Hi Kory,
So we met two important theories or machines:
Classical Logic CL and Peano Arithmetic PA.
As collection of theorems, the first is a subset of
the second:
PA



CL
Now I have chosen PA to set the things. Any theory or
machine rich enough to prove elementary theorems about
numbers will do.
What will be important is that anybody (including possible
machines) should be able to verify "mechanically" if a given
proof is a proof, at least in principle (in case a given proof
is 2^64 steps long).
We will interview PA about its possible (consistent) extensions.
It is Godel who showed how to make such an interview,
by showing how to translate the metapredicate of "provability"
into an arithmetical sentence.
Let me be a little more specific. Don't hesitate to tell me you know
all this, but it could help some others, and it could help
in front of futur misunderstandings.
PA has a language. Here it is:
L_PA = {v, &, >, not, t, f, A, E, =, +, *,
SUC, 0, (, ), x,y,z,x1, y1, z1, x2, y2, ...}
We could code those symbol by the odd integer above 1. So
the Godel number of "v" is 3, and this I abbreviate by ng(v)=3.
So ng(&)=5, ng(>)=7, ng(not)=9, ng(t)=11, ..., ng(x)=33, ng(y)=35, etc.
So exactly in the sense that you can define the divisability predicate
Div(x,y) (which is true when x divide y) only with the available symbol
as in
Div(x,y) = Ez(x*z = y) (there exist a number z such that
x multiplied by z gives y)
you can define the metapredicate ISAVARIABLE(x), which is
true when x is a number representing a variable, as Godel number.
Actually, with the code we have chosen:
ISAVARIABLE(x) = ODD(x) & BIGGEROREQUAL(x,3)
= Ey(x=2y+1) & Ez(x = z+33)
Now we want to talk about formula also with PA. I recall
that there are precise formula formation rule like
if A is a formula then (not A) is a formula
if A and B are formula then (A & B) is a formula,
etc.
To code a formula by a number, a traditional way relies
on the well known fundamental theorem of arithmetic
saying that all number have a unique decomposition
in product of prime numbers.
So the formula "Ez(x*z = y)" will be coded by the number
2^(ng(E))*3^(ng(z))*5^(ng(())*7^ng(x)*11^(ng(*))*13^(ng(z))* ...
(I let you continue)
(I have also avoid the quotation mark for not wasting ink,
but ng('(') is more readable than ng((), mmhh..
Now, a proof in, or by, PA ["in" if you think as PA
being a theory; "by" if you think as PA as a machine
(a theorem prover)] is just a sequence of formula
such that they are either axioms or has been obtained
from the axioms by a finite number of application of
the inference rule.
So, although it could be hard to *find* a proof (as
unpedagogical exercice try to prove the formula (A > A)
in the systeme send yesterday), it is easy to verify
a proof once the proof is given.
The point is that we can translate a proof into a number
by relying a second time on the fundamental theorem
of arithmetic. A proof is just a finite sequence of
formula F1 F2 F3 F4 F5 ... FN (actually it will be proof
of FN if F1, F2, ... are axioms or comes from axioms
and preeceeding formula by application of the inference
rule). Its (Godel) number will be
2^ng(F1)*3^ng(F2)*5^ng(F3) ....
Now, like we have define in PA language ISAVARIABLE(x)
we can translate metapredicates like ISANAXIOME(x),
HASBEENDERIVEDFROMMODUSPONENS(x,y,z), meaning
z is derivable from x and y by modus ponens.
Etc. Etc. So that we can defined in PA arithmetical language
ISAPROOFOF(x,y) saying that y is a translation (a godel
number of) of a proof of formula (with Godel number) x.
This was the B(x,y) of yesterday.
And then the Godel formula BEW(x), that is PROVABLE(x),
is just EyB(x,y) i.e. it exist a number y such that y is a proof
of x, to make it short.
Now, a machine will be said to be consistent,if the machine
does not prove f. "f" is a symbol with the intended meaning of
an absurdity, or a contradiction. in arithmetic f could be defined
by the formula (1 = 0), well actually (SUC(0) = 0).
So the consistency of the machine, or of the theory, or
of PA (here) can be defined by the formula (not BEW(f)).
Godel second incompleteness theorem is that
if a machine is consistent then the machine cannot prove
she is consistent. Now I will explain in which sense we can
say that to be consistent is the same has having a consistent
extension, so you see that Godel second incompleteness
explain why, as I said to John, when we ask the machine
if she *do* have at least one consistent extension, she
remains silent.
Well that is a little bit discouraging isn' it? How could we
hope the machine to be able to give us the geometry
of its consistent extensions when she remain silent on
the very question of having consistent extensions at all?
But if you are patient enough with PA, she will justify
her silence by proving that: if she is consistent then she cannot
prove her consistency:
PA proves (not BEX(f)) >(not BEW (not BEX(f))
I will prove Godel's theorem on the way, but I think
I will first give you a map of the territory including
two paths which make possible to sum up
the "mathematical confirmation" in one precise
mathematical formula. Well I don't know.
Any question or comment?
Bruno
http://iridia.ulb.ac.be/~marchal/
Received on Thu Apr 29 2004  13:05:32 PDT