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 September 9th, 2017, 03:17 PM #11 Senior Member   Joined: Jun 2015 From: England Posts: 697 Thanks: 199 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, 10:33 PM   #12
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Quote:
 Originally Posted by studiot 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
This is the standard usage in math texts. They mean a set of sets but for clarity or to avoid redundancy they say family or collection. This convention is pretty much universal.

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 10th, 2017 at 12:21 AM.

 September 10th, 2017, 01:43 AM #13 Senior Member   Joined: Jun 2015 From: England Posts: 697 Thanks: 199 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, 06:47 AM   #14
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Quote:
 Originally Posted by studiot If T is a set then why is is necessary to specify explicitly that T contains the empty set?
In this context, the word "contains" means "contains as an element" rather than "contains as a subset". That is, it is being specified that $\varnothing \in T$.

September 10th, 2017, 09:15 AM   #15
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 Originally Posted by cjem In this context, the word "contains" means "contains as an element" rather than "contains as a subset". That is, it is being specified that $\varnothing \in T$.
Sorry I was being rather vague since 'contains' is another one of those non defined words.

I meant that the empty set is a subset of every set, not that it is a member of every set.

September 10th, 2017, 09:34 AM   #16
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 Originally Posted by studiot Sorry I was being rather vague since 'contains' is another one of those non defined words. I meant that the empty set is a subset of every set, not that it is a member of every set.
When the definition says "T contains the empty set", it means that the empty set is a member of T.

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 09:37 AM.

 September 10th, 2017, 10:28 AM #17 Senior Member   Joined: Jun 2015 From: England Posts: 697 Thanks: 199 Ah clarity, thanks.
 October 17th, 2017, 05:08 PM #18 Math Team   Joined: Jan 2015 From: Alabama Posts: 2,826 Thanks: 753 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/russell-paradox/.
 October 17th, 2017, 07:09 PM #19 Math Team     Joined: May 2013 From: The Astral plane Posts: 1,659 Thanks: 652 Math Focus: Wibbly wobbly timey-wimey stuff. It seems that I have missed the other thread. @Zylo: As a stand alone thread this is a bit non-informative. 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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