A standard model of Peano arithmetic with no conservative elementary extension

AbstractThe principal result of this paper answers a long-standing question in the model theory of arithmetic [R. Kossak, J. Schmerl, The Structure of Models of Peano Arithmetic, Oxford University Press, 2006, Question 7] by showing that there exists an uncountable arithmetically closed family A of subsets of the set ω of natural numbers such that the expansion NA≔(N,A)A∈A of the standard model N≔(ω,+,×) of Peano arithmetic has no conservative elementary extension, i.e., for any elementary extension NA∗=(ω∗,…) of NA, there is a subset of ω∗ that is parametrically definable in NA∗ but whose intersection with ω is not a member of A. We also establish other results that highlight the role of countability in the model theory of arithmetic.Inspired by a recent question of Gitman and Hamkins, we furthermore show that the aforementioned family A can be arranged to further satisfy the curious property that forcing with the quotient Boolean algebra A/FIN (where FIN is the ideal of finite sets) collapses ℵ1 when viewed as a notion of forcing

Categorical characterizations of the natural numbers require primitive recursion

Simpson and the second author asked whether there exists a characterization of the natural numbers by a second-order sentence which is provably categorical in the theory RCA$^*_0$. We answer in the negative, showing that for any characterization of the natural numbers which is provably true in WKL$^*_0$, the categoricity theorem implies $\Sigma^0_1$ induction. On the other hand, we show that RCA$^*_0$ does make it possible to characterize the natural numbers categorically by means of a set of second-order sentences. We also show that a certain $\Pi^1_2$-conservative extension of RCA$^*_0$ admits a provably categorical single-sentence characterization of the naturals, but each such characterization has to be inconsistent with WKL$^*_0$+superexp.Comment: 17 page