On the complexity of models of arithmetic

1982 ◽  
Vol 47 (2) ◽  
pp. 403-415 ◽  
Author(s):  
Kenneth McAloon
Keyword(s):  
Standard Model ◽  
Initial Segment ◽  
Peano Arithmetic ◽  
Nonstandard Model ◽  
Recursively Enumerable ◽  
The Standard Model ◽  
Models Of Arithmetic

AbstractLet P0 be the subsystem of Peano arithmetic obtained by restricting induction to bounded quantifier formulas. Let M be a countable, nonstandard model of P0 whose domain we suppose to be the standard integers. Let T be a recursively enumerable extension of Peano arithmetic all of whose existential consequences are satisfied in the standard model. Then there is an initial segment M′ of M which is a model of T such that the complete diagram of M′ is Turing reducible to the atomic diagram of M. Moreover, neither the addition nor the multiplication of M is recursive.

Flipping properties in arithmetic

1982 ◽  
Vol 47 (2) ◽  
pp. 416-422 ◽  
Author(s):  
L. A. S. Kirby
Keyword(s):  
Standard Model ◽  
Set Theory ◽  
Initial Segment ◽  
Peano Arithmetic ◽  
Combinatorial Properties ◽  
Nonstandard Model ◽  
Standard Part ◽  
The Standard Model ◽  
Nonstandard Models ◽  
Image Position

Flipping properties were introduced in set theory by Abramson, Harrington, Kleinberg and Zwicker [1]. Here we consider them in the context of arithmetic and link them with combinatorial properties of initial segments of nonstandard models studied in [3]. As a corollary we obtain independence resutls involving flipping properties.We follow the notation of the author and Paris in [3] and [2], and assume some knowledge of [3]. M will denote a countable nonstandard model of P (Peano arithmetic) and I will be a proper initial segment of M. We denote by N the standard model or the standard part of M. X ↑ I will mean that X is unbounded in I. If X ⊆ M is coded in M and M ≺ K, let X(K) be the subset of K coded in K by the element which codes X in M. So X(K) ⋂ M = X.Recall that M ≺IK (K is an I-extension of M) if M ≺ K and for some c∈K,In [3] regular and strong initial segments are defined, and among other things it is shown that I is regular if and only if there exists an I-extension of M.

Two theorems on degrees of models of true arithmetic

1984 ◽  
Vol 49 (2) ◽  
pp. 425-436 ◽  
Author(s):  
Julia Knight ◽  
Alistair H. Lachlan ◽  
Robert I. Soare
Keyword(s):  
Standard Model ◽  
Binary Operation ◽  
Peano Arithmetic ◽  
Turing Degree ◽  
First Order ◽  
Nonstandard Model ◽  
Basis Theorem ◽  
The Standard Model ◽  
The Universe ◽  
Countable Models

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.

UNIVERSAL ROSSER PREDICATES

2017 ◽  
Vol 82 (1) ◽  
pp. 292-302 ◽  
Author(s):  
MAKOTO KIKUCHI ◽  
TAISHI KURAHASHI
Keyword(s):  
Standard Model ◽  
Initial Segment ◽  
Nonstandard Model ◽  
Complete Extension ◽  
The Standard Model ◽  
Nonstandard Models ◽  
Incompleteness Theorems ◽  
Image Position ◽  
Gödel’S Incompleteness Theorems

AbstractGödel introduced the original provability predicate in the proofs of Gödel’s incompleteness theorems, and Rosser defined a new one. They are equivalent in the standard model ${\mathbb N}$ of arithmetic or any nonstandard model of ${\rm PA} + {\rm Con_{PA}} $, but the behavior of Rosser’s provability predicate is different from the original one in nonstandard models of ${\rm PA} + \neg {\rm Con_{PA}} $. In this paper, we investigate several properties of the derivability conditions for Rosser provability predicates, and prove the existence of a Rosser provability predicate with which we can define any consistent complete extension of ${\rm PA}$ in some nonstandard model of ${\rm PA} + \neg {\rm Con_{PA}} $. We call it a universal Rosser predicate. It follows from the theorem that the true arithmetic ${\rm TA}$ can be defined as the set of theorems of ${\rm PA}$ in terms of a universal Rosser predicate in some nonstandard model of ${\rm PA} + \neg {\rm Con_{PA}} $. By using this theorem, we also give a new proof of a theorem that there is a nonstandard model M of ${\rm PA} + \neg {\rm Con_{PA}} $ such that if N is an initial segment of M which is a model of ${\rm PA} + {\rm Con_{PA}} $ then every theorem of ${\rm PA}$ in N is a theorem of $\rm PA$ in ${\mathbb N}$. In addition, we prove that there is a Rosser provability predicate such that the set of theorems of $\rm PA$ in terms of the Rosser provability predicate is inconsistent in any nonstandard model of ${\rm PA} + \neg {\rm Con_{PA}} $.

