On Monday, March 16, 2020 at 1:06:17 PM UTC-4, Ken Kubota wrote: > > In terms of expressiveness, schemes are another way of expressing what > usually requires a higher order. > If you want to determine the order of a logic, it doesn't make sense to > allow schemes. > What can be expressed in first-order logic (FOL) with schemes requires at > least second-order logic (without schemes). > So it is adequate to state that PA requires second-order logic (i.e., > first-order logic without extensions like schemes is not sufficient). > > Schemes are used in FOL to describe the axioms. They aren't the axioms themselves.
Propositional calculus also uses schemes to describe its axioms. Note that the axioms themselves - an infinite number of them - are not schemes. If the use of schemes in the description of axioms automatically makes a logic second order, then there can be no such thing as first-order logic per your definition, since you can't even specify the underlying propositional calculus without them. Your definition is not the one used in the literature to define first-order logic. Norm -- You received this message because you are subscribed to the Google Groups "Metamath" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/metamath/c1857263-11fa-4230-97ca-69cc7280b5ba%40googlegroups.com.
