Sramana 2011 01

SR No.: 525075
Original Sutra Author: N/A
Author: Sundarshanlal Jain, Shreeprakash Pandey
Publisher: Parshvanath Vidhyashram Varanasi
Publication Year: 2011
Total Pages: 172
Language: Hindi
Classification: Magazine, India_Sramana, & India
File Size: 15 MB

Page 76

58: Śramana, Vol 62, No. 1 January-March 2011
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2n, ē }, and the set of all squares of positive integers {1, 4, 9, ē n2, ē }; thus, in contrast to finite sets, two infinite sets, one of which is a subset of the other, can have the same transfinite cardinal number, in this case, aleph0. It can be proved that all countably infinite sets, among which are the set of all rational numbers (any number that can be expressed as a ratio of two integers e.g. 1/10, 2/ 35987, ē) and the set of all algebraic numbers, have the cardinal number aleph0. Since the union of two countably infinite sets is a countably infinite set, aleph0+ aleph0 = aleph0; moreover, aleph0, aleph0 = aleph0, so that in general, n, aleph0= aleph0 and alephOn = aleph0, where n is any finite number. It can also be shown, however, that the set of all real numbers (includes irrational numbers such as p or the square root of 2), designated by c (for "continuum"), is greater than aleph0; the set of all points on a line and the set of all points on any segment of a line are also designated by the transfinite cardinal number c. An even larger transfinite number is 2c, which designates the set of all subsets of the real numbers, i.e., the set of all {a,b}-valued functions whose domain for a and b is the real numbers. Transfinite ordinal numbers are also defined for certain ordered sets, two such being equivalent if there is a one-to-one correspondence between the sets, which preserves the ordering. The transfinite ordinal number of the positive integers is designated by ?30
Thus Bajzelj has her point. An infinite number of jivas can occupy a fixed amount of space so long as their centerpoints are distinct from one another, their forms overlap, and the size of the largest form is not larger than the space it occupies. The addition of any number to aleph0 is still aleph0. That is, no matter how many jivas are liberated, the volume of their residence does not have to change to accommodate them.