GodelsBetaFunction < Courses

Suppose we want to code a sequence of numbers that are all less than 1000. One easy way to do it, is to write numbers in base 1000 notation, and just use the sequence. Thus, if we want to code a sequence of numbers all less than 9, we can do it this way: 21173758 (no initial 0s).

That almost works, but how do we tell which is the th number in the sequence? We have to count off members of the sequence, which itself seems to require using a sequence, which we don't yet have.

The way we'll solve this problem is to use 1221314753677588 instead of just 21173758, putting the numbering into the sequence.

So to code a sequence of length at most of numbers that are at most , we can do it using a number thought of as written in base .

To work base 10, we would be interested in powers of 10, and so we'd need a way to say that a number is a power of 10. Even that isn't so easy. But, it is easy for powers of a prime number : A number is a power of if and only if it and all of its divisors except 1 are divisible by . So, we only work with numbers base for prime .

Given any sequence of numbers at most of length at most we find two numbers: a prime bigger than and the number that codes the seqence as indicated above, in numbers written base . We then need a function $\beta (t,p,i)$ which is definable in the language of arithmetic and whose value is the th member of the sequence coded by base . That is Gödel's beta function, and it is what we will spend the next class writing out in some detail.

p. 177

$\beta$ -Function Lemma. There is a function $\beta:\Bbb{N}^{3}\longrightarrow\Bbb{N}$ with the following properties: %RENDERLIST{"empty"}%

a. For every sequence $(a_{0},\dots ,a_{r})$ over $\mathBbb{N}$ there are $t,p\in\mathBbb{N}$ such that for all $i\le r$
$\beta (t,p,i)=a_i$
b. $\beta$ is definable in the language of arithmetic, that is, there is a formula $\phi_{\beta}(v_{0},v_{1},v_{2},v_{3})$ such that for all $t,p,i,a\in\Bbb{N}$ ,

$\mathcal{N}\models\phi_{\beta}(t,p,i,a){\text{ if and only if }}\beta (t,p,i)=a.$

Proof: Let be a prime number bigger than $a_{0},\dots a_{r},r+1$ . Let be

$1 p^{0}+a_{0} p^{1}+2p^{2}+a_{1}p^{3}+\cdots(i+1)p^{2i}+a_{i}p^{2i+1}+\cdots (r+1)p^{2r}+a_{r}p^{2r+1}.$

$a=a_{i}$ if and only if there are $b_{0},b_{1},b_{2}$ such that %RENDERLIST{"empty"}%

1. $t=b_{0}+b_{1}((i+1)+ap+b_{2}p^{2})$
2.
3. $b_{0}<b_{1}$
4. $b_{1}$ is an even power of , that is, is a square and for all $d\not=1$ such that divides , divides .

$\square$

-- ProfessorShaughanLavine? - 30 Mar 2005