Disturbing arithmetic

Extract

One long-range objective of logic is to find models of arithmetic with noteworthy properties, perhaps properties that imply some long-standing number theoretic conjectures. In areas of mathematics such as algebra or set theory, new models are often made by extending old models, that is, by adjoining new elements to already existing models. Usually the extension retains most of the characteristics of the old model with at least one exception that makes the new model interesting. However, such a scheme is difficult in the area of arithmetic. Many interesting properties of the fine structure of arithmetic are diophantine and hence unchangeable in extensions. For instance, one cannot change a prime number into a composite one by adjoining new elements.

One could possibly get around this diophantine difficulty in one of two ways. One way is to change the usual language of addition and multiplication to an equivalent language that does not transmit so much information to extensions. For instance, multiplication is definable from the squaring function, as one sees from the identity 2xy = (x + y)² − x² − y², and the squaring function in turn is definable either from the unary square predicate (as one sees from the fact that n = m² if n and n + 2m + 1 are successive squares) or from the divisor relation (as one sees from the fact that n = m² if n is the smallest number such that m divides n and m + 1 divides n + m). Either of these two alternatives to multiplication might make for interesting extensions.