Gurevich and Shelah have shown that Peano Arithmetic cannot be interpreted in the monadic second-order theory of short chains (hence, in the monadic second-order theory of the real line). We show here that it is consistent that there is no interpretation even in the monadic second-order theory of all chains.
If M is a nonstandard model of Peano Arithmetic, then M is lofty iff M has a simple elementary extension that is recursively saturated. This had previously been known for countable M.
According to Chaitin, Gödel once told him "it doesn't matter which paradox you use [to prove the First Incompleteness Theorem]". In this paper I will present a few infinitary paradoxes and show how to "translate" them to some undecidable sentences in Peano arithmetic, like what Gödel did to the Liar paradox. The results partly verify Gödel's claim.
In \cite{MacResField} the second author gave a systematic analysis of definability and decidability for rings $\mathcal M/p\mathcal M$, where $\mathcal M$ is a model of Peano Arithmetic and $p$ is a prime in $\mathcal M$. In the present paper we extend those results to the more difficult case of $\mathcal M/p^k\mathcal M$, where $\mathcal M$ is a model of Peano Arithmetic, $p$ is a prime in $\mathcal M$, and $k>1$. In \cite{MacResField} work of Ax on finite fields was used, here we use in addition work of Ax on ultraproduct of $p$-adics.
The parameter-free part $\text{PA}_2^\ast$ of $\text{PA}_2$, the 2nd order Peano arithmetic, is considered. We make use of a product/iterated Sacks forcing to define an $ω$-model of $\text{PA}_2^\ast + \text{CA}(Σ^1_2)$, in which an example of the full Comprehension schema $\text{CA}$ fails. Using Cohen's forcing, we also define an $ω$-model of $\text{PA}_2^\ast$, in which not every set has its complement, and hence the full $\text{CA}$ fails in a rather elementary way.
We show that there is an arithmetical formula F such that ZF proves that F is independent of PA and yet, unlike other arithmetical independent statements, the truth value of F cannot at present be established in ZF or in any other trusted metatheory. In fact we can choose an example of such a formula F such that ZF proves that F is equivalent to the twin prime conjecture.
In 1994 Jech gave a model theoretic proof of Gödel's second incompleteness theorem for Zermelo-Fraenkel set theory in the following form: ZF does not prove that ZF has a model. Kotlarski showed that Jech's proof can be adapted to Peano Arithmetic with the role of models being taken by complete consistent extensions. In this note we take another step in the direction of replacing proof-theoretic by model-theoretic arguments. We show, without passing through the arithmetized completeness theorem, that the existence of a model of PA of complexity $Σ^0_2$ is independent of PA, where a model is ide…
We investigate $IPA$ - real closed fields, that is, real closed fields which admit an integer part whose non-negative cone is a model of Peano Arithmetic. We show that the value group of an $IPA$ - real closed field is an exponential group in the residue field, and that the converse fails in general. As an application, we classify (up to isomorphism) value groups of countable recursively saturated exponential real closed fields. We exploit this characterization to construct countable exponential real closed fields which are not $IPA$ - real closed fields.
In the Handbook of Mathematical Logic, the Paris-Harrington variant of Ramsey's theorem is celebrated as the first result of a long 'search' for a purely mathematical incompleteness result in first-order arithmetic. This paper questions the existence of any such search and the status of the Paris-Harrington result as the first mathematical incompleteness result. In fact, I argue that Gentzen gave the first such result, and that it was restated by Goodstein in a number-theoretic form.
