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Authors
David Pierce
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This paper grew out of the observation that the possibilities of proof by induction and definition by recursion are often confused. The paper reviews the distinctions. The von Neumann construction of the ordinal numbers includes a construction of natural numbers as a special kind of ordinal. In any case, the natural numbers can be understood as composing a free algebra in a certain signature, {0,s}. The paper here culminates in a construction of, for each algebraic signature S, a class ON_S that is to the class of ordinals as S is to {0,s}. In particular, ON_S has a subclass that is a free algebra in the signature S.
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Several authors have conjectured that Conways field of surreal numbers, equipped with the exponential function of Kruskal and Gonshor, can be described as a field of transseries and admits a compatible differential structure of Hardy-type. In this paper we give a complete positive solution to both problems. We also show that with this new differential structure, the surreal numbers are Liouville closed, namely the derivation is surjective.
We study the topological version of the partition calculus in the setting of countable ordinals. Let $alpha$ and $beta$ be ordinals and let $k$ be a positive integer. We write $betato_{top}(alpha,k)^2$ to mean that, for every red-blue coloring of the collection of 2-sized subsets of $beta$, there is either a red-homogeneous set homeomorphic to $alpha$ or a blue-homogeneous set of size $k$. The least such $beta$ is the topological Ramsey number $R^{top}(alpha,k)$. We prove a topological version of the ErdH{o}s-Milner theorem, namely that $R^{top}(alpha,k)$ is countable whenever $alpha$ is countable. More precisely, we prove that $R^{top}(omega^{omega^beta},k+1)leqomega^{omega^{betacdot k}}$ for all countable ordinals $beta$ and finite $k$. Our proof is modeled on a new easy proof of a weak version of the ErdH{o}s-Milner theorem that may be of independent interest. We also provide more careful upper bounds for certain small values of $alpha$, proving among other results that $R^{top}(omega+1,k+1)=omega^k+1$, $R^{top}(alpha,k)< omega^omega$ whenever $alpha<omega^2$, $R^{top}(omega^2,k)leqomega^omega$ and $R^{top}(omega^2+1,k+2)leqomega^{omegacdot k}+1$ for all finite $k$. Our computations use a variety of techniques, including a topological pigeonhole principle for ordinals, considerations of a tree ordering based on the Cantor normal form of ordinals, and some ultrafilter arguments.
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.
Given a dense additive subgroup $G$ of $mathbb R$ containing $mathbb Z$, we consider its intersection $mathbb G$ with the interval $[0,1[$ with the induced order and the group structure given by addition modulo $1$. We axiomatize the theory of $mathbb G$ and show it is model-complete, using a Feferman-Vaught type argument. We show that any sufficiently saturated model decomposes into a product of a standard part and two ordered semigroups of infinitely small and infinitely large elements.
The present article surveys surreal numbers with an informal approach, from their very first definition to their structure of universal real closed analytic and exponential field. Then we proceed to give an overview of the recent achievements on equipping them with a derivation, which is done by proving that surreal numbers can be seen as transseries and by finding the `simplest structure of H-field, the abstract version of a Hardy field. All the latter notions and their context are also addressed, as well as the universality of the resulting structure for surreal numbers.
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