dmy (01/29/83)
The various "orders of infinity" are called TRANSFINITE numbers, not TRANSCENDENTAL numbers. Those are something else. And it is only in the realm of mathematics that it is proper to speak of infinity without qualifying the term as "potential infinity" or "actual infinity". In mathematics, infinity is carefully defined. Outside of mathematics, one can speak of things as being potentially infinite (such as the size of the universe, or the human capacity for knowledge) in the sense that these things have no inherent, pre-determined limit; but nothing can be actually infinite -- that so-called concept is self-contradictory. Actual "quantities", that can be described in words, such as "the number of grains of sand on all the beaches in the world" or "the total number of atoms in the universe" or "the total number of legal tournament chess games" are all finite. Potential quantities, such as "the total number of sub-atomic particles of any granularity, currently known or that will ever be discovered" or "the total number of chess games that COULD be played if repeated positions are ignored" may be infinite, but only in the "potential" sense -- their "infiniteness" can never be "achieved in reality". Consider that "infinite" means "without end" or "without completion". And any time that anything is defined or described as having some quality or property in "infinite measure", where this is intended to be meant in the "actual" sense, it is either a case of poetic license, or an error -- the thing doesn't have that property, or it doesn't exist. --dmy-- (David Yablon, 82/1/29, No.Virg.)