no set of axioms suffices to specify the standard model of arithmetic (i.e. to distinguish it from other models).
Then what do you mean when you say "integers"^H^H "natural numbers", if no set of premises suffices to talk about it as opposed to something else?
Anyway, no countable set of first-order axioms works. But a finite set of second-order axioms work. So to talk about the natural numbers, it suffices merely to think that when you say "Any predicate that is true of zero, and is true of the successor of every number it is true of, is true of all natural numbers" you made sense when you said "any predicate".
It is this sort of minor-seeming yet important technical inaccuracy that separates "The Big Questions" from "Good and Real", I'm afraid.
"Any predicate that is true of zero, and is true of the successor of every number it is true of, is true of all integers"
Natural numbers, rather. (Minor typo.)
A monthly thread for posting rationality-related quotes you've seen recently (or had stored in your quotesfile for ages).
ETA: It would seem that rationality quotes are no longer desired. After several days this thread stands voted into the negatives. Wolud whoever chose to to downvote this below 0 would care to express their disapproval of the regular quotes tradition more explicitly? Or perhaps they may like to browse around for some alternative posts that they could downvote instead of this one? Or, since we're in the business of quotation, they could "come on if they think they're hard enough!"