In the last two web pages we learnt that is able to prove all true sentences and that formulas represent all r.e. relations. It is now time to put these facts together.
A function is -represented in if there is a formula such that for all and for , is true (and hence provable in ) and also
A set is -represented in if there is a formula such that for all
If you think about it, it should be clear that only recursive sets can be represented in . This is because one could search for proofs of or to determine whether . Similarly, a total function , if represented, must be recursive. In fact every recursive set and recursive function can be so represented.
Every total recursive function is -represented in .
Let be a total recursive function. We write for a tuple of variables and for a tuple of natural numbers. Similarly we write a bar to denote a tuple (pair, triple, quadruple, etc.) of unknown or undefined length.
We saw on an earlier web page that there is a
formula such that for all
holds iff is true.
For the tuple , denotes the term where is the length of the tuple. Consider the formula defined to be,
saying that not only does hold but the choice of is least possible. Note that the quantifier here means that each variable quantified over is bounded by the term so it is a bounded quantifier.
First note that our new defines in as before. For if there are such that and we may choose to minimise hence .
To show that consider some and . Then for the chosen in the last paragraph we have since this is and true, and hence . Therefore and as with in it must be true: . (For if not, and this is also a true statement hence true in .) Therefore as this is the only value of the function , as required.
Every recursive set is -represented in .
If is recursive then we know its characteristic function is represented in the sense of the previous result. This means that there is a formula such that if then and if then . Also, since , if then and also hence . Therefore is represented as a set by the formula .
The key lemma here, called the Diagonalisation lemma or the Diagonal lemma is due to Gödel. It is rather powerful and used for much more than just to prove the incompleteness theorems. In some sense it generalises Cantor's diagonal method that was used to prove that neither the set of reals nor the power set of the naturals is countable. First we review the idea of Gödel-numbering.
Gödel-numbering is a system of numbering expressions of a first order language.
Since a first order language can be considered as a countable set of symbols
grammatical production rules) the set of strings (of finite
length) formed from these symbols is also countable. It follows that there is an injection
from strings to natural numbers,
called Gödel-numbering. With the -function we can be more precise:
if we have a mapping of symbols
we can define
where is the length of
and is the least code of the sequence
in the sense of
the -function: .
The important thing is that this operation is computable in a very natural sense:
every syntactical operation of terms and formulas of the language required
to define the notion of
proof corresonds to a recursive operation
on Gödel-numbers. (This is a lot of detailed checking and somewhat outside the
scope of these notes.)
For any formula with a single free variable there is a sentence of such that
Moreover, if is then we can find such which is also .
Define the recursive function to be if , the Gödel-number of a formula of one single free variable , and to be otherwise. Thus is the Gödel-number of a sentence equivalent to where . Let be a formula representing in . Let be , which you can think of as . Let , so you may think of as , and where is .
Tarski's theorem on Truth
Let . Then for no formula of is it the case that
for all sentences of .
If such exists in some , find such that and obtain a contradiction.
Why does the preceding argument not show that there is no definition of truth? In fact there is some formula of such that for all sufficiently strong models (satisfying a few additional axioms as well as ) it the case that
for all sentences of . (Follow the above class and point out where the formula class is wrong.) What class of formulas can you convincingly prove not to define truth?