A result on implications of Σ1-sentences and its application to normal form theorems

1981 ◽  
Vol 46 (3) ◽  
pp. 634-642
Author(s):  
Jean-Yves Girard ◽  
Peter Päppinghaus
Keyword(s):  
Normal Form ◽  
Reflection Principle ◽  
The Other ◽  
Other Hand ◽  
Primitive Recursive Arithmetic ◽  
Incompleteness Theorems ◽  
False Premise ◽  
Primitive Recursive ◽  
Gödel’S Incompleteness Theorems

In this paper we are concerned with the formal provability of certain true implications of Σ1-sentences. Old completeness and incompleteness results already give some information about this. For example by Σ1-completeness of PRA (primitive recursive arithmetic) every true implication of the form D → E, where D is a Σ0-sentence and E a Σ1-sentence, or D a Σ1-sentence and E a true Σ1-sentence, is provable in PRA. On the other hand, by Gödel's incompleteness theorems one can define for every suitable theory S a false Σ1-sentence Ds such that for every false Σ0-sentence E the true implication Ds → E is not provable in S, but is provable in PRA + Cons. So one sees that for a suitable fixed conclusion the provability of true implications of Σ1-sentences depends on the content of the premise.Now we ask, if and how for a suitable fixed premise the provability of true implications of Σ1-sentences depends on the conclusion. As remarked above, by Σ1-completeness of PRA this question is settled, if the premise is true. For a false premise it is answered in § 1 as follows: Let D be a false Σ1-sentence, S an extension of PRA, and S+ ≔ PRA + IAΣ1 + RFNΣ1S). Depending on D and S one can define a Σ1-sentence Es such that S+ ⊢ D → Es, but S ⊬ D → Es provided that S+ is not strong enough to refute D. (IAΣ1 denotes the scheme of the induction axiom for Σ1-formulas, and RFNΣ1(S) the uniform Σ1-reflection principle for S.)

scholarly journals Things that can and things that can’t be done in PRA

1998 ◽  
Vol 5 (18) ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Mathematical Reasoning ◽  
The Other ◽  
Function Parameter ◽  
Recursive Functions ◽  
Other Hand ◽  
Elementary Analysis ◽  
Primitive Recursive Arithmetic ◽  
Primitive Recursive ◽  
Different Levels ◽  
Bounded Sequences

It is well-known by now that large parts of (non-constructive) mathematical reasoning can be carried out in systems T which are conservative over primitive recursive arithmetic PRA (and even much weaker systems). On the other hand there are principles S of elementary analysis (like the Bolzano-Weierstrass principle, the existence of a limit superior for bounded sequences etc.) which are known to be equivalent to arithmetical<br />comprehension (relative to T ) and therefore go far beyond the strength of PRA (when added to T ). <br />In this paper we determine precisely the arithmetical and computational strength (in terms of optimal conservation results and subrecursive characterizations of provably recursive functions) of weaker function parameter-free schematic versions S− of S, thereby<br />exhibiting different levels of strength between these principles as well as a sharp borderline between fragments of analysis which are still conservative over PRA and extensions which just go beyond the strength of PRA.

scholarly journals On mathematical instrumentalism

2005 ◽  
Vol 70 (3) ◽  
pp. 778-794 ◽  
Author(s):  
Patrick Caldon ◽  
Aleksandar Ignjatović
Keyword(s):  
Equational Theory ◽  
Reverse Mathematics ◽  
The Other ◽  
Conservative Extension ◽  
Philosophical View ◽  
Other Hand ◽  
Primitive Recursive Arithmetic ◽  
Speed Up ◽  
Primitive Recursive

AbstractIn this paper we devise some technical tools for dealing with problems connected with the philosophical view usually called mathematical instrumentalism. These tools are interesting in their own right, independently of their philosophical consequences. For example, we show that even though the fragment of Peanos Arithmetic known as IΣ1 is a conservative extension of the equational theory of Primitive Recursive Arithmetic (PRA). IΣ1 has a super-exponential speed-up over PRA. On the other hand, theories studied in the Program of Reverse Mathematics that formalize powerful mathematical principles have only polynomial speed-up over IΣ1.

Closing the GAP—A 5Å Scanning Microscope

1969 ◽  
Vol 27 ◽  
pp. 6-7
Author(s):  
A. V. Crewe
Keyword(s):  
Normal Form ◽  
The Other ◽  
Resolving Power ◽  
Scanning Microscope ◽  
Conventional Microscope ◽  
Other Hand ◽  
Closing The Gap ◽  
Thermal Sources ◽  
Better Than ◽  
Transmission Microscope

We have become accustomed to differentiating between the scanning microscope and the conventional transmission microscope according to the resolving power which the two instruments offer. The conventional microscope is capable of a point resolution of a few angstroms and line resolutions of periodic objects of about 1Å. On the other hand, the scanning microscope, in its normal form, is not ordinarily capable of a point resolution better than 100Å. Upon examining reasons for the 100Å limitation, it becomes clear that this is based more on tradition than reason, and in particular, it is a condition imposed upon the microscope by adherence to thermal sources of electrons.

scholarly journals An application of graphical enumeration to PA

2003 ◽  
Vol 68 (1) ◽  
pp. 5-16
Author(s):  
Andreas Weiermann
Keyword(s):  
Normal Form ◽  
Asymptotic Analysis ◽  
Natural Number ◽  
Proof Theory ◽  
Peano Arithmetic ◽  
First Order ◽  
Primitive Recursive Arithmetic ◽  
Independence Result ◽  
Primitive Recursive

AbstractFor α less than ε0 let Nα be the number of occurrences of ω in the Cantor normal form of α. Further let ∣n∣ denote the binary length of a natural number n, let ∣n∣h denote the h-times iterated binary length of n and let inv(n) be the least h such that ∣n∣h ≤ 2. We show that for any natural number h first order Peano arithmetic, PA, does not prove the following sentence: For all K there exists an M which bounds the lengths n of all strictly descending sequences 〈α0, …, αn〉 of ordinals less than ε0 which satisfy the condition that the Norm Nαi of the i-th term αi is bounded by K + ∣i∣ · ∣i∣i.As a supplement to this (refined Friedman style) independence result we further show that e.g., primitive recursive arithmetic, PRA, proves that for all K there is an M which bounds the length n of any strictly descending sequence 〈α0, …, αn〉 of ordinals less than ε0 which satisfies the condition that the Norm Nαi of the i-th term αi is bounded by K + ∣i∣ · inv(i). The proofs are based on results from proof theory and techniques from asymptotic analysis of Polya-style enumerations.Using results from Otter and from Matoušek and Loebl we obtain similar characterizations for finite bad sequences of finite trees in terms of Otter's tree constant 2.9557652856.…

scholarly journals The Refined Best Reply Correspondence and Backward Induction

2019 ◽  
Vol 20 (1) ◽  
pp. 52-66
Author(s):  
Dieter Balkenborg ◽  
Christoph Kuzmics ◽  
Josef Hofbauer
Keyword(s):  
Normal Form ◽  
Fixed Points ◽  
The Other ◽  
Backward Induction ◽  
Perfect Recall ◽  
Extensive Form ◽  
Remarkable Property ◽  
Extensive Form Games ◽  
Other Hand ◽  
Perfect Equilibria

Abstract Fixed points of the (most) refined best reply correspondence, introduced in Balkenborg et al. (2013), in the agent normal form of extensive form games with perfect recall have a remarkable property. They induce fixed points of the same correspondence in the agent normal form of every subgame. Furthermore, in a well-defined sense, fixed points of this correspondence refine even trembling hand perfect equilibria, while, on the other hand, reasonable equilibria that are not weak perfect Bayesian equilibria are fixed points of this correspondence.

scholarly journals Normalisation holomorphe de structures de Poisson

2009 ◽  
Vol 30 (4) ◽  
pp. 1165-1199
Author(s):  
PHILIPP LOHRMANN
Keyword(s):  
Singular Point ◽  
Normal Form ◽  
Lie Algebra ◽  
Poisson Structure ◽  
Linear Part ◽  
The Other ◽  
Diophantine Condition ◽  
Quadratic Part ◽  
Other Hand ◽  
The One

AbstractWe show that a Poisson structure whose linear part vanishes can be holomorphically normalized in a neighbourhood of its singular point $0\in \Bbb C^n$ if, on the one hand, a Diophantine condition on a Lie algebra associated to the quadratic part is satisfied and, on the other hand, the normal form satisfies some formal conditions.

scholarly journals [The Aesthetics of A Sufist Poet, Ahmad Kamal Abdullah, Known as “Kemala”,] Jamaliah Syair al-Ghazal as-Sufi inda as-Syair Malazia

2018 ◽  
Vol 17 (1) ◽  
pp. 113-151
Author(s):  
Aisyah Ishak ◽  
Rosni Samah
Keyword(s):  
Middle East ◽  
Normal Form ◽  
Analytical Study ◽  
The Other ◽  
Musical Language ◽  
Love Poetry ◽  
Other Hand ◽  
Philosophy Of Love

