Interface talk:Axioms of first-order logic in terms of substitution built on equality

freeness and theorems
Question: Is "(x is-not-free-in x = x)" true or false in this axiomatization? Norm 19:05, 7 March 2012 (UTC)


 * Neither it nor its negation are provable. At least, that was my intention. This might seem like an odd choice to make (in the sense of "shouldn't we clarify this one way or the other?"), but it doesn't seem like being able to prove or disprove the above formula is needed for one's ability to use free variables the way one needs to. At least, that has been my experience to date. Kingdon 05:49, 8 March 2012 (UTC)


 * Actually, it appears we can prove (x is-not-free-in (x = x)). In predicate calculus we can prove ( ( ∀ x x = x ) ↔ x = x ).  By BoundForAllNotFree, (x is-not-free-in (∀ x x = x)).  From buildNotFree, (x is-not-free-in x = x) follows. This shows that "is-not-free-in" is slightly different from the usual literature definition.  This isn't necessarily inconsistent, but more likely merely a different definition that in the end will lead to the same theorems (in which "is-not-free-in" doesn't occur). Norm 15:41, 8 March 2012 (UTC)


 * Good catch! I've added a formalized version of your proof to First-order logic in terms of substitution built on equality. Doing without  isn't necessarily appealing, given that it is, I think, what allows us to define new predicates or operations and just define an ax-13/ax-14 style equality axiom, not separate axioms for   and substitution. Not that is is a problem if   behaves like the set.mm , but it is not something I realized I was doing. As for inconsistency, since the axioms here are theorems of the metamath-inspired treatment of first-order logic which wikiproofs started with, the risks seem no worse than that other axiomization. To double-check what I just said, I have now exported this axiom set from First-order logic with quantifiability.  That one is based on axioms which follow set.mm closely, but with the significant addition of the Interface:Axiom of quantifiability. Kingdon 05:17, 10 March 2012 (UTC)