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September 9th, 2017, 02:17 PM   #11
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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
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September 9th, 2017, 09:33 PM   #12
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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 9th, 2017 at 11:21 PM.
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September 10th, 2017, 12:43 AM   #13
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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?
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September 10th, 2017, 05:47 AM   #14
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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 $\varnothing \in T$.
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September 10th, 2017, 08:15 AM   #15
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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.
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September 10th, 2017, 08:34 AM   #16
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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.
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Last edited by cjem; September 10th, 2017 at 08:37 AM.
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September 10th, 2017, 09:28 AM   #17
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Ah clarity, thanks.
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