1. Axiomatic Number Theory and Gödel’s Incompleteness Theorems
Eric Ottman
Syracuse University
April 8, 2017

2. A Tiny Bit of History
In 1874, Georg Cantor published his first article on set theory, including, among other things, his famous “diagonal argument” proving uncountability of the real numbers
In 1901, Bertrand Russell discovered a paradox in Cantor’s version of set theory, now known as Russell’s Paradox, involving the “set of all sets that are not members of themselves” (call this “set” S – S must not be contained in S, but then S must be contained in S, and so on…)
In light of this, Russell and many others worked to develop paradox- free “axiomatic” versions of set theory (and number theory, and geometry, etc.)

3. Axiomatic Theories
A typical axiomatic theory consists of the following information:
A collection of symbols, which I will denote in boldface
For example: 0, =, Ɐ, Ǝ, ᴧ, ᴠ, ¬, a, ‘
A set of syntax rules dictating which strings of characters form valid statements (nothing here about what’s true or false)
For example: If A and B are valid statements, so are
(A ᴧ B), (A ᴠ B), and ¬(A)

4. Axiomatic Theories
A list of axioms – statements that are true and do not require proof. For example: Ɐa:(0+a=a) and Ɐa:(0·a=0)
A list of “rules of inference” – specific rules by which one may manipulate true statements to obtain new true statements.
For example: The string “¬¬” may be deleted from any theorem or added to the beginning of any theorem.
A “proof” is a list of statements, each of which is either an axiom or the result of applying a rule of inference to one or more previous lines.
A “theorem” is a non-axiom statement that is the last line of a proof.

5. Gödel’s Incompleteness Theorems
In 1931, Kurt Gödel proved that any sufficiently powerful (I’ll make this notion a little more precise later) axiomatic system of number theory must be either incomplete (i.e., able to express a true statement that is not a theorem) or inconsistent (i.e., able to express a false statement that is a theorem). This is the first incompleteness theorem.
The second incompleteness theorem states that an axiomatic system cannot “prove its own consistency.”
The proof of the first theorem is the focus of the rest of the talk.

6. The Key Idea: Gödel Numbering
For the rest of the talk, fix any sufficiently strong (again, explained later) axiomatic number theory.
To each symbol of the theory, assign a positive integer – for example, we could assign 3 to =, 4 to a, 5 to ‘, and so on. Also assign a positive integer to the “line break” character.
Once this is done, use this numbering to establish a correspondence between statements of the theory and positive integers – Gödel used prime factorizations: with the numbering above, the statement a=a becomes 24· 33· 54 = 270,000.

7. Gödel Numbering
Note that since prime factorization is unique, every positive integer corresponds to at most one statement (or list of statements) of our theory.
We can also assign Gödel numbers to entire proofs in this way, since we have assigned a number to line breaks.
This gives us a very clever way to allow the system to “talk about itself,” which is essential if we are to construct a statement we know to be a non-theorem – as we all know, it is very difficult in general to determine whether or not a given statement is a theorem.
Denote the Gödel number of a statement A by [A].

8. Gödel Numbering
The main point here is that in a sense, as long as our theory is sufficiently powerful, it can now express and detect its own rules of inference!
These kinds of statements tend to be incredibly long and complex, but we don’t necessarily need to be able to write them down; we just need to know they exist.

9. Two Key Statements
Now we can get our theory to talk about provability of its own statements.
Definition 1: Let Proof(a, a’) denote the statement in our theory interpreted as follows:
a is the Gödel number of a proof whose last line has Gödel number a’.
If our theory is reasonably powerful, this should be possible – working “outside the system,” one can write an algorithm that tests this property that is guaranteed to terminate in finitely many steps (involving checking that a represents a valid proof and that a’ is indeed its last line). If our theory can’t express this algorithm, it’s certainly not good enough to express all of number theory!

10. Two Key Statements
Now we can also express the statement “a’ is the Gödel number of a theorem:”
Ǝa:(Proof(a, a’))
Definition 2: If a is the Gödel number of some statement containing one free variable, define Subst(a, a’, a’’) to be the statement in our theory interpreted as:
a’’ is the Gödel number of the statement obtained by substituting the value a’ into the statement whose Gödel number is a.
Again, this can be checked “outside the system” by an algorithm that terminates after finitely many steps, so it should be expressible in our theory.

11. The Punchline
Let U be the following statement (or the equivalent in whatever theory we’re working with):
¬Ǝa:Ǝa’:(Proof(a, a’) ᴧ Subst(a’’, a’’, a’))
Note that the first two arguments of Subst are the same; we are able to substitute a statement’s own Gödel number into itself – this kind of self- reference is also the basis of Russell’s Paradox. Also note that this statement has one free variable, namely a’’.
Now let G be the following related statement:
¬Ǝa:Ǝa’:(Proof(a, a’) ᴧ Subst([U], [U], a’))
In other words, G is the result of substituting [U] for a’’ in U. This turns out to be the statement that proves the theorem!

12. Interpreting G U: ¬Ǝa:Ǝa’:(Proof(a, a’) ᴧ Subst(a’’, a’’, a’))
G: ¬Ǝa:Ǝa’:(Proof(a, a’) ᴧ Subst([U], [U], a’))
G asserts that there are no a and a’ satisfying certain properties.
The second property is that a’ is the result of substituting [U] for a’’ in U. Clearly this is possible (really it just assigns a particular value to a’), so the interesting part is the first property.
The first property essentially says that a’ is the Gödel number of a theorem. But in reality G is precisely the statement with Gödel number a’! (It’s obtained by replacing a’’ by [U] in U).

13. Interpreting G So G can really be interpreted as: G is not a theorem.
So if G is indeed not a theorem, then it is a true statement of number theory that is not provable in our theory, so the theory is incomplete.
If G is a theorem, then it represents a false statement, so the theory is inconsistent.