On Tue, 20 Aug 2013, Lax Clarke wrote: > > > Please correct me if I'm wrong please: > > > This is the order of bootstrapping the foundations of mathematics: > > Boot strapping has nothing to do with mathematics. > > It's part of computer science. > Ok. I just want to know the order of learning I guess. Or say a really smart and "rigor-loving" alien landed in our backyard, in what order would we explain thing to him in (assuming the alien learns English first). > > > 1) Naive logic (like the ones the Greeks played with). > > > 2) Use 1) to talk about Naive Set theory (like Halmos' book). > > > 3) Use 2) above to define Mathematical Logic / First-Order Logic > > > 4) Use 3) above to define axiomatic set theory. > > A metalanguage is used to describe a formal language. > 1, 2 would be metalanguages? These would be used to build up formal > languages 3,4? The metalanguage is usually a simple native language with simple logic which usually includes simple induction. The metalanguage is used to describe a formal language and it's logic, such as rules of inference. It could be used to describe a FOL. If axioms are added then it would be describing a theory. The theory could be one of the set theories ML, ZF, ZFC or NBG. Abbreviations are part of the metalanguage and within a simple book, say ML, abbreviations can quickly represent hundreds or thousands, perhaps millions, of primitive symbols.