To study the Gödel incompleteness theorems we need to choose and study an appropriate logical theory and set of formal rules, and in this an appropriate minimal theory of arithmetic. For the logical theory, we take first-order logic. There are a few choices for the minimal first-order theory or arithmetic, including and also theories in a purely relational language, which are convenient for some reasons, but I prefer to concentrate on a slightly stronger theory because it has a clear algebraic meaning. This page introduces the reader to .
As explained in
The Mathematics of Logic, there is a system -
first-order logic - which is described precisely by proof rules that
could be programmed into a computer to check, which is expressive
enough for everyday mathematics, and in which all everyday
mathematical proofs can be formalised.
We write to say that from additional non-logical axioms the system can derive , i.e. there is a formal proof of in the system with additional non-logical axioms . You might think that to make progress with such a system it would be necessary to produce many many detailed and boring formal proofs. (They are all long!) But it happens that is equivalent to a related notion, , and we can use this instead. means that every structure (an algebraic object like a group or a ring) that makes true must make true.
is called the Soundness Theorem, as it says the system
correct, and the implication
known as the Completeness Theorem as it says that is
strong enough or
complete so as to formalise all informal
mathematical proofs. Both these results are discussed in
Mathematics of Logic and the Completeness Theorem is proved fully
there. A sketch proof of the Soundness Theorem is also give, and
further details of it are available on these web pages. Our approach
here is to use throughout, and appeal to Soundness and
Completeness to see that what we said about applies
equally well to . In particular, we can concentrate on
the semantic meaning of our non-logical axioms, rather than the formal
semantics of how they are to be written down.
theory we are about to introduce is an algebraic theory in what
I consider as the
usual language of arithmetic, .
This language has two constants, and , two binary
function symbols and , and a binary relation
. These symbols are all thought of in the usual way as zero,
one, addition, multiplication and order, and in particular they are
interpreted in in the usual way. The
structure , or more properly
is called the standard model of arithmetic.
(Note that in all this discussion zero will be considered as a natural
number.) Other popular languages omit one or two of these symbols
(for example, is strictly speaking not needed as it is
usually possible to define it from addition) or add the successor
The axioms that follow say what it means for an -structure to be the nonnegative part of a discretely ordered ring. The set of these axioms could more accurately and precisely be given the name to indicate the precise algebraic content and the choice of language we made. This is a bit long-windeed to write and we will usually write or just . The same theory is often called .
Most of these axioms should be familiar as axioms for an ordered ring. The last one makes it the nonnegative part of such a ring, and the penultimate one (together with properties of addition) makes it a discretely ordered ring.
Unless explicitly stated otherwise, all structures in this section of the web pages will be considered to be models of , and all -theories will contain the axioms of . Therefore we can use the standard abbreviations for terms, such as for since by commutativity and associativity the first expression is unambiguous in .
We also use for , for and so on.
The canonical closed terms, or numerals, for natural numbers are , , , , and so on. We denote these , , , , etc., or just when no confusion arises.
Explain how, given a model of , you may construct from it in a natural way a model of all axioms except the last, which is to be replaced by . Thus justify calling the axioms for the nonnegative parts of discretely ordered rings.
The next few results show that the map is an embedding of the standard model onto an initial segment of every model of . Many of the proofs are missing or just hinted at. They are all easy. What is perhaps not so easy is to attempt to prove these results in the right order. Without following the order given here you might find them much harder.
It is provable in that .
Suppose and . Then .
Induction on .
Observe that because of the order axioms. Also if or then as or respectively, it is impossible that in any model of we also have . So there is a strong converse to the above theorem: if and in some model we have then in the naturals too. Similarly if then in the naturals too. Therefore the mapping is a one-to-one map from to our model of for which holds in if and only if holds in .
Suppose and . Then .
Induction on .
Suppose and . Then .
Induction on , using the previous result.
For each it is provable in that .
Induction on .
Because of these results we are justified is identifying each with the value of the canonical term in each model of .
We are going to give some very important definitions, and examine some classes of sentences provable in .
If is a term of we use the abbreviation for . We use for . These quantifiers are said to be bounded.
A formula of is bounded or if all quantifiers occurring in it are bounded. That is, is the least collection of formulas of containing all atomic formulas and closed under the syntactical boolean operations of , , , and under bounded quantification.
We say a sentence is true if it is true in the standard model .
Suppose is a closed term of with value in the standard model. Then .
By induction on the term and how it is made from .
Suppose is a sentence of , i.e. is bounded and has no free variables. Suppose is true. Then .
Exercise. Use an induction on sentences, showing that for every such sentence , either or . Then observe that, for any sentence , by the soundness theorem if then . The theorem follows as if is true we cannot have .
A formula is if it is of the form for some tuple of variables and some formula . We allow the tuple to be empty, so that any formula is by definition also .
Using this, we can extend the previous theorem slightly, though its proof is needed to get the slightly more general result.
Suppose is a sentence of and is true. Then .
If is and true, let such that . Then (using canonical terms to represent these natural numbers) is a true sentence hence provable in . Hence by -introduction .
A formula is if it is of the form for some tuple of variables and some formula . Again, may be empty, so that any formula is by definition also .
Suppose is a sentence of and is false. Then is refutable in , i.e. .
It turns out that there are sentences which are neither provable in nor refutable in . The existence of such sentences will follow from our proof of the Gödel incompleteness theorems.
Note that most of the axioms of are not . Examine the arguments above and find an infinite set of sentences of such that: (a) is a recursive theory; (b) every sentence in is a true sentence; and (c) every true sentence is provable in .