In introductory books about natural numbers, a common kind of assertion - often left as exercise to the reader - is that certain forms of induction on $\mathbb{N}$ (regular/ordinary, complete/strong) are equivalent one to each other and to the well-ordering principle. This means that if P1 and P2 are two of these principles, then, under all the other usually adopted postulates for the set of natural numbers (e.g., Peano axioms other than the induction axiom), P1 implies P2 and vice-versa. In this paper, we shows that, for a reasonable formalization, based on Peano arithmetic, some of the alleg…
We show that the theory $IΣ_1$ of $Σ_1$-induction proves the following statement: For all $n\geq 2$, the uniform $Σ_1$-reflection principle over the theory $IΣ_n$ is equivalent to the totality of the function $F_{ω_n}$ at stage $ω_n$ of the fast-growing hierarchy. The method applied is a formalization of infinite proof theory. The literature contains several proofs which place the quantification over $n$ in the meta-theory (and also prove the separate cases $n=0,1$). In contrast, the author knows of no explicit argument that would allow us to internalize the quantification while keeping the me…
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 natur…
The notion of slow provability for Peano Arithmetic ($\mathsf{PA}$) was introduced by S.D. Friedman, M. Rathjen, and A. Weiermann. They studied the slow consistency statement $\mathrm{Con}_{\mathsf{s}}$ that asserts that a contradiction is not slow provable in $\mathsf{PA}$. They showed that the logical strength of $\mathsf{PA}+\mathrm{Con}_{\mathsf{s}}$ lies strictly between that of $\mathsf{PA}$ and $\mathsf{PA}$ together with its ordinary consistency: $\mathsf{PA}\subsetneq \mathsf{PA}+\mathrm{Con}_{\mathsf{s}}\subsetneq \mathsf{PA}+\mathrm{Con}$. This paper is a further investigation into …
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 …
Induction is typically formalized as a rule or axiom extension of the LK-calculus. While this extension of the sequent calculus is simple and elegant, proof transformation and analysis can be quite difficult. Theories with an induction rule, for example Peano arithmetic do not have a {\em Herbrand} theorem. In this work we extend an existing meta-theoretic formalism, so called proof schemata, a recursive formulation of induction particularly suited for proof analysis, to Peano arithmetic. This relationship provides a meaningful conservative reflection principle between \PA and an alternative p…
Harvey Friedman shows that, over Peano Arithmetic, the consistency statement for a finitely axiomatised theory $A$ can be characterised as the weakest statement $C$ over Peano Arithmetic such that ${\sf PA}+C$ interprets $A$. We study which base theories $U$ have the property that, for any finitely axiomatised $A$, there is a weakest $C$ such that $U+C$ interprets $A$. We call such theories Friedman-reflexive. We show that a very weak theory, Peano Corto, is Friedman-reflexive. We do not get the usual consistency statements here, but bounded, cut-free or Herbrand consistency statements. We pro…
Let $\widetilde{\mathbb{Q}_p}$ be the field of $p$-adic numbers in the language of rings. In this paper we consider the theory of $\widetilde{\mathbb{Q}_p}$ expanded by two predicates interpreted by multiplicative subgroups $α^\mathbb{Z}$ and $β^\mathbb{Z}$ where $α, β\in\mathbb{N}$ are multiplicatively independent. We show that the theory of this structure interprets Peano arithmetic if $α$ and $β$ have positive $p$-adic valuation. If either $α$ or $β$ has zero valuation we show that the theory of $(\widetilde{\mathbb{Q}_p}, α^\mathbb{Z}, β^\mathbb{Z})$ does not interpret Peano arithmetic. In…
We calculate the possible Scott ranks of countable models of Peano arithmetic. We show that no non-standard model can have Scott rank less than $ω$ and that non-standard models of true arithmetic must have Scott rank greater than $ω$. Other than that there are no restrictions. By giving a reduction via $Δ^{\mathrm{in}}_{1}$ bi-interpretability from the class of linear orderings to the canonical structural $ω$-jump of models of an arbitrary completion $T$ of $\mathrm{PA}$ we show that every countable ordinal $α>ω$ is realized as the Scott rank of a model of $T$.
