Chpt 4 section 1 explanation needed

Hi

On page 66 of the text book, there is something I would like explained.

For unlabelled X <- X + 1 they have:
<0, <1,1>> I understand where this comes from.
They then say <0, <1,1>> = <0, 5>. I assume they get the 5 from
2¹ + 3¹. This is where stuff stops making sense. Why is it 2 + 3 and not 2 * 3. I am sure the Goedel numbers are . and not +.

Anyhow, lets assume it is a +. They then say <0, 5> = 10. Okay... 1* 0 + 2* 5 = 10. (Is this correct - because if you assume this to be correct then the rest of the page makes no sense).


The rest of the page:

labeled [A] X <- X + 1 gives <1, <1, 1>>. This I understand.
<1, <1, 1>> = <1,5> This I understand too if ¹ + 3¹ is used instead of ¹ . 3¹.

Then the part that really stumps me is: <1, 5> = 21. How do they get that??

Any help would be appreciated!

Thanks
Celene

I got an answer from the lecturer so for anyone else who may get stuck:

On page 59 of the textbook we find <x,y>=2x(2y+1)-1.

This means that <0,5>=20(2(5)+1)-1, which is equal to 1(10+1)-1, which simplifies to 10. Similarly, <1,5>=21(2(5)+1)-1, which simplifies to 2(11)-1 or 21.