scholarly journals ULTRAHIGH ENERGY NEUTRINOS

2003 ◽  
Vol 18 (22) ◽  
pp. 4085-4096 ◽  
Author(s):  
SHARADA IYER DUTTA ◽  
MARY HALL RENO ◽  
INA SARCEVIC
Keyword(s):  
Black Hole ◽  
Standard Model ◽  
Cross Section ◽  
Neutrino Oscillations ◽  
Ultrahigh Energy ◽  
Nonstandard Model ◽  
Air Shower ◽  
Energy Neutrino ◽  
The Standard Model ◽  
Event Rates

The ultrahigh energy neutrino cross section is well understood in the standard model for neutrino energies up to 1012 GeV, Tests of neutrino oscillations (νμ ↔ ντ) from extragalactic sources of neutrinos are possible with large underground detectors. Measurements of horizontal air shower event rates at neutrino energies above 1010 GeV will be able to constrain nonstandard model contributions to the neutrino-nucleon cross section, e.g., from mini-black hole production.

On models of arithmetic—Answers to two problems raised by H. Gaifman

1975 ◽  
Vol 40 (1) ◽  
pp. 41-47 ◽  
Author(s):  
Alex Wilkie
Keyword(s):  
Diophantine Equation ◽  
Initial Segment ◽  
Original Model ◽  
New Model ◽  
Recursive Functions ◽  
Nonstandard Model ◽  
Definable Functions ◽  
Models Of Arithmetic

In a recent paper [3] H. Gaifman investigated some model theoretic consequences of Matijasevič's theorem [5], and posed some further problems which naturally arise. We provide here partial answers to two of these problems, the results having been previously announced in the postscript of [3].Firstly, it is shown in [3] that if M1 and M2 are models of the Peano axioms P and M1 ⊆ M2, then M1 is closed under the recursive functions of M2. The converse of this statement is false. Moreover, Gaifman asks: Is every initial segment of a model M of P which is closed under the recursive functions of M (or the ∑n-definable functions) also a model of P? We show that this is false and our method gives, en route, another proof of a theorem of Rabin [7] stating the P is not implied by any consistent set of ∑n sentences for any n.Secondly, we partially answer a question posed on p. 129 of [3] by proving (some-what more than) every countable nonstandard model of P has an end extension in which a diophantine equation, not solvable in the original model, has a solution. We can, in fact, take the new model to be isomorphic to the original one. This generalises (apart from the countability restriction) a theorem of Rabin [6].

ILLUSORY MODELS OF PEANO ARITHMETIC

2016 ◽  
Vol 81 (3) ◽  
pp. 1163-1175 ◽  
Author(s):  
MAKOTO KIKUCHI ◽  
TAISHI KURAHASHI
Keyword(s):  
Initial Segment ◽  
Peano Arithmetic ◽  
Nonstandard Model

AbstractBy using a provability predicate of PA, we define ThmPA(M) as the set of theorems of PA in a model M of PA. We say a model M of PA is (1) illusory if ThmPA(M) ⊈ ThmPA(ℕ), (2) heterodox if ThmPA(M) ⊈ TA, (3) sane if M ⊨ ConPA, and insane if it is not sane, (4) maximally sane if it is sane and ThmPA(M) ⊆ ThmPA(N) implies ThmPA(M) = ThmPA(N) for every sane model N of PA. We firstly show that M is heterodox if and only if it is illusory, and that ThmPA(M) ∩ TA ≠ ThmPA(ℕ) for any illusory model M. Then we show that there exists a maximally sane model, every maximally sane model satisfies ¬ConPA+ConPA, and there exists a sane model of ¬ConPA+ConPA which is not maximally sane. We define that an insane model is (5) illusory by nature if its every initial segment being a nonstandard model of PA is illusory, and (6) going insane suddenly if its every initial segment being a sane model of PA is not illusory. We show that there exists a model of PA which is illusory by nature, and we prove the existence of a model of PA which is going insane suddenly.

scholarly journals IMPLICATIONS OF THE HIGGS DISCOVERY FOR GRAVITY AND COSMOLOGY

2013 ◽  
Vol 22 (12) ◽  
pp. 1342017 ◽  
Author(s):  
DEJAN STOJKOVIC
Keyword(s):  
Standard Model ◽  
Particle Physics ◽  
Hadron Collider ◽  
Strongly Coupled ◽  
Nonstandard Model ◽  
Large Hadron Collider Data ◽  
The Standard Model ◽  
Light Scalar ◽  
Perturbative Quantum ◽  
Invisible Decays

