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Topic: Chain of subsets (Read 1225 times) 

tim
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Posts: 81


Chain of subsets
« on: Oct 1^{st}, 2002, 1:13am » 
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At first, I thought that of course it must be countable: after all, each link in the chain must add at least one element, and there are only a countable number to add. Then I started to have second thoughts. For example, if the chain counts up through all the even numbers, then all remaining multiples of three, etc. That would break the simple mapping I had in mind. Solution: Finally, I found a counterexample. It was ridiculously contorted at first, being a mapping from [0,1) into sequences of binary digits, which in turn can be broken down into a recursive function of unions of powers of primes. Then I hit a really simple one: Every real number has a Dedekind cut, the set of all rationals less than it. These sets of rationals form a chain. Rationals can be bijectively mapped with the positive integers. Therefore an uncountable chain of sets of positive integers exists.


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Pietro K.C.
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Re: Chain of subsets
« Reply #1 on: Oct 1^{st}, 2002, 9:33am » 
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Hats off to Tim! Really beautiful solution.

« Last Edit: Oct 1^{st}, 2002, 9:34am by Pietro K.C. » 
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