It’s not just about 8 examples—with any number of examples it would be perfectly valid to insert something like 6 = 1. And so there’s an additional axiom in Peano arithmetic that has to explicitly rule it out (if you’re talking about numbers that way). Not super-shocking.
My interpretation of the original quote was to take “see that 5 + 4 is not 6” as “prove that you cannot prove that 5 + 4 = 6″, in other words, “prove that Peano’s arithmetic is consistent”. Maybe I was too influenced by this.
It’s not just about 8 examples—with any number of examples it would be perfectly valid to insert something like 6 = 1. And so there’s an additional axiom in Peano arithmetic that has to explicitly rule it out (if you’re talking about numbers that way). Not super-shocking.
My interpretation of the original quote was to take “see that 5 + 4 is not 6” as “prove that you cannot prove that 5 + 4 = 6″, in other words, “prove that Peano’s arithmetic is consistent”. Maybe I was too influenced by this.
I think that’s a way better interpretation :D