Disturbing arithmetic

E-ISSN： 1943-5886|50|2|375-379

ISSN： 0022-4812

Source： The Journal of Symbolic Logic, Vol.50, Iss.2, 1985-06, pp. : 375-379

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Abstract

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.