> We both agree that there is a clear distinction between formulae that are true in some model (specified, or inferred from context) and formulae that are true in all models; [...]
Sure. But I feel we are deviating from the subject. We have obviously been educated differently so it is pointless to argue about that, but there is a language issue. You insist on comparing what I mean by "true" (alone) with "true in a model". However, that's an apple to orange comparison. We should be comparing what I mean by "true" (alone) with what you mean by "true" (alone), and by that, you mean: "true in the standard model". (I don't think your references validate that use, although I don't have access to all of them at the moment.) The obvious problems with that are:
- I don't think there is such a thing as a standard model in set theory (actually you cannot prove that a model of set theory exists).
- When most mathematicians say something like "X is true", what they mean is "X can be proved from the axioms of set theory", which in logical terms means "X is valid". Are you really arguing against that?
- And of course (back to the original point), you get that confusing idea that "undecidable" means "true but unprovable" (I had never heard of the incompleteness theorem being presented that way before.). I argue "undecidable" is "neither provable nor disprovable".
EDIT: "X is valid" should read "X is valid in set theory".
Sure. But I feel we are deviating from the subject. We have obviously been educated differently so it is pointless to argue about that, but there is a language issue. You insist on comparing what I mean by "true" (alone) with "true in a model". However, that's an apple to orange comparison. We should be comparing what I mean by "true" (alone) with what you mean by "true" (alone), and by that, you mean: "true in the standard model". (I don't think your references validate that use, although I don't have access to all of them at the moment.) The obvious problems with that are:
- I don't think there is such a thing as a standard model in set theory (actually you cannot prove that a model of set theory exists).
- When most mathematicians say something like "X is true", what they mean is "X can be proved from the axioms of set theory", which in logical terms means "X is valid". Are you really arguing against that?
- And of course (back to the original point), you get that confusing idea that "undecidable" means "true but unprovable" (I had never heard of the incompleteness theorem being presented that way before.). I argue "undecidable" is "neither provable nor disprovable".
EDIT: "X is valid" should read "X is valid in set theory".