In Godel, Escher, Bach: An Eternal Golden Braid, Douglas Hofstadter outlines his axiomatic “Typographical Number Theory” and sets certain problems for the reader – one is to show that b (a variable) is a power of 2.
My reasoning runs like this: if a number is a power of two then it has no prime factors other than 2 (and 1, if we count that as a prime) – see the earlier entry on the fundamental theorem of arithmetic.
A prime (a) in TNT:
First Update: Now realise there is a simpler way of stating a prime in TNT – namely that there is no two numbers bigger than one the product of which is the number we seek:
So, we have a definition for a prime, a, and we can set a to a prime other than 2.
I think this is a sufficient definition for “b is a power of 2”, as we know that all numbers are products of primes and we have ruled out any prime other 2 being a factor of b. Is that right?
On further thought I don’t think that’s enough, because we need to specify that b is a multiple of 2.
So what about this?
So has that nailed it?
- How I discovered the fundamental theorem of arithmetic by chance (cartesianproduct.wordpress.com)