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As far as I can tell this is not a misconception, I am just not communicating clearly enough. I figured I could make the point at a higher level. You can const

As far as I can tell this is not a misconception, I am just not communicating clearly enough. I figured I could make the point at a higher level. You can construct a statement in an axiomatic system complex enough to express arithmetic: 1. Which is true, 2. AND which can be proved to be true under the assumption the system is consistent, 3. and which can not be proven true within that axiomatic system because you can't prove consistency of the system itself within the axiomatic system itself. The crass positivist position would be that such undecidable statements are meaningless, i.e. neither true nor false. The incompleteness theorems provide a strong counter-example: a statement which can not be proved within that axiomatic system, but is true (if the system is consistent). The positivists must then amend their position to include things which can't be proved within the axiomatic system but can be proved true. This resulted in the watered down late-positivism.

griffzhowl 1y ago
Yes, the misconception is to think Godel's theorem shows that there are true but unprovable statements. In fact, what it means for a mathematical statement to be true is that it's provable. Godel showed that the concept of provability is much more subtle than following from a single axiom system, but it doesn't show that there's some kind of transcendent form of mathematical truth that doesn't depend on proof, in my view.
- EthanHeilman 1y ago
  How would you define provable without first assuming a set of axioms?
  - griffzhowl 1y ago
    Good question. I don't think there's an easy answer, except for something like: A statement is provable if it's possible to write a proof for it. But you're unlikely to find that satisfactory... Spelling out the bounds of what can possibly constitute a proof is a big question in mathematical logic, model theory, and proof theory, and I'm far from an expert in these subjects. I think any proof has to be based on some set of axioms, but Godel showed that not all provable statements can be formally derived from a single set of axioms. Another way to put it might be that the space of all possible proofs can't be defined by a single axiom system with its rules of inference, but I'm not sure about this, because as I understand it we should always be able to tell mechanically, given a putative proof, whether it really proves the theorem. In any case, I believe all these subtleties and difficulties also have to be faced if we try to spell out precisely what we mean by mathematical truth. In my view, this is the same question: mathematically true = it's possible to prove it.

griffzhowl 1y ago

Yes, the misconception is to think Godel's theorem shows that there are true but unprovable statements. In fact, what it means for a mathematical statement to be true is that it's provable. Godel showed that the concept of provability is much more subtle than following from a single axiom system, but it doesn't show that there's some kind of transcendent form of mathematical truth that doesn't depend on proof, in my view.

EthanHeilman 1y ago
How would you define provable without first assuming a set of axioms?
- griffzhowl 1y ago
  Good question. I don't think there's an easy answer, except for something like: A statement is provable if it's possible to write a proof for it. But you're unlikely to find that satisfactory... Spelling out the bounds of what can possibly constitute a proof is a big question in mathematical logic, model theory, and proof theory, and I'm far from an expert in these subjects. I think any proof has to be based on some set of axioms, but Godel showed that not all provable statements can be formally derived from a single set of axioms. Another way to put it might be that the space of all possible proofs can't be defined by a single axiom system with its rules of inference, but I'm not sure about this, because as I understand it we should always be able to tell mechanically, given a putative proof, whether it really proves the theorem. In any case, I believe all these subtleties and difficulties also have to be faced if we try to spell out precisely what we mean by mathematical truth. In my view, this is the same question: mathematically true = it's possible to prove it.