This paper shows that predicative Frege arithmetic naturally interprets some weak but non-trivial arithmetical theories. The weak theories in question are all relational versions of Tarski, Mostowski, and Robinson's R and Q, i.e., they are formulated using predicates Pxy, Axyz, and Mxyz in place of the usual function symbols Sx, x+y, and x×y. We lose the existence and uniqueness of successor, sum, and product, as generalizations, but retain these in each particular case (much as we lose the recursion clauses for addition in R, but retain them in each particular case). In saying that the interpretation is "natural", I mean that it relies only upon "definitions" of arithmetical notions that are themselves "natural", that is, that have some claim to be "definitions" in something other than a purely formal sense.The published version can be found here; the pre-publication version, here. If you need a copy of the published version but don't have access without paying, then send me an email.
Wednesday, November 19, 2014
"Frege Arithmetic and 'Everyday Mathematics'" Published
My paper "Frege Arithmetic and 'Everyday Mathematics'" has been published in Philosophia Mathematica 22 (2014), pp. 279-307. Abstract: