December 15th, 2016, 11:20 AM  #21 
Senior Member Joined: Sep 2015 From: USA Posts: 1,791 Thanks: 923 
scanning all these they all seem to stem from the same basic unease of infinite sets having zero measure. I really like the argument in the xkcd comment. If you pick a random natural number, the next one you pick is almost surely greater than the one you picked. This right here precludes the idea of a uniform distribution on the naturals. The probability of picking any natural number we can represent given the matter available in the universe or less is 0 as there are infinitely many numbers greater than this. 
December 15th, 2016, 03:18 PM  #22 
Senior Member Joined: Jun 2014 From: USA Posts: 347 Thanks: 26  
February 23rd, 2017, 06:34 PM  #23 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 227 Thanks: 2  Arbitrary Set
Can anyone explain me what is an Arbitrary set ? and what is an Arbitrary union and Intersection ? if possible with definitions Thank you 
February 23rd, 2017, 07:09 PM  #24  
Senior Member Joined: Aug 2012 Posts: 1,784 Thanks: 482  Quote:
In general an arbitrary set is an arbitrary set. Any old set at all. Like an arbitrary restaurant, it's just any restaurant that's open when you happen to be hungry. Any old restaurant, any old set. Arbitrary unions and intersections are just unions and intersections without restriction. For example if I have a collection of sets, in some applications we want to consider only finite intersections, or perhaps countable intersections. Without any restrictions we can consider arbitrary intersections. Intersections of a perhaps uncountable collection of sets. I hope this vague answer is helpful but if you can say more about why you're asking I can give a more specific answer in the appropriate context.  
February 23rd, 2017, 10:49 PM  #25 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 227 Thanks: 2  Arbitrary Set
Thank you Maschke. I have started reading topology. Here I come across Arbitrary sets more often. Do you mean an orbitrary set is a collection of sets of which cardinality cannot be expressed in a natural number ? Can an arbitrary set be infinite and finite ? 
February 24th, 2017, 07:50 AM  #26 
Senior Member Joined: Aug 2012 Posts: 1,784 Thanks: 482  An arbitrary set could be finite OR infinite, yes. It couldn't be infinite AND finite, that would be a logical contradiction. A particular set is not necessarily a collection of sets unless they say so; although technically in set theory all sets are collections of sets. Perhaps if you give a specific context ...

February 24th, 2017, 07:52 AM  #27  
Senior Member Joined: Nov 2015 From: hyderabad Posts: 227 Thanks: 2  Quote:
okay. Thanks for the information ðŸ˜Š  

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