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How is infinity defined?

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How is infinity defined?

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There are different sizes of infinity. Two sets are said to have equal cardinality (=equal size) if there exists a one-on-one correspondence between them. A set A is said to have less or equal cardinality than B if there exists an injection [itex]A\rightarrow B[/itex]. So a set A has strictly less cardinality if there exists an injection [itex]A\rightarrow B[/itex] but there does not exists a bijection [itex]A\rightarrow B[/itex].

So, it is very easy to see why the naturals have less cardinality than the reals. Indeed, consider

[tex]\mathbb{N}\rightarrow \mathbb{R}:~n\rightarrow n[/tex],

this is an injection. So the cardinality of N is less (or equal!!) to the cardinality of R. But, in fact, the cardinality is striclty less. For that, we need to show that there does not exist a bijection between N and R, and this is what Cantor's diagonal argument shows.

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Ok I see , I am very much enjoying this conversation .

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I'm glad you find this forum informative!Ok I see , I am very much enjoying this conversation .

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thanks for the recommendation

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