TITLE: Are there first-order statements that second order PA proves that first order PA does not?
QUESTION [5 upvotes]: Are there first-order statements that second order PA proves that first order PA does not? Is this known one way or the other? Could you share an example? (edit: to clarify, by 'second order PA' I don't mean any first order theory)

REPLY [8 votes]: This question more or less reduces to asking for natural arithmetic statements unprovable in PA.  A very interesting but lesser-known example that I learned about recently concerns fusible numbers. The term "fusible" may be unfamiliar but it arises as a natural generalization of a well-known puzzle involving measuring lengths of time by burning fuses (finite pieces of string) of different lengths.  The puzzle is an old one and was not specifically constructed to produce unprovable statements, so the unprovable statements involving fusible numbers are arguably the most "natural" PA-unprovable statements around.