Order-types of models of Peano arithmetic: a short survey

Here, PA is the first-order theory in the language with 0, 1,+, ·, < containing finitely many basic axioms true in N together with the first-order induction axiom scheme. Any model of PA has an initial segment isomorphic to N, and we will always identify N with this initial segment. The theory PA is well-known not to be complete, and has 2א0 complete extensions. ‘True arithmetic’—the theory Th(N) of the standard model N—is one of these extensions, and sometimes needs to be treated differently from the others. For example, nonstandard models of Th(N) do not have any nonstandard definable elements, whereas any model of ‘false arithmetic’ (a model of PA not satisfying Th(N)) always has nonstandard definable elements. Kaye’s book [5] provides a good background to the model theory of PA, resplendency and recursive saturation, and should be consulted for definitions and results not contained here. We shall also assume knowledge of a certain amount of standard model theory throughout this paper, as can be found in any of the standard texts. The main question above is easily solved for countable models (see Section 3 below), so the issue is what the possible order-types of uncountable models of ∗Department of Mathematics, Yliopistonkatu 5, University of Helsinki, 00014, Finland. The author was supported by a grant of the University of Helsinki. School of Mathematics and Computer Science, Istanbul Bilgi University, Kustepe, Sisli, 80310, Istanbul, Turkey. The author was supported by a NATO-PC Advanced Fellowship via TÜBITAK, the Scientific and Technical Research Council of Turkey. †School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, U.K. http://www.mat.bham.ac.uk/R.W.Kaye/