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   Author  Topic: Chain of subsets  (Read 1225 times)
tim
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Chain of subsets  
« on: Oct 1st, 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 1st, 2002, 9:33am »
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  Hats off to Tim! Smiley
 
   Really beautiful solution.
« Last Edit: Oct 1st, 2002, 9:34am by Pietro K.C. » IP Logged

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