So here, at last, is the answer to the second of the five questions I posed back on 3 July.
Question 2: If anything is possible, then is it possible that nothing is possible?
Logically yes, of course. If anything is possible then naturally it is possible for nothing to be possible as we have to assume that the null set is a valid entity — just like zero is valid in our number system.
But then we have to ask whether nothing is possible in reality. And the answer would have to be that no it is not possible, because there are things all around us which are not nothing, so nothing cannot be possible.
Reductio ad absurdum then says that in our universe, as we understand it, anything cannot be possible, because we have just shown that nothing is not possible; whereas if anything is possible, nothing must also be possible.
Confused? Welcome to the world of the logician. Now go and read Alice in Wonderland for a gentler introduction.
More when you brain has had a chance to recover!
This post reminded me of an essay on the subject by Martin Gardner (of ‘Annotated Alice’ fame). It took me a while to track down, but it’s in Mathematical Magic Show (ISBN-10: 0140165568), available online from £0.01.
In fact ‘Nothing’ occupies the whole of chapters 1 and 2, such was the flood of correspondence it generated in Scientific American in 1975.
PS. Claim: The null set exists. Proof: Suppose for contradiction that it doesn’t exist. Collect all supposed null sets (e.g. the set whose sole member is the King of France). There aren’t any null sets by hypothesis. So there are no members in your collection. Hey presto, you’ve constructed a null set. Ta-da.