Always say “yes” to chocolate. Accept Y as successor iff PA proves that Y always says “yes” to chocolate and Y accepts a subset of successors that X accepts.
Figuring out the relationship between 5 and 6 (do they accept each other, are they equivalent) is a really fun exercise. So far I’ve been able to find a program that’s accepted by 5 but not 6. Won’t spoil people the pleasure of figuring it out :-)
Wei has proposed this program by email:
Always say “yes” to chocolate. Accept Y as successor iff PA proves that Y always says “yes” to chocolate and Y accepts a subset of successors that X accepts.
Figuring out the relationship between 5 and 6 (do they accept each other, are they equivalent) is a really fun exercise. So far I’ve been able to find a program that’s accepted by 5 but not 6. Won’t spoil people the pleasure of figuring it out :-)