This article discussed about the aesthetics of a Sufist poet, Ahmad Kamal Abdullah, known as “Kemala”, who is the most famous Modern Malaysian poet. There are a lot of his spiritual and mystical poems collected in several collections or “diwan”. Therefore, this study aims to decipher his mystical poems focused on both of its forms and contents; which studied the form of selected poems consisting of five points: musical, language, intertextuality, symbols and image. On the other hand, it also studied the contents in order to know the ideas that were raised, his emotions, and also its meanings and significance. Finally, the article displayed the results; firstly in terms of its form, the selected poems presented by the poet use several rhymes in one poem, which showed that they are contemporary poems not bonded by a single rhyme, which are the normal form for classical poems. Besides that, the article found that the poet used elements of repetition and question style frequently in his poetry and it is basically related to his emotions and the strength of his feelings. In addition, the poet used the vocabularies and phrases that indicate the meanings of love, besides revealing the element of intertextuality which were affected by Quranic verses, stories of the prophets, the historical incidents, the philosophy of love by Imam al-Ghazali, and also the classical Malay poetry. After that, the study showed that the poet used two types of imageries in his mystical poems; metaphoric and real images, selected by Kemala in order to reveal his feelings of love. Secondly: in term of its contents, it is heavily focused on three important topics; the poetry of love for Allah SWT, the poetry of love for Prophet Muhammad SAW, and the Sufisme love poetry. Last but not least, we can find from this analytical study that Kemala is a Sufist poet revealing his feelings and ideas intellectually and creatively, as well as a Ghazal poet due to selected texts that qualified the Ghazal Poem which is well known in Middle East. Keywords: Mystical Ghazal- Aesthetics- Love- Kemala   تتناول هذه المقالة عن جماليات شعر الغزل الصوفي عند  أحمد كمال عبدالله (Ahmad Kamal Abdullah) المعروف بـ "كمالا" (Kemala)، وهو من أشهر الشعراء الماليزيين في العصر الحديث. واشتهر بقصائده الروحية والصوفية التي جمعت في دواوين عدة بعناوين خاصة. ويعبر في قصائده عما في قلبه من القضايا أو الظواهر من الحوادث والقصص التي حدثت حوله محليا أو عالميا. ولذلك، تهدف هذه المقالة إلى دراسة النصوص الصوفية له شكلا ومضمونا، فمن حيث الشكل تدرس المستوى الموسيقي، والمستوى اللغوي، والتناص، والرمز، والصورة. وأما دراسة مضامين النصوص فهي تركز على  تجلية جماليتها، ومعرفة أفكار الشاعر المطروحة فيها، وعواطفه، وأيضا دلالة تلك الأفكار. وتوصلت المقالة إلى عدة نتائج؛ وأهمها؛ أن النصوص من ظواهرها الشكلية تكونت من القوافي غير الموحدة المعروفة في شعر الحر، مع إيجاد ظاهرة التكرار فيها مؤديا إلى معانيها الخاصة مرتبطة بالواقع النفسي عند الشاعر، وكثرة استخدام أساليبب الاستفهام، واستخدام المفردات والعبارات التي تدل على الحب والشوق والعشق، ومع وجود ظاهرة التناص من قصص الأنبياء والحوادث التاريخية وفلسفة الحب والشوق عند الإمام الغزالي، وبجانب الاستفادة من معاني الآية القرأنية والتراث الشعري القديم. كما استخدم الشاعر صورا تشبيهية وفنية في نقل أفكاره إلى المتلقي بطريقة فعالة تثير القلب واستخدام رموز متعددة. وأن نصوص الغزل الصوفي عند كمالا قد تحدثت عن ثلاثة موضوعات وهي شعر الحب الإلهي، والحب النبوي، والسكر الصوفي. والخلاصة، يعد كمالا شاعرا صوفيا مبدعا مثقفا، وكذلك شاعر الغزل الذي يقدم حبه الخالص للمحبوب والعشق به بكل قدسية،  وبطريقة فعالة بالأساليب الجذابة.   الكلمات المفتاحية: الغزل الصوفي – جماليات – الحب – العشق – كمالا

A hierarchy for the 1-section of any type two object

1974 ◽  
Vol 39 (1) ◽  
pp. 88-94 ◽  
Author(s):  
S. S. Wainer
Keyword(s):  
Affirmative Answer ◽  
The Other ◽  
Uniqueness Property ◽  
Hyperarithmetical Hierarchy ◽  
Other Hand ◽  
Primitive Recursion ◽  
Ordinal Notations ◽  
Definition Of ◽  
Image Position ◽  
Primitive Recursive

Shoenