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(1): Here (Why is the empty set bounded?) it is argued that the empty set is bounded.

(2): Here (Proving finite unions and arbitrary intersections of bounded sets are bounded) it is a argued that an arbitrary intersection of bounded sets is bounded.

(3): Here (https://proofwiki.org/wiki/Intersection_of_Empty_Set) and here (Empty intersection and empty union) it is argued that the intersection of the empty set is the universal set.

How can we reconcile (1) and (2) with (3)?

Labbsserts
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  • (3) it’s not intersection of empty set, rather it’s the intersection of empty set of subsets of a universal set that’s defined as the universal set itself. Maybe it’s easier to think of it as empty collection of subsets of the universal set. – Pranay Nov 25 '24 at 13:21
  • The intersection of the empty set is not the empty set, so (1) is irrelevant to the matter. – Trebor Nov 25 '24 at 13:40

1 Answers1

2

  • (1) seems irrelevant to the issue - it is about the empty set as a subset of the universe, not as an empty family of sets.

  • (2) says that the intersection of an arbitrary non-empty family of bounded sets is a bounded set - which is correct.

  • (3) says that the intersection of the empty family of subsets of a fixed universe equals that universe, typically an unbounded set - which is also correct.

There is no contradiction between (2) and (3), because (2) is about the intersection of non-empty families of (bounded) sets and (3) is about the intersection of the empty family of sets. The former is always bounded, the latter is typically not.

Adayah
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