many people are quite sure that (e.g.) a Goedel sentence for their favourite formalization of arithmetic is either true or false (and by the latter they mean not-true).
Those people seem a bit silly, then. If you say “The Godel sentence (G) is true of the smallest model (i.e. the standard model) of first-order Peano Arithmetic (PA)” then this truth follows from G being unprovable: if there were a proof of G in the smallest model, there would be a proof of G in all models, and if there were a proof of G in all models, then by Godel’s completeness theorem G would be provable in PA. To insist that the Godel sentence is true in PA—that it is true wherever the axioms of PA are true—rather than being only “true in the smallest model of PA”—is just factually wrong, flat wrong as math.
The people I’m thinking of—I was one of them, once—would not say either “G is true in PA” or “G is true in such-and-such a model of PA”. They would say, simply, “G is true”, and by that they would mean that what G says about the natural numbers is true about the natural numbers—you know, the actual, real, natural numbers. And they would react with some impatience to the idea that “the actual, real, natural numbers” might not be a clearly defined notion, or that statements about them might not have a well-defined truth value in the real world.
I think most people who know Goedel’s theorem say “G is true” and are “unreflective platonists,” by which I mean that they act like the natural numbers really exist, etc, but if you pushed them on it, they’d admit the doubt of your last couple of sentences.
Similarly, most people (eg, everyone on this thread), state Goedel’s completeness theorem platonically: a statement is provable if it is true in every model. That doesn’t make sense without models having some platonic existence. (yes, you can talk about internal models, but people don’t.) I suppose you could take the platonic position that all models exist without believing that it is possible to single out the special model. (Eliezer referred to “the minimal model”; does that work?)
Those people seem a bit silly, then. If you say “The Godel sentence (G) is true of the smallest model (i.e. the standard model) of first-order Peano Arithmetic (PA)” then this truth follows from G being unprovable: if there were a proof of G in the smallest model, there would be a proof of G in all models, and if there were a proof of G in all models, then by Godel’s completeness theorem G would be provable in PA. To insist that the Godel sentence is true in PA—that it is true wherever the axioms of PA are true—rather than being only “true in the smallest model of PA”—is just factually wrong, flat wrong as math.
This thread needs a link to Tarski’s undefinability theorem.
Also, you’re assuming the consistency of PA.
The people I’m thinking of—I was one of them, once—would not say either “G is true in PA” or “G is true in such-and-such a model of PA”. They would say, simply, “G is true”, and by that they would mean that what G says about the natural numbers is true about the natural numbers—you know, the actual, real, natural numbers. And they would react with some impatience to the idea that “the actual, real, natural numbers” might not be a clearly defined notion, or that statements about them might not have a well-defined truth value in the real world.
In other words, Platonists.
I think most people who know Goedel’s theorem say “G is true” and are “unreflective platonists,” by which I mean that they act like the natural numbers really exist, etc, but if you pushed them on it, they’d admit the doubt of your last couple of sentences.
Similarly, most people (eg, everyone on this thread), state Goedel’s completeness theorem platonically: a statement is provable if it is true in every model. That doesn’t make sense without models having some platonic existence. (yes, you can talk about internal models, but people don’t.) I suppose you could take the platonic position that all models exist without believing that it is possible to single out the special model. (Eliezer referred to “the minimal model”; does that work?)