True arithmetic is the name given to the "real world"(tm) model of Peano arithmetic (PA). Gödel's theorem implies that there exist many sentences in the language of arithmetic which are unprovable in PA. For any such unprovable sentence S, either S is true of the "real world" or it is false and ~S is true of the "real world". Since S is unprovable in PA, you can have a consistent model of PA in which S is true, and you can also have a consistent model of PA in which ~S is true. True arithmetic is a model in which S is true iff S "really" is true.

As an example, consider the Gödel sentence G which says (essentially) "this statement cannot be proven in PA". Note that G cannot be proved in PA. Hence we know that G really is true in true arithmetic: it says that it cannot be proven, and it cannot be proven. (But in a nonstandard model of PA ~G may be provable; explanations will sometime be given, elsewhere (but not yet)).

Which raises the important question: what do we mean by "real world" here?

If you're an idealistic Platonist (like the oversigned), you take "real world" to be the real world: Natural Numbers (I use the Capitalised Term to refer to the Real Natural Numbers, as opposed to the natural numbers about which PA talks) really do exist in the world. Any sentence S about natural numbers is either true or false of the Natural Numbers (of course, we usually don't know which, and even if we think we do we might not know how to prove it). True arithmetic describes these Natural Numbers, so if S is true of the Natural Numbers then S is true in true arithmetic, and if S is false of the Natural Numbers then ~S is true in true arithmetic. Since the real world is consistent, only one of S,~S will be true of true arithmetic.

But what if you're more literal-minded, and don't believe in such a direct connection between mathematics (or at least arithmetic) and the real world? All is not lost! There exists a model of PA inside every model of ZFC, say, or whatever your favourite axiomatic system is for doing mathematics. So take whatever model it is in which you do mathematics (since you're doing mathematics, you believe ZFC is consistent; since you believe it's consistent, Gödel's completeness theorem says it has a model, so you have at least one to choose from). In that model, take true arithmetic to be just the model of the natural numbers.

Logically sound, but often somewhat dodgy for actual proclamations of what is true.

Log in or register to write something here or to contact authors.