A first order theory T is said to be "tight" if for any two deductively closed extensions U and V of T (both of which are formulated in the language of T), U and V are bi-interpretable iff U = V. By a theorem of Visser, PA (Peano Arithmetic) is tight. Here we show that Z_2 (second order arithmetic), ZF (Zermelo-Fraenkel set theory), and KM (Kelley-Morse theory of classes) are also tight theories.
We construct a theory definitionally equivalent to first-order Peano arithmetic PA and a non-standard computable model of this theory. The same technique allows us to construct a theory definitionally equivalent to Zermelo-Fraenkel set theory ZF that has a computable model.
Linear arithmetics are extensions of Presburger arithmetic (Pr) by one or more unary functions, each intended as multiplication by a fixed element (scalar), and containing the full induction schemes for their respective languages. In this paper we construct a model M of the 2-linear arithmetic LA_2 (linear arithmetic with two scalars) in which an infinitely long initial segment of "Peano multiplication" on M is 0-definable. This shows, in particular, that LA_2 is not model complete in contrast to theories LA_1 and LA_0=Pr that are known to satisfy quantifier elimination up to disjunctions of p…
Fix a countable nonstandard model $\mathcal M$ of Peano Arithmetic. Even with some rather severe restrictions placed on the types of minimal cofinal extensions $\mathcal N \succ \mathcal M$ that are allowed, we still find that there are $2^{\aleph_0}$ possible theories of $(\mathcal N,M)$ for such $\mathcal N$'s.
Learning, Realizability and Games in Classical Arithmetic
在这篇论文中，我们提供了数学证据，学习的概念可以用来给一个新的和直观的计算语义的经典证明中的各种片段的谓词算术。首先，我们将Kreisel修正的可实现性扩展到一阶算术的经典片段，即Heyting算术加上EM 1（排除中间公理仅限于Sigma^0_1公式）。我们引入了一个新的可实现性语义，我们称之为“基于交互式学习的可实现性”。我们的实现者是自我纠正的程序，它们从错误中学习，并随着时间的推移而进化。其次，我们扩展类的学习为基础的实现者的PCF的经典版本PCF类，然后，比较Coquand游戏语义的可实现性的概念，并证明了充分的合理性和完整性的结果。特别是，我们发现有一个一对一的实现者和递归获胜策略之间的对应关系，在1-回溯版本的塔斯基游戏。第三，我们提供了一个完整的和充分详细的建设性分析的学习，因为它出现在基于学习的实现HA+ EM 1，Avigad的更新程序和Peano算术PA的替代方法。我们提出了新的建设性的技术来约束学习过程的长度，我们应用它们来重新证明-通过我们的理论-哥德尔的经典结果，可证明的总功能PA可以表示在哥德尔的系统T。最后，我们给出了一个公理化的学习，需要计算解释谓词经典二阶算术。我们的工作是Avigad的推广，并将更新过程的概念推广到超限情形。超限更新过程必须学习不可计算函数的超限序列的值，以便从经典证明中提取证据。
In the theory of conditional sets, many classical theorems from areas such as functional analysis, probability theory or measure theory are lifted to a conditional framework, often to be applied in areas such as mathematical economics or optimization. The frequent experience that such theorems can be proved by `conditionalizations' of the classical proofs suggests that a general transfer principle is in the background, and that formulating and proving such a transfer principle would yield a wealth of useful further conditional versions of classical results, in addition to providing a uniform a…
Models of PA: when two elements are necessarily order automorphic
我们希望看到对于PA（Peano算术）的某个完备化T的模型M，M限制{<}在多大程度上决定M，比如直到同构。我们提出了PA的非标准模型M的“最小”集{（a，b）：n < a < b for n in N和线性阶{c：c <_Ma}，{c：c <_Mb}是同构的}的特征，特别是包括{（a，b）：对于M中没有c，对于N中的每一个n，我们有M个模型（对于N中的所有n）（存在c）[2 < c^n < a wedge b < a c^n]}和对于某些模型等于{（a，b）：a < b < a^n for some n in N}。
