Abstract
Publisher Summary This chapter discusses the classical and the ω-complete arithmetic. It deals with two formal systems for the theory of natural numbers, both of which are applied second-order functional calculi with equality and the description operator. The two systems have the same primitive symbols, rules of formation, and axioms, differing only in the rules of inference. The classes of number formulas (nfs) and propositional formulas (pfs) are defined inductively as the least classes of formal expressions. Together the nfs and pfs are called formulas. pf containing no free variables is called closed, and a formula in which no quantifier binding a function variable occurs is called elementary. The chapter provides a description of semantics after making meta-mathematical observation that the notions and axioms which are assumed in meta-mathematics comprise all the ordinary notions and axioms of set theory, so that such non-elementary notions can be used in definitions as classes and functions of arbitrarily high types.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Studies in Logic and the Foundations of Mathematics
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
