

Angst and the Empty Set - dnetesn
http://nautil.us/issue/16/nothingness/angst-and-the-empty-set

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kazinator
Of course, the empty set not "nothing": it is an object.

A set which contains the empty set contains one element, so its cardinality is
1.

If the empty set is nothing, then the set containing the empty set would
contain nothing, and therefore its cardinality would be 0.

Speaking of 0, it's the additive identity element in number theory, and 1 is
the multiplicative identity. Neither of them are nothing. The connection
between 0, the empty set and nothing is that the empty set contains nothing
(no elements), and its cardinality is 0. There is also a connection to
propositional logic: we can universally quantify a negated predicate to say
that nothing exists which satisfies a certain property.

One question is, can we work with an actual nothing which is not "wrapped" in
the empty set.

We could try to postulate a nothing object such that when a set is constructed
which contains that object as an element, it is the empty set (and still has
cardinality zero). Just like every set has the empty set as its subset, every
set has the nothing object as an element.

This nothing could simultaneously serve as an additive identity, and as a
multiplicative identity: a plus nothing is a, a times nothing is a. (This
works in Lisp, by the way: (* 42) is 42, and so is (+ 42). We can pretend
there is a second operand there after 42, and that operand is nothing: it
doesn't extend the length of the argument list by one. Also, nothing plus
nothing should be zero. Nothing times nothing should be 1. This is needed so
that (a * nothing) * nothing is equal to a * (nothing * nothing) so
associativity of multiplication is preserved. (Lisp is with us here again: (+)
-> 0, (*) -> 1.)

It's a matter of going through all the grunt work to see whether the formalism
can be used in a way that is consistent: does its presence lead to theorem-
wrecking contradictions.

The question of whether we can work with an "unboxed" nothing, of course, does
not address the philosophical question. Any formalized nothing that we use is
actually a something: a syntactic and semantic concept denoting the idea of
nothingness. When we think of it, we are not thinking nothing; our minds are
not blank. It is not wrapped in an empty set, but it's still boxed in a
framework. We can have variables that hold nothing, and so on.

