Type theory and formal proof pdf

In a way, I am very egocentric. I believe in a proof if I understand it, if it's clear. A computer will also make mistakes, but they are much more difficult to find. Must a proof be elegant? Do I have to understand the proof or is it enough to see that every step is correct? What if only one person in the world understands the proof. Is it a proof?

In case you are considering to adopt this book for courses with over 50 students, please contact ties. This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory higher-order logic. It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive notation facilitates proofs of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory.