Order Peano Research Articles

On the Significance of Parameters in the Choice and Collection Schemata in the 2nd Order Peano Arithmetic

We make use of generalized iterations of the Sacks forcing to define cardinal-preserving generic extensions of the constructible universe L in which the axioms of ZF hold and in addition either (1) the parameter-free countable axiom of choice ACω* fails, or (2) ACω* holds but the full countable axiom of choice ACω fails in the domain of reals. In another generic extension of L, we define a set X⊆P(ω), which is a model of the parameter-free part PA2* of the 2nd order Peano arithmetic PA2, in which CA(Σ21) (Comprehension for Σ21 formulas with parameters) holds, yet an instance of Comprehension CA for a more complex formula fails. Treating the iterated Sacks forcing as a class forcing over Lω1, we infer the following consistency results as corollaries. If the 2nd order Peano arithmetic PA2 is formally consistent then so are the theories: (1) PA2+¬ACω*, (2) PA2+ACω*+¬ACω, (3) PA2*+CA(Σ21)+¬CA.

Improved Bagging Algorithm for Pattern Recognition in UHF Signals of Partial Discharges

This paper presents an Improved Bagging Algorithm (IBA) to recognize ultra-high-frequency (UHF) signals of partial discharges (PDs). This approach establishes the sample information entropy for each sample and the re-sampling process of the traditional Bagging algorithm is optimized. Four typical discharge models were designed in the laboratory to simulate the internal insulation faults of power transformers. The optimized third order Peano fractal antenna was applied to capture the PD UHF signals. Multi-scale fractal dimensions as well as energy parameters extracted from the decomposed signals by wavelet packet transform were used as the characteristic parameters for pattern recognition. In order to verify the effectiveness of the proposed algorithm, the back propagation neural network (BPNN) and the support vector machine (SVM) based on the IBA were adopted in this paper to carry out the pattern recognition for PD UHF signals. Experimental results show that the proposed approach of IBA can effectively enhance the generalization capability and also improve the accuracy of the recognition for PD UHF signals.

Peano Differentiation via Integration

In a little known paper, Haslam-Jones defined a collection of higher order derivatives in terms of an integral and Legendre polynomials. One member of this collection is equivalent to the higher order Peano derivatives. The purpose of this paper is to present a more direct proof of this equivalence.

Expanding the additive reduct of a model of Peano arithmetic

AbstractLet M be a model of first order Peano arithmetic (PA) and I an initial segment of M that is closed under multiplication. LetM0 be the {0, 1,+}‐reduct ofM. We show that there is another model N of PA that is also an expansion of M0 such that a · M a = a · N a if and only if a ∈ I for all a ∈ M.

DERIVATES, APPROXIMATE DERIVATES AND POROSITY DERIVATES OF n-CONVEX FUNCTIONS

It is shown that if $f$ is $n$-convex then the four $n$th order Peano derivates of $f$ are respectively equal to the corresponding $n$th order approximate Peano derivates and the porosity Peano derivates of $f$. It is further shown that the same result holds for the de la Vall\'ee Poussin derivates, and the symmetric and unsymmetric Riemann derivates.

On Peano and Riemann derivatives

For a given real function of one real variablef the conceptsn-th order Peano and (modified) Riemann derivatives are introduced. The conjecture is formulated that Peano derivative ofn-th order exists if and only if all Riemann derivatives of order less or equal ton exist and then then-th order Peano and Riemann derivative coincide. It is shown that this conjecture is equivalent to an assertion about the value of certain functional determinant of order 2n+1. This assertion is checked forn≤8. The general case remains an open question.

Extending n times differentiable functions of several variables

It is shown that n times Peano differentiable functions defined on a closed subset of \(\mathbb{R}^m \) and satisfying a certain condition on that set can be extended to n times Peano differentiable functions defined on \(\mathbb{R}^m \) if and only if the nth order Peano derivatives are Baire class one functions.

EQUIVALENCE OF (n + 1)-th ORDER PEANO AND USUAL DERIVATIVES FOR n-CONVEX FUNCTIONS

A real-valued function $f$ defined on an interval of $\mathbb{R}$ is said to be $n$-convex if all its $n$-th order divided differences are not negative. Let $f$ be such a function defined in a right neighborhood of $t_0 \in \mathbb{R}$ whose usual right derivatives, $f^{(r)}_+,$ $1\le r\le n,$ exist in that neighborhood and whose $(n+1)$-th order Peano derivative, $f_{n+1}(t_0)$, exists at $t_0$. Under these assumptions we prove that $f$ also possesses $(n+1)$-th order usual right derivative $f^{(n+1)}_+(t_0)$ at $t_0$. This result generalizes the known case for convex (that is 1-convex) functions. The latter appears in works of B. Jessen studying the curvature of convex curves and of J. M. Borwein, M. Fabian, D. Noll studying the second order differentiability of convex functions on abstract spaces.

Large discrete parts of the E-tree

Let PA be first order Peano arithmetic, let Λ be the lattice of Π1 sentences modulo PA, and let S be the poset of prime filters of Λ ordered by reverse inclusion. We show there are large convex discrete parts of S; in particular there are convex parts which form a completed Baire tree or an Aronszajn tree.The elements of S, which we call nodes, correspond to the extensions of PA which are complete for sentences. Equivalently, for each model of PA the Π1-theory ∀() of is a node, and every node occurs in this form. Note that the Π1-theory ∀() of the standard model (i.e. the filter of true Π1 sentences) is the unique root of S.This poset S, which is sometimes called the E-tree, was first studied in [1] where it is shown that:(1) The poset is tree-like, i.e. the set of predecessors of any node is linearly ordered.(2) The poset has branches, each of which is closed under unions and intersections; in particular each branch has a maximum member.(3) There are branches on which ∀() does not have an immediate successor. Further properties of the E-tree are given in [2]−[7]. In particular in [4] Misercque shows that:(4) There are branches on which ∀() does have an immediate successor.(5) There are nodes with both an immediate predecessor and an immediate successor.The two results (3) and (4) show that there are fundamentally different branches of S, and (5) shows that parts of branches may be discrete.

Saturation of homogeneous resplendent models

The complete diagram of a structure , denoted by Dc(), is the set of all sentences true in the structure (, a)a∈. A structure is said to be resplendent if for every sentence θ involving a new relation symbol R in addition to symbols occurring in Dc(), if θ is consistent with Dc(), then there is a relation P on such that (see[1]).Baldwin asked whether a homogeneous recursively saturated structure is necessarily resplendent. Here it is shown that this need not be the case. It is shown that if is an uncountable homogeneous resplendent model of an unstable theory, then must be saturated. The proof is related to the proof in [5] that an uncountable homogeneous recursively saturated model of first order Peano arithmetic must be saturated. The example for Baldwin's question is an uncountable homogeneous model for a particular unstable theory, such that is recursively saturated and omits some type. (The continuum hypothesis is needed to show the existence of such a model in power ℵ1.)The proof of the main result requires two lemmas.

Bounded existential induction

The present work may perhaps be seen as a point of convergence of two historically distinct sequences of results. One sequence of results started with the work of Tennenbaum [59] who showed that there could be no nonstandard recursive model of the system PA of first order Peano arithmetic. Shepherdson [65] on the other hand showed that the system of arithmetic with open induction was sufficiently weak to allow the construction of nonstandard recursive models. Between these two results there remained for many years a large gap occasioned by a general lack of interest in weak systems of arithmetic. However Dana Scott observed that the addition alone of a nonstandard model of PA could not be recursive, while more recently McAloon [82] improved these results by showing that even for the weaker system of arithmetic with only bounded induction, neither the addition nor the multiplication of a nonstandard model could be recursive.Another sequence of results starts with the work of Lessan [78], and independently Jensen and Ehrenfeucht [76], who showed that the structures which may be obtained as the reducts to addition of countable nonstandard models of PA are exactly the countable recursively saturated models of Presburger arithmetic. More recently, Cegielski, McAloon and the author [81] showed that the above result holds true if PA is replaced by the much weaker system of bounded induction.However in both the case of the Tennenbaum phenomenon and in that of the recursive saturation of addition the problem remained open as to how strong a system was really necessary to generate the required phenomenon. All that was clear a priori was that open induction was too weak to produce either result.

