Re: Is {{}} a valid construct?
Date: Fri, 02 Feb 2007 17:55:49 GMT
Message-ID: <FWKwh.1456$R71.19445_at_ursa-nb00s0.nbnet.nb.ca>
Jan Burse wrote:
> Neo wrote:
>
>> Can an empty set contain an empty set? Is it valid according to set >> theory? If so, what does it correlate to in the real world? (when last >> explained to me, I was left holding two empty bags of potatoes but no >> clear understanding :) >>
> If a set contains an element, bit it the empty set or not,
> its not the empty set.
>
> So put your one empty bag of potatoes into your other
> empty bag of potatoes. And wush your second bag is not
> empty anymore. You see, it contains the first empty bag.
>
> Bye
Sigh, I know I waste my breath but here goes:
The empty set is the canonical (and only) set of cardinality zero: {}
The set containing only the empty set is the canonical (but not the only) set of cardinality one: {{}}
The set containing both the empty set and a set containing the empty set is the canonical set of cardinality two: {{},{{}}}
And so on.
One can research this further by searching 'formalism' in mathematics or by searching 'foundations of mathematics'.
Neo needs to pay particularly close attention that the set {{}} is not empty because it contains {}.
Another way of writing {} is ∅ ie. ∅ === {}
Perhaps it would clarify if I rewrote the above sets as: ∅, {∅}, {∅,{∅}}
∅ is the empty set
{∅} is not empty because it contains ∅
{∅,{∅}} is not empty because it contains both ∅ and {∅}
etc. Received on Fri Feb 02 2007 - 18:55:49 CET