The Gödel incompleteness theorems do not just apply to
arithmetic, but they apply to any formal theory that interprets
arithmetic. The idea of
interpretation is a very difficult one to pin
down, perhaps because there is not one single idea here. In a more
advanced treatment, interpretations may be best looked at in the
language of category theory (when they are arrows in a category
of theories) but this also begs questions as well as solving some.
This web page will try to make a few short and simple definitions and observations on interpretations. Quite a lot of the material here is simplified or over-simplified and many proofs will be omitted.
One formal theory interprets another theory if there is some faithful way of translating the expressions and statements of in such a way that all deductions in can be performed on their translations in . This is a very general idea an one that is difficult to pin down and define precisely. There are two very different ideas: one is to define interpretations in terms of the syntax of the theories and the other is to define them in terms of their semantics or models.
Whatever precise definition is used we will use the notation to mean that is an interpretation of the theory in the theory and for the translation of under . The has low precidence, so means .
At the most general level of all, we might imagine and are formal systems whose provable utterances are strings of symbols , from alphabets and . It will be convenient to assume that these alphabets are finite, as they can be by encoding each symbol as some string of s and s such as fo a certain number of s. Then there is a notion of computability on strings from these alphabets (for example by encoding strings as numbers) and notions of provability and . For a syntactic idea of interpretation we could define an interpretation to be a recursive (i.e. computable) function taking strings of symbols from to strings of symbols from . We'll require a couple of things for this interpretation. First, the translations of all string provable in must be provable in , that is
Also the theory will
contain some version of propositional logic (in all cases of interest
will in fact be a first order theory) and hence has a
implies. We require also that,
With this given, the Gödel incompleteness theorems apply to the
theory that interprets or other such theories.
We still need to be consistent and recursive, but the
best way to describe a theory being
recursive in general is to say
that its set of theorems form a recursively enumerable (r.e.) set.
Here is the first theorem.
Suppose is a recursive intepretation as above. Suppose that the set of theorems of is r.e. Suppose also that is -consistent in the sense that it is not the case that for any , for all and . Then there is some such that and .
Gödel-number the expressions of as usual. As the set of theorems of is r.e. there is a formula such that for each , we have that holds if and only if . As the interpretation is recursive there is a formula such that for all of the language of , and for all other . Now let be such that
Apply the Gödel argument to show that and .
The second theorem is slightly more complicated. We suppose that there is a provability predicate for in which is, there is an expression of the language of such that the following hold for each in the language of .
The statement of expressing the consistency of , or , is the expression of , , and this can of course be interpreted in .
Suppose that the set of theorems of is r.e., that is a recursive intepretation of a first order theory extending , and that is a provability predicate as above. Suppose is consistent in the sense that Then .
Similar to that given before in the page on Gödel's second incompleteness theorem. The main steps are sketched here.
First let be the statement of such that
Then it can be shown that implies .
Next let be the statement of stating . Then applying the other rules for provability predicate and provability in and we find that , hence and . It follows that if then and hence contradicting the previous part.
In most casess we will be interpreting arithmetic, such as or some other theory in the same lanuguage in another first order theory . The most natural way to do this is to specify (in that theory ) the domain of the model of arithmetic and the definitions of addition and multiplication. These will be sets definable in the language of . Usually the domain is a definable set of -tuples for some modulo a definable equivalence relation (which needs to be shown to be, provably in , an equivalence relation on ). Then the definition of addition and multiplication have to be shown (in ) to be well-defined of course. This is the usual kind of interpretation of one theory in another. The reader will have to check that such definitions do indeed lead to a translation of formulas of one language into the other. (This isn't quite a triviality. It is not completely easy to see what to do about terms in the language of arithmetic, which you are trying to interpret in some purely relational language.)
In many cases, the equivalence relation can be omitted and all we need is the domain which might if we are lucky to be defined by a single formula is a single free variable. This will be the case in the main examples we will need: where some theory of sets interprets arithmetic, such as or .