[FOM] Attack/defense on Godel
Harvey Friedman
friedman at math.ohio-state.edu
Sat May 3 04:21:28 EDT 2003
Here I want to give my thoughts on what I take to be an interesting
attack on the Godel theorems, and how it can be repelled.
I think that the attack is much more interesting against Godel's
second incompleteness theorem than against the first.
It is said that Godel's second theorem tells us that, for instance,
PA does not prove that PA is consistent.
We can state this with some care about just what it takes to prove
it, as follows.
*If PA is consistent then PA does not prove that PA is consistent.
Furthermore, this is proved within a very weak fragment of PA.*
The attack goes like this.
#Godel shows that some particular formal sentence A that is clearly
related to the assertion that PA is consistent, is not provable in PA.
However, this particular formal sentence A should not (automatically)
be identified with the actual assertion that PA is consistent. In
particular, it involves rather complicated arithmetic definitions,
and is certainly not any kind of direct translation of the assertion
that PA is consistent.#
How does one defend against this attack? It seems to me to be closely
related to what Floyd must be talking about??
I think it is best to move to adjust the formal systems under
discussion in order to get around such an attack.
*****************************
I claim that Godel's 2nd incompleteness theorem, in proper
generality, shows the following.
A FAITHFUL FORMALIZATION OF FINITE MATHEMATICS HAS BEEN GIVEN THAT IS
GENERALLY ACCEPTED AS BEING INCLUSIVE. THIS FORMALIZATION INCLUDES A
FAITHFUL FORMALIZATION, B, OF THE ASSERTION THAT ZFC IS CONSISTENT.
HOWEVER, B CANNOT BE PROVED IN THIS FAITHFUL FORMALIZATION.
FURTHERMORE, THIS FACT IS ESTABLISHED WITHIN FINITE MATHEMATICS
TOGETHER WITH THE ASSUMPTION THAT THE FAITHFUL FORMALIZATION OF
MATHEMATICS IS CONSISTENT. IN ADDITION, ZFC CAN BE REPLACED BY THE
FAITHFUL FORMALIZATION OF FINITE MATHEMATICS.
The faithful formalization of finite mathematics that is at least
inclusive, that I have in mind, is the system T, whose language has
epsilon,=,N,0,S, and whose axioms areas follows. Here N is the unary
relation for "x is a natural number". Also S is the binary relation
for "x,y are natural numbers and y is the successor of x".
1. If x,y are not natural numbers and have the same elements, then
they are equal.
2. Pairing.
3. Union.
4. Power set.
5. Foundation for arbitrary formulas.
6. Separation for arbitrary formulas.
7. Replacement.
8. Choice.
9. Every nonempty set has an epsilon maximal element (i.e., every set
is finite).
9. Successor axioms for N,0,S.
10. Induction scheme on N using N,0,S, but arbitrary formulas.
This system T really does faithfully mirror how a mathematician would
carefully and fully rigorously discuss the syntax of ZFC, and also of
T. In particular, it avoids any kind of Godel numbering. One
literally talks about finite sequences of letters. Finite sequences
are defined as functions with domain an initial segment of the
natural numbers, etc.
One small difficulty is the concept of "letter". There are two
approaches. One is to simply give an adhoc assignment for the
finitely many letters used for the alphabet of ZFC and T. This is
fully defensible, since mathematicians do this kind of thing all the
time when trying to be very precise about technical material. The
other approach is to consider the letters to be any distinct
entities, and take the consistency statement to assert that it holds
for any choice of entities for letters. Also one proves that if it
holds for some choice of entities for letters, it holds for all
choices of entities for letters.
More information about the FOM
mailing list