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Mathematics
General Math
Are there an infinite number of infinities?
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[QUOTE="FactChecker, post: 6830096, member: 500115"] Good question. Your argument is a good start, but there are different sizes of infinity, called "cardinality". They are indicated by ##\aleph_0, \aleph_1, ...## You can't use regular arithmetic when infinity is involved. You have to use one-to-one (1-1) mappings. If there is a 1-1 mapping between two infinite sets, then they have the same cardinality. Otherwise, they have different cardinality. The cardinality of the natural numbers is ##\aleph_0##. Consider the natural numbers, 1,2,3,4,... and the even natural numbers, 2,4,6,8,... They have the same cardinality because the mapping ##n \rightarrow 2n## is a 1-1 mapping between the natural numbers and the even numbers. You can define a 1-1 mapping between the natural numbers and an infinite number (cardinality ##\aleph_0##) of infinite sets (cardinality ##\aleph_0##). On the other hand, [URL='https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument']Cantor has shown[/URL] that there is no 1-1 mapping between the natural numbers and the real numbers in the interval [0.1]. The real numbers in [0,1] is a set with a larger cardinality. It is a larger type of infinity, ##\aleph_1## (EDIT or larger). [/QUOTE]
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Mathematics
General Math
Are there an infinite number of infinities?
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