The discovery of the Higgs boson is one of the greatest discoveries in this century. The standard model is finally complete. Apart from its significance in particle physics, this discovery has profound implications for gravity and cosmology in particular. Many perturbative quantum gravity interactions involving scalars are not suppressed by powers of Planck mass. Since gravity couples anything with mass to anything with mass, then Higgs must be strongly coupled to any other fundamental scalar in nature, even if the gauge couplings are absent in the original Lagrangian. Since the Large Hadron Collider data indicate that the Higgs is very much standard model-like, there is very little room for nonstandard model processes, e.g. invisible decays. This severely complicates any model that involves light enough scalar that the Higgs can kinematically decay to. Most notably, these are the quintessence models, models including light axions, and light scalar dark matter models.

Large discrete parts of the E-tree

1988 ◽  
Vol 53 (3) ◽  
pp. 980-984 ◽  
Author(s):  
Harold Simmons
Keyword(s):  
Standard Model ◽  
Peano Arithmetic ◽  
Unique Root ◽  
First Order ◽  
Aronszajn Tree ◽  
The Standard Model

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.

The intersection of nonstandard models of arithmetic

1972 ◽  
Vol 37 (1) ◽  
pp. 103-106 ◽  
Author(s):  
Andreas Blass
Keyword(s):  
Standard Model ◽  
Single Element ◽  
Main Result Theorem ◽  
Order Language ◽  
The Standard Model ◽  
Elementary Embeddings ◽  
Nonstandard Models ◽  
Result Theorem ◽  
Canonical Image ◽  
Models Of Arithmetic

If two nonstandard models of complete arithmetic are elementarily embedded in a third, then their intersection may be considerably smaller than either of them; indeed, the intersection may be only the standard model. For example, if D and E are nonprincipal ultrafilters on ω, then the nonstandard models D-prod and E-prod (where is the standard model) have canonical elementary embeddings into D-prod (E-prod , and the intersection of their images is easily seen to be the (canonical image of the) standard model. In this paper, we shall prove that, under certain conditions, this phenomenon will not occur. Our main result (Theorem 3) is that the intersection of countably many pairwise cofinal models is itself cofinal with these models, provided that at least one of them is generated by a single element. (Precise definitions will be given below.)The theorems in this paper were first formulated in terms of ultrafilters, then rephrased (using the methods of Chapter III of [1]) as statements about ultra-powers of , and finally generalized to their present form. Since the theorems and their proofs are now entirely model-theoretic, they are presented here separately from the study of ultrafilters in which they originated. That study, including applications of the present results, will appear in [2].Let L be the first-order language whose n-place relation symbols are all the relations R ⊆; ωn and whose n-place function symbols are all the functions f: ωn → ω. Let be the standard model for L; its universe is ω and every nonlogical symbol of L denotes itself. Let be an elementary extension of . The relation (or function) denoted by R (or f) in will be called *R (or *f).

Torre models in the isols

1994 ◽  
Vol 59 (1) ◽  
pp. 140-150 ◽  
Author(s):  
Joseph Barback
Keyword(s):  
Number Theory ◽  
Structural Property ◽  
Recursive Function ◽  
Peano Arithmetic ◽  
The Other ◽  
Close Connection ◽  
New Variety ◽  
Nonstandard Model ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets

AbstractIn [14] J. Hirschfeld established the close connection of models of the true AE sentences of Peano Arithmetic and homomorphic images of the semiring of recursive functions. This fragment of Arithmetic includes most of the familiar results of classical number theory. There are two nice ways that such models appear in the isols. One way was introduced by A. Nerode in [20] and is referred to in the literature as Nerode Semirings. The other way is called a tame model. It is very similar to a Nerode Semiring and was introduced in [6]. The model theoretic properties of Nerode Semirings and tame models have been widely studied by T. G. McLaughlin ([16], [17], and [18]).In this paper we introduce a new variety of tame model called a torre model. It has as a generator an infinite regressive isol with a nice structural property relative to recursively enumerable sets and their extensions to the isols. What is then obtained is a nonstandard model in the isols of the fragment of Peano Arithmetic with the following property: Let T be a torre model. Let f be any recursive function, and let fΛ be its extension to the isols. If there is an isol A with fΛ(A) ϵ T, then there is also an isol B ϵ T with fΛ(B) = fΛ(A).

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