- We have witnessed the struggle with the limits of reason
in the case of infinity, and seen both the emergence of a theory
of infinite sets and also the Cantorian antinomy with the
first version of such a theory.
- A natural inclination is to take a step back, and attempt
to get some certainty by expunging conflicting intuitions and
examining the formal nature of reason.
- This goal inspires both the project of formalizing
(axiomatizing) as much as possible of mathemetical theory, and
also attempt to prove that various such axiomatizations of
math -- perhaps ideally all of math -- are consistent.
That is, we would like to both create a powerful and clearly
defined theory (of, say, infinite sets) and also prove
that such a theory is consistent.
- Russell and Whitehead's Principia Mathematica was
the most ambitious, if not the most rigorous or succinct,
attempt to do axiomatize arithmetic. It is an axiomatization
of arithmetic using type theory.
Background to Russell's Approach
- Russell recognizes that there are several contradictions which
pose a specific problem for attempts to make reasoning, especially
pure logical reasoning, clear and unproblematic. Here are just a
- The Cretan: "I am lying."
- The Liar: "This sentence is false."
- Russell's paradox: the set of all sets that are not
members of themselves.
- Berry paradox: "the least integer not nameable in
fewer than nineteen syllables"
- Burali-Forti paradox (this is quite similar to
what we have called the Cantor Antinomy).
- What do these have in common? A kind of self-reference
or reflexiveness! (For the Berry paradox, this is not so
obvious, but it is because it refers to the number of
syllables in our language, and so it refers both to numbers
and to the language we are using to talk about these numbers).
- It is not enough just to rule out immediate
self-reference: we can create looping versions of the
paradoxes with as many steps as you like (Here's a two step
Cretan: "The following sentence is false. The previous
sentence is true." Try to make a three step!)
- Russell's solution is a theory of types.
Types: Each statement has a type. (Roughly:) a
statement just about numbers, our basic objects, is of the
first type. A statement about collections of numbers is
of type 2. A statement about collections of collections
of numbers is of type 3, and so on.
Type restriction: in the Principia Mathematica
system, we only allow statements of type n to refer to things
of type less than n.
- Solution: we don't have to go into the details to see
intuitively how Russell's system works to avoid the paradoxes.
Consider the first three listed above.
1 and 2. The Cretan: "I am lying." If our system included
something which allowed us to say, for example, statement
P is true, we could use that to say something which
meant, "This statement is not true," and create something
like the Cretan paradox. But that would be impossible in
type theory: such a statement would have to refer to its
own type level (in order to say "this statment..."),
which is forbidden. Thus any form of this paradox will
not be well-formed.
3. Russell's paradox: the set of all sets that are not
members of themselves. Since in type theory a set can
only collect things of lower types, no set can contain
itself, and any claim about such an object is not well
4. Berry paradox: "the least integer not nameable in
fewer than nineteen syllables." In a type theory formal
system, any property analogous to number of syllables
will refer to lower types; it cannot be used to describe
its own statement.
5. Burali-Forti paradox (this is quite similar to
what we have called the Cantor Antinomy). (Cantor's
antinomy is ruled out because we no longer have a
set of everything -- such a set would also refer to
its own type and higher types, since it would include
itself and higher types.)
Russell's system, perhaps with a little house cleaning, may give us
everything we need in order to do arithmetic. Several questions
- Is this system complete? That is, is every
true well-formed formula of this system also provable
in this system?
- (Hilbert:) Is this system consistent? That
is, are we sure that we cannot prove some sentence P and
also ~P? (Note that in this system, if it is inconsistent
then it is trivially complete.)
- (Hilbert:) Is this system decidable? Can we
have a certain mechanical test of any well-formed formula
to decide whether it is a theorem or not?