September 9th, 2017, 02:17 PM  #11 
Senior Member Joined: Jun 2015 From: England Posts: 853 Thanks: 258 
So why is everybody so determined to use undefined terms instead of the word set? A family of subsets A collection of subsets but never A set of subsets 
September 9th, 2017, 09:33 PM  #12  
Senior Member Joined: Aug 2012 Posts: 2,007 Thanks: 574  Quote:
Note that there is no possible ambiguity with proper classes. By the powerset axiom, the family or collection of subsets of a given set is itself a set. So instead of saying, "Let X be a set of subsets of Y" they say "Let X be a collection ..." I don't think anyone would say, "Let X be a collection of sets," because that IS ambiguous. Is X a set or a proper class? We have no way of knowing. But when the collection in question is a subset of the powerset of some set, we know it's a class and there is no ambiguity. I think I didn't really answer your question. I only agreed that everyone does this. I don't know how it got started. I think "set of sets" sounds a little awkward and "family" doesn't. Last edited by skipjack; September 9th, 2017 at 11:21 PM.  
September 10th, 2017, 12:43 AM  #13 
Senior Member Joined: Jun 2015 From: England Posts: 853 Thanks: 258 
Thank you Maschke. If T is a set then why is is necessary to specify explicitly that T contains the empty set? Surely a more fundamental axiom or (result?) is that every set contains the empty set, therefore T contains it? 
September 10th, 2017, 05:47 AM  #14 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 226 Thanks: 75 Math Focus: Algebraic Number Theory, Arithmetic Geometry  
September 10th, 2017, 08:15 AM  #15  
Senior Member Joined: Jun 2015 From: England Posts: 853 Thanks: 258  Quote:
I meant that the empty set is a subset of every set, not that it is a member of every set.  
September 10th, 2017, 08:34 AM  #16  
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 226 Thanks: 75 Math Focus: Algebraic Number Theory, Arithmetic Geometry  Quote:
As you've explained, it's unnecessary to specify that the empty set is a subset of T. Fortunately, the definition does not do this. Last edited by cjem; September 10th, 2017 at 08:37 AM.  
September 10th, 2017, 09:28 AM  #17 
Senior Member Joined: Jun 2015 From: England Posts: 853 Thanks: 258 
Ah clarity, thanks.

October 17th, 2017, 04:08 PM  #18 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 
One prefers not to use the phrase "set of sets" because of the difficulty proving that there is such a thing as a result of "Russel's paradox", https://plato.stanford.edu/entries/russellparadox/.

October 17th, 2017, 06:09 PM  #19 
Math Team Joined: May 2013 From: The Astral plane Posts: 1,888 Thanks: 765 Math Focus: Wibbly wobbly timeywimey stuff. 
It seems that I have missed the other thread. @Zylo: As a stand alone thread this is a bit noninformative. Okay, so you've got your definition of a topology (however lax you might be about using the specific terms, but no matter.) So far so good. But what is your point? Dan 

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