Primitive Recursive Arithmetic: Arithmetic, Quantification, Thoralf Skolem, Finitism, Foundations of Mathematics, Ordinal Analysis, Peano Axioms, Natural Number

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Primitive recursive arithmetic, or PRA, is a quantifier-free formalization of the natural numbers. It was first proposed by Skolem as a formalization of his finitist conception of the foundations of arithmetic, and it is widely agreed that all reasoning of PRA is finitist. Many also believe that all of finitism is captured by PRA, but others believe finitism can be extended to forms of recursion beyond primitive recursion, up to ¿0, which is the proof-theoretic ordinal of Peano arithmetic. PRA's proof theoretic ordinal is ¿¿, where ¿ is the smallest transfinite ordinal. PRA is sometimes called Skolem arithmetic. The language of PRA can express arithmetic propositions involving natural numbers and any primitive recursive function, including the operations of addition, multiplication, and exponentiation. PRA cannot explicitly quantify over the domain of natural numbers. PRA is often taken as the basic metamathematical formal system for proof theory, in particular for consistency proofs such as Gentzen's consistency proof of first-order arithmetic.