Adding dense sets?

  • Thread starter cragar
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  • #26
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How is infinity defined?
 
  • #27
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A set is infinite if it isn't finite. And a set X is finite, if it is empty or if there exists a one-on-one correspondence [itex]X\rightarrow \{1,...,n\}[/itex] for a natural number n.
 
  • #28
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ok and why can we say that there are more reals than naturals . I mean they are both infinite. I have seen cantors diagonal argument.
 
  • #29
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ok and why can we say that there are more reals than naturals . I mean they are both infinite. I have seen cantors diagonal argument.
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.
 
  • #30
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Ok I see , I am very much enjoying this conversation .
 
  • #31
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Ok I see , I am very much enjoying this conversation .
I'm glad you find this forum informative! :biggrin:
 
  • #32
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You may want to read "The pea and the sun" by Wapner. It has some very informative things on infinity and it's paradoxes...
 
  • #33
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thanks for the recommendation
 

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