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 Paradoxes
The
second in Francis Moorcroft's series looking at some the classic
philosophical paradoxes.
No.
2 Russell's Paradox
Francis Moorcroft
The
British Library sends out instructions that every library in the
country has to make a catalogue of all its books. Each librarian
makes their catalogue and are then faced with a choice: the catalogue
is, after all, a book in their library; should the title of the
catalogue be included in the catalogue itself or not? Some librarians
decide to include it, others not to.
In
the course of time the catalogues are sent to the British Library
and the chief librarian there has the job of making a catalogue
of these catalogues. But they find that they have two different
sorts of catalogues to deal with: those that mention themselves
inside the catalogue and those that don’t. That is, they have
catalogues that contain their own titles and catalogues that don't.
So the chief librarian decides to make two different catalogues
corresponding to these two different kinds. With the catalogue of
all those catalogues that include themselves the librarian has the
choice to include the title of the catalogue in itself or not. No
problem there. But with the catalogue of catalogues that don't
include themselves the librarian is faced with a dilemma: should
they include the title of the catalogue in the catalogue or not?
if they do then it is not a catalogue that does not contain
its own title and so it shouldn't be included; if they don't put
it in then it is a catalogue that doesn't contains its own title
and so should be included. Either way, it should contain itself
if it doesn't and shouldn't contain itself if it does!
This
paradox is a version of Russell's Paradox. It came about
from Bertrand Russell thinking about the notion of a set, or class,
or collection of things, and whether a set can be a member of itself
or not. For example, think about the totality of cats in the world:
this is the set of cats. Is this set a member of itself or not.
Clearly not – it’s a set, an abstract object, not a cat.
But now think about all of the things in the world that are not
cats - dogs, chairs, books, violin sonatas, . . . and sets. This
set is a member of itself. Now it is far more usual for a set not
to be a member of itself than for it to be a member of itself. Let's
call sets that don't belong to themselves normal. Now Russell's
problem is: is the set of all normal sets a member of itself or
not? If it is then it isn't. But if it isn't then it is...
Russell's
own solution to the problem is his Theory of Types. This can be
explained as follows. Imagine that five people get together to form
a five-a-side football team. This team then joins a local league,
and the league in turn is part of a regional association. Clearly,
an individual can only be member of a team and cannot be a member
of a league or an association: it is the wrong type of thing,
as only teams can be members of leagues and leagues members of associations.
Similarly Russell thought that individuals, type C, were members
of sets or classes of type 1. These sets could only be members of
sets of type 2, and sets of type 2 could only be members of the
higher type 3, and so on. Specifically, a set could not belong to
another set of the same type, as it has to belong to a set of the
next highest type, and so a set can never be a member of itself.
While
the Theory of Types does work as a solution to the Paradox many
logicians feel that it has an ad hoc air about it, and isn’t
as intuitively obvious as is required for the foundations of mathematics:
after all, Russell's solution may seem plausible for football teams
but mathematics does consider sets that are formed from individuals,
sets of individuals, sets of sets of individuals and so on, and
counting out such sets seems to be a high price to pay for avoiding
the Paradox.
Readers
may wonder whether Russell's Paradox is a problem only in the foundations
of mathematics or if it actually occurs in real life. I recently
saw a policy concerning disability which contained a number of clauses
naming specific disabilities which were included under the policy.
The last clause, however, contained the statement 'Other disabilities
not on this is list'. Now consider a specific disability not named
in the list: if it was not on this list then this clause includes
it in the list so it is on the list; if it is on the list
then the clause says it is not on the list. Without care,
paradoxes can occur where we least expect them.
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