Re: Is {{}} a valid construct?

From: Bob Badour <bbadour_at_pei.sympatico.ca>
Date: Fri, 02 Feb 2007 17:55:49 GMT
Message-ID: <FWKwh.1456$R71.19445@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 - 11:55:49 CST