding would lead to a notion of transfinite numbers. Jains did consider space, time, matter as comprising of infinite elements but did they really have a notion of transfinite numbers. Dhavalā, a treatise on enumeration of souls, states that the number which does not get exhausted by continuous subtraction of one after one is ananta." Further Dhavala states that number which is not exhausted by continuous subtraction even till endless time has been called ananta by great teacher (maharși). This is a clear and exact formulation of notion of transfinite number called ananta. It is interesting to note that term ananta literally means 'un-ending' and it was used only in this sense in non-Jain literature but Jains gave it also a technical meaning of transfinite number or mathematical actual infinity.
Though it is clear that ananta is a transfinite number but such a clarity does not exist about asamkhyāta number. Confusion regarding exact meaning of asamkhyāta number is inbuilt in Jain literature and we would see later that this confusion stems from (a) confusion of mathematical and physical notions of actual inifinity, (b) confusion in Jain thought on nature of time (kāla). Now we will elaborate on this confusion.
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6 Mathematical actual infinity or transfinite number is like a number higher than all natural numbers. Such number would be beyond finite natural numbers hence is called transfinite numbers.
Dhavala [III, p 267]/jo rasi egegaruve avanijjamane puna na samappaiso rasi ananto/ Also Dhavala [III, p 25]/vaenitthantassa anantattavirohado/'which gets exhausted by subtraction is against nature of ananta'.
Dhavala [IV, p 338]/sante vae na nitthadi kalenanamtaena vi jo rasi so ananto tti viniddittho mahesina/'that number which is not exhausted by continuous subtraction even till endless time has been called ananta by great teachers'. Dhavala [IV, p 338]/so ananto vuccadi, jo samkhejjasamkhejjarasivvae santa anantena vi kalena na nitthadi/'ananta is that which by subtraction of samkhyata (numerable) and asamkhyata (innumerable) numbers for even till unending time is not exhausted'.
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Notion of 'actual infinity'; signifies an entity which actually is composed of infinite elements. As distinct from it 'potential infinity' means potentially infinite like natural numbers which can be counted on and on giving impression of being potentially unending or infinite but not of one entity which has infinite component numbers.
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