[FOM] Consistency of formal systems
Matt Insall
montez at fidnet.com
Fri Jun 13 10:45:26 EDT 2003
On another list, a friend, Hitoshi Kitada, posted the following argument,
which I post here with his permission. My question to this list is where
did it go wrong? I think I know where there is a mistake, but I have
difficulty explaining why. Can someone help me explain where the problem
is? (After the quote, I will state the line at which I think there is
a first error.
<<Begin Quote>>
Consider Rosser's formula A_q(q), which means
``To any proof of A_q(q), there exists a proof of ~A_q(q) with an equal
or lesser Goedel number." ( ~ denotes negation )
Then one can prove that, if the formal system S(0) of number theory is
consistent, then neither A_q(q) nor ~A_q(q) is provable.
Add one of A_q(q) or ~A_q(q) as a new axiom of S(0), and name it S(1).
Then by Rosser's theorem, S(1) is consistent, if S(0) is consistent.
Let A_q(1)(q(1)) be the new Rosser's formula having the same meaning as
above, but now in S(1). Then arguing similarly we have that neither
A_q(1)(q(1)) nor ~A_q(1)(q(1)) is provable in S(1).
Continue this procedure, and construct a system S(omega) that has
countably infinite new axioms having either form A_q(n)(q(n)) or
~A_q(n)(q(n)) for n > or = 0.
Then S(pmega) is consistent. Further with some reasoning we can show
that the Rosser's formula A_q(omega)(q(omega)) in S(omega) is also
primitive recursively defined, so Goedel number q(omega) is
well-defined.
Continue this procedure transfinite inductively to reach the
cardinality of continuum. We do not lose the primitive recursive
definition of Rosser's formula at each step. But at some step we have to
reach the continuum if we assume something like continuum hypothesis.
Then we must have exhausted all of the formulas of S(0) at some step
with an ordinal beta before we reach the continuum. At that beta, we
have an inconsistent S(beta).
We started with an assumption that S(0) is consistent, and at each step
we had a consistent system S(alpha), so the above-obtained S(beta) must
be consistent in this construction. This is a contradiction.
Then by reductio ad absurdum, one has that the first assumption that S(0)
is consistent is wrong, and he has that the number theory S(0) is
inconsistent.
What was wrong? If the argument is not deceiving us, a possible cause of
the contradiction is continuum hypothesis or an equivalent: for some
ordinal alpha, the cardinality of alpha is equal to the continuum
cardinality.
If this is the cause of the contradiction, we seem to have to reconsider
ZFC.
<<End Quote>>
The place I first notice what I think is an error is
``Then we must have exhausted all of the formulas of S(0) at some step
with an ordinal beta before we reach the continuum. At that beta, we
have an inconsistent S(beta).''
I say this assumes that at each step, we not only add one formula,
but we also deductively close the result. Otherwise, it is conceivable
that after some beta, the tower of S(alphas) stabilizes (i.e. for
each alpha and gamma >= beta, S(alpha)=S(gamma)). But in fact, I am
not convinced that this makes some S(beta) inconsistent. I believe that
at some beta, S(beta) must be complete and consistent, and the least
such beta must be a special type of denumerable limit ordinal.
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