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.

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.

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.

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.

An Interview with the Physicist that Her Head in the Universe and Both Feet on the Earth

2019 ◽  
Author(s):  
Adib Rifqi Setiawan
Keyword(s):  
Standard Model ◽  
Particle Physics ◽  
Space Time ◽  
Additional Dimension ◽  
The Earth ◽  
The Standard Model ◽  
The Universe

Put simply, Lisa Randall’s job is to figure out how the universe works, and what it’s made of. Her contributions to theoretical particle physics include two models of space-time that bear her name. The first Randall–Sundrum model addressed a problem with the Standard Model of the universe, and the second concerned the possibility of a warped additional dimension of space. In this work, we caught up with Randall to talk about why she chose a career in physics, where she finds inspiration, and what advice she’d offer budding physicists. This article has been edited for clarity. My favourite quote in this interview is, “Figure out what you enjoy, what your talents are, and what you’re most curious to learn about.” If you insterest in her work, you can contact her on Twitter @lirarandall.

An Interview with the Physicist that Her Head in the Universe and Both Feet on the Earth

2019 ◽  
Author(s):  
Adib Rifqi Setiawan
Keyword(s):  
Standard Model ◽  
Particle Physics ◽  
Space Time ◽  
Additional Dimension ◽  
The Earth ◽  
The Standard Model ◽  
The Universe

Put simply, Lisa Randall’s job is to figure out how the universe works, and what it’s made of. Her contributions to theoretical particle physics include two models of space-time that bear her name. The first Randall–Sundrum model addressed a problem with the Standard Model of the universe, and the second concerned the possibility of a warped additional dimension of space. In this work, we caught up with Randall to talk about why she chose a career in physics, where she finds inspiration, and what advice she’d offer budding physicists. This article has been edited for clarity. My favourite quote in this interview is, “Figure out what you enjoy, what your talents are, and what you’re most curious to learn about.” If you insterest in her work, you can contact her on Twitter @lirarandall.

scholarly journals Complete leading-order standard model corrections to quantum leptogenesis

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Paul Frederik Depta ◽  
Andreas Halsch ◽  
Janine Hütig ◽  
Sebastian Mendizabal ◽  
Owe Philipsen
Keyword(s):  
Standard Model ◽  
Gauge Coupling ◽  
Thermal Bath ◽  
Leading Order ◽  
Boltzmann Equations ◽  
The Standard Model ◽  
Majorana Neutrinos ◽  
Infinite Loop ◽  
First Time ◽  
The Universe

Abstract Thermal leptogenesis, in the framework of the standard model with three additional heavy Majorana neutrinos, provides an attractive scenario to explain the observed baryon asymmetry in the universe. It is based on the out-of-equilibrium decay of Majorana neutrinos in a thermal bath of standard model particles, which in a fully quantum field theoretical formalism is obtained by solving Kadanoff-Baym equations. So far, the leading two-loop contributions from leptons and Higgs particles are included, but not yet gauge corrections. These enter at three-loop level but, in certain kinematical regimes, require a resummation to infinite loop order for a result to leading order in the gauge coupling. In this work, we apply such a resummation to the calculation of the lepton number density. The full result for the simplest “vanilla leptogenesis” scenario is by $$ \mathcal{O} $$ O (1) increased compared to that of quantum Boltzmann equations, and for the first time permits an estimate of all theoretical uncertainties. This step completes the quantum theory of leptogenesis and forms the basis for quantitative evaluations, as well as extensions to other scenarios.

scholarly journals Theoretical uncertainties for cosmological first-order phase transitions

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Djuna Croon ◽  
Oliver Gould ◽  
Philipp Schicho ◽  
Tuomas V. I. Tenkanen ◽  
Graham White
Keyword(s):  
Phase Transitions ◽  
Standard Model ◽  
Scale Dependence ◽  
Effective Field ◽  
Bubble Nucleation ◽  
Perturbative Approach ◽  
First Order ◽  
The Standard Model ◽  
First Order Phase ◽  
Renormalisation Scale

Abstract We critically examine the magnitude of theoretical uncertainties in perturbative calculations of fist-order phase transitions, using the Standard Model effective field theory as our guide. In the usual daisy-resummed approach, we find large uncertainties due to renormalisation scale dependence, which amount to two to three orders-of-magnitude uncertainty in the peak gravitational wave amplitude, relevant to experiments such as LISA. Alternatively, utilising dimensional reduction in a more sophisticated perturbative approach drastically reduces this scale dependence, pushing it to higher orders. Further, this approach resolves other thorny problems with daisy resummation: it is gauge invariant which is explicitly demonstrated for the Standard Model, and avoids an uncontrolled derivative expansion in the bubble nucleation rate.

scholarly journals Electroweak-like baryogenesis with new chiral matter

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Kohei Fujikura ◽  
Keisuke Harigaya ◽  
Yuichiro Nakai ◽  
Ruoquan Wang
Keyword(s):  
Phase Transition ◽  
Gauge Symmetry ◽  
Cp Violation ◽  
Standard Model ◽  
Dipole Moments ◽  
Dark Matter Candidate ◽  
Lepton Asymmetry ◽  
First Order ◽  
The Standard Model ◽  
Concrete Realization

Abstract We propose a framework where a phase transition associated with a gauge symmetry breaking that occurs (not far) above the electroweak scale sets a stage for baryogenesis similar to the electroweak baryogenesis in the Standard Model. A concrete realization utilizes the breaking of SU(2)R× U(1)X→ U(1)Y. New chiral fermions charged under the extended gauge symmetry have nonzero lepton numbers, which makes the B − L symmetry anomalous. The new lepton sector contains a large flavor-dependent CP violation, similar to the Cabibbo-Kobayashi-Maskawa phase, without inducing sizable electric dipole moments of the Standard Model particles. A bubble wall dynamics associated with the first-order phase transition and SU(2)R sphaleron processes generate a lepton asymmetry, which is transferred into a baryon asymmetry via the ordinary electroweak sphaleron process. Unlike the Standard Model electroweak baryogenesis, the new phase transition can be of the strong first order and the new CP violation is not significantly suppressed by Yukawa couplings, so that the observed asymmetry can be produced. The model can be probed by collider searches for new particles and the observation of gravitational waves. One of the new leptons becomes a dark matter candidate. The model can be also embedded into a left-right symmetric theory to solve the strong CP problem.

scholarly journals Relativistic Cosmology with an Introduction to Inflation

Universe ◽  
2021 ◽  
Vol 7 (8) ◽  
pp. 276
Author(s):  
Muhammad Zahid Mughal ◽  
Iftikhar Ahmad ◽  
Juan Luis García Guirao
Keyword(s):  
Standard Model ◽  
Early Universe ◽  
Large Scale ◽  
Initial Conditions ◽  
Big Bang ◽  
Relativistic Cosmology ◽  
The Standard Model ◽  
Big Bang Model ◽  
The Big Bang ◽  
The Universe

In this review article, the study of the development of relativistic cosmology and the introduction of inflation in it as an exponentially expanding early phase of the universe is carried out. We study the properties of the standard cosmological model developed in the framework of relativistic cosmology and the geometric structure of spacetime connected coherently with it. The geometric properties of space and spacetime ingrained into the standard model of cosmology are investigated in addition. The big bang model of the beginning of the universe is based on the standard model which succumbed to failure in explaining the flatness and the large-scale homogeneity of the universe as demonstrated by observational evidence. These cosmological problems were resolved by introducing a brief acceleratedly expanding phase in the very early universe known as inflation. The cosmic inflation by setting the initial conditions of the standard big bang model resolves these problems of the theory. We discuss how the inflationary paradigm solves these problems by proposing the fast expansion period in the early universe. Further inflation and dark energy in fR modified gravity are also reviewed.

Ring-diagram summations in the finite-temperature effective potential

1993 ◽  
Vol 71 (5-6) ◽  
pp. 227-236 ◽  
Author(s):  
M. E. Carrington
Keyword(s):  
Phase Transition ◽  
Standard Model ◽  
Effective Potential ◽  
Finite Temperature ◽  
Electroweak Phase Transition ◽  
First Order ◽  
Ring Diagram ◽  
First Order Phase Transition ◽  
The Standard Model ◽  
The One

There has been much recent interest in the finite-temperature effective potential of the standard model in the context of the electroweak phase transition. We review the calculation of the effective potential with particular emphasis on the validity of the expansions that are used. The presence of a term that is cubic in the Higgs condensate in the one-loop effective potential appears to indicate a first-order electroweak phase transition. However, in the high-temperature regime, the infrared singularities inherent in massless models produce cubic terms that are of the same order in the coupling. In this paper, we discuss the inclusion of an infinite set of these terms via the ring-diagram summation, and show that the standard model has a first-order phase transition in the weak coupling expansion.

Sign in / Sign up

Export Citation Format

Share Document