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Register a new userThe Pentagon Lattice as a Substructure Lattice of Models of Peano Arithmetic
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Authors
James H. Schmerl
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If $M prec N$ are models of Peano Arithmetic and Lt$(N/M)$ is the pentagon lattice $N_5$, then $N$ is either a cofinal or an end extension of $M$. In contrast, there are $M prec N$ that are models of PA* (PA in a language with countably many new predicate symbols) such that Lt$(N/M) cong N_5$ and $N$ is neither a cofinal nor an end extension of $M$.
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Simpson showed that every countable model $mathcal{M} models mathsf{PA}$ has an expansion $(mathcal{M}, X) models mathsf{PA}^*$ that is pointwise definable. A natural question is whether, in general, one can obtain expansions of a non-prime model in which the definable elements coincide with those of the underlying model. Enayat showed that this is impossible by proving that there is $mathcal{M} models mathsf{PA}$ such that for each undefinable class $X$ of $mathcal{M}$, the expansion $(mathcal{M}, X)$ is pointwise definable. We call models with this property Enayat models. In this paper, we study Enayat models and show that a model of $mathsf{PA}$ is Enayat if it is countable, has no proper cofinal submodels and is a conservative extension of all of its elementary cuts. We then show that, for any countable linear order $gamma$, if there is a model $mathcal{M}$ such that $mathrm{Lt}(mathcal{M}) cong gamma$, then there is an Enayat model $mathcal{M}$ such that $mathrm{Lt}(mathcal{M}) cong gamma$.
We study notions of genericity in models of $mathsf{PA}$, inspired by lines of inquiry initiated by Chatzidakis and Pillay and continued by Dolich, Miller and Steinhorn in general model-theoretic contexts. These papers studied the theories obtained by adding a random predicate to a class of structures. Chatzidakis and Pillay axiomatized the theories obtained in this way. In this article, we look at the subsets of models of $mathsf{PA}$ which satisfy the axiomatization given by Chatzidakis and Pillay; we refer to these subsets in models of $mathsf{PA}$ as CP-generics. We study a more natural property, called strong CP-genericity, which implies CP-genericity. We use an arithmetic version of Cohen forcing to construct (strong) CP-generics with various properties, including ones in which every element of the model is definable in the expansion, and, on the other extreme, ones in which the definable closure relation is unchanged.
A subset of a model of ${sf PA}$ is called neutral if it does not change the $mathrm{dcl}$ relation. A model with undefinable neutral classes is called neutrally expandable. We study the existence and non-existence of neutral sets in various models of ${sf PA}$. We show that cofinal extensions of prime models are neutrally expandable, and $omega_1$-like neutrally expandable models exist, while no recursively saturated model is neutrally expandable. We also show that neutrality is not a first-order property. In the last section, we study a local version of neutral expandability.
Inspired by a mathematical riddle involving fuses, we define the fusible numbers as follows: $0$ is fusible, and whenever $x,y$ are fusible with $|y-x|<1$, the number $(x+y+1)/2$ is also fusible. We prove that the set of fusible numbers, ordered by the usual order on $mathbb R$, is well-ordered, with order type $varepsilon_0$. Furthermore, we prove that the density of the fusible numbers along the real line grows at an incredibly fast rate: Letting $g(n)$ be the largest gap between consecutive fusible numbers in the interval $[n,infty)$, we have $g(n)^{-1} ge F_{varepsilon_0}(n-c)$ for some constant $c$, where $F_alpha$ denotes the fast-growing hierarchy. Finally, we derive some true statements that can be formulated but not proven in Peano Arithmetic, of a different flavor than previously known such statements: PA cannot prove the true statement For every natural number $n$ there exists a smallest fusible number larger than $n$. Also, consider the algorithm $M(x)$: if $x<0$ return $-x$, else return $M(x-M(x-1))/2$. Then $M$ terminates on real inputs, although PA cannot prove the statement $M$ terminates on all natural inputs.
We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $mathcal{D}_{h}$. In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable locally finite lattice) is isomorphic to an initial segment of $mathcal{D}_{h}$. Corollaries include the decidability of the two quantifier theory of $% mathcal{D}_{h}$ and the undecidability of its three quantifier theory. The key tool in the proof is a new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of $omega _{1}^{CK}$. Somewhat surprisingly, the set theoretic analog of this forcing does not preserve $omega _{1}$. On the other hand, we construct countable lattices that are not isomorphic to an initial segment of $mathcal{D}_{h}$.
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