Two theorems on degrees of models of true arithmetic

Let PA be the theory of first order Peano arithmetic, in the language L with binary operation symbols + and ·. Let N be the theory of the standard model of PA. We consider countable models M of PA such that the universe ∣M∣ is ω. The degree of such a model M, denoted by deg(M), is the (Turing) degree of the atomic diagram of M. The results of this paper concern the degrees of models of N, but here in the Introduction, we shall give a brief survey of results about degrees of models of PA.Let D0 denote the set of degrees d such that there is a nonstandard model of M of PA with deg(M) = d. Here are some of the more easily stated results about D0.(1) There is no recursive nonstandard model of PA; i.e., 0 ∈ D0.This is a result of Tennenbaum [T].(2) There existsd ∈ D0such thatd ≤ 0′.This follows from the standard Henkin argument.(3) There existsd ∈ D0such thatd < 0′.Shoenfield [Sh1] proved this, using the Kreisel-Shoenfield basis theorem.(4) There existsd ∈ D0such thatd′ = 0′.Jockusch and Soare [JS] improved the Kreisel-Shoenfield basis theorem and obtained (4).(5) D0 = Dc = De, where Dc denotes the set of degrees of completions of PA and De the set of degrees d such that d separates a pair of effectively inseparable r.e. sets.Solovay noted (5) in a letter to Soare in which in answer to a question posed in [JS] he showed that Dc is upward closed.

Expansions of models and turing degrees

If is a countable recursively saturated structure and T is a recursively axiomatizable theory that is consistent with Th(), then it is well known that can be expanded to a recursively saturated model of T [7, p. 186]. This is what has made recursively saturated models useful in model theory. Recursive saturation is the weakest notion of saturation for which this expandability result holds. In fact, if is a countable model of Pr = Th(ω, +), then can be expanded to a model of first order Peano arithmetic P just in case is recursively saturated (see [3]).In this paper we investigate two natural sets of Turing degrees that tell a good deal about the expandability of a given structure. If is a recursively saturated structure, I() consists of the degrees of sets that are recursive in complete types realized in . The second set of degrees, D(), consists of the degrees of sets S such that is recursive in S-saturated. In general, I() ⊆ D(). Moreover, I() is obviously an “ideal” of degrees. For countable structures , D() is “closed” in the following sense: For any class C ⊆ 2ω, if C is co-r.e. in S for some set S such that , then there is some σ ∈ C such that . For uncountable structures , we do not know whether D() must be closed.

On the role of Ramsey quantifiers in first order arithmetic

The purpose of this paper is to study a formal system PA(Q2) of first order Peano arithmetic, PA, augmented by a Ramsey quantifier Q2 which binds two free variables. The intended meaning of Q2xx′φ(x, x′) is that there exists an infinite set X of natural numbers such that φ(a, a′) holds for all a, a′ Є X such that a ≠ a′. Such an X is called a witness set for Q2xx′φ(x, x′). Our results would not be affected by the addition of further Ramsey quantifiers Q3, Q4, …, Here of course the intended meaning of Qkx1 … xkφ(x1,…xk) is that there exists an infinite set X such that φ(a1…, ak) holds for all k-element subsets {a1, … ak} of X.Ramsey quantifiers were first introduced in a general model theoretic setting by Magidor and Malitz [13]. The system PA{Q2), or rather, a system essentially equivalent to it, was first defined and studied by Macintyre [12]. Some of Macintyre's results were obtained independently by Morgenstern [15]. The present paper is essentially self-contained, but all of our results have been directly inspired by those of Macintyre [12].After some preliminaries in §1, we begin in §2 by giving a new completeness proof for PA(Q2). A by-product of our proof is that for every regular uncountable cardinal k, every consistent extension of PA(Q2) has a k-like model in which all classes are definable. (By a class we mean a subset of the universe of the model, every initial segment of which is finite in the sense of the model.)