(Also posted in mathstackexchange prior to this).
I stumbled across a set of axioms for first order logic a bit ago. Intrigued, I decided to try to write it all down and organise what I read. After I did that, it seemed to me as though one could quite simply define 'truth' using these axioms and deduction rules.
If β, γ, δ are wfs, x, y are variables, and t is a term, then the following wfs are called logical axioms:
- β → (γ → β)
- (β → (γ → δ)) → ((β → γ) → (β → δ))
- (¬β → ¬γ) → ((¬β → γ) → β)
- ((∀x)β) → β[t/x] if t is free for x in β
- (∀x)(β → γ) → (β → (∀x)γ) if β contains no free occurrence of x
- (∀x)(x = x)
- (∀x)(∀y)((x = y) → (β → β[y/x]))
(Then give one's particular set of mathematical axioms, if one intends for them to be thought of as true).
We then define a subclass of wfs which we call the true statements. If α, β are wfs, and x is a variable, then:
All logical axioms are true, as are all mathematical axioms. If α and α → β are true, then β is true. If β is true, then (∀x)β is true. Finally, a statement φ is false if and only if ¬φ is true.
When I showed this to some of my philosophy of mathematics friends, they thought for quite a while before deciding that the above did not suffice to define truth within such a formal system. When pressed as to why not, they weren't entirely sure, though a few possible places where issues could arise were discussed. We couldn't figure out what sort of true statement would not fall under such a definition, however.
My question is therefore - does the above fail to define truth? If so, why does it fail? Are there any amendments that could be made to fix the above? What sort of true statements would not be "true" in the above sense? What have I defined above?
Additionally, any sources for further reading on this topic would be of great, great interest to me, and would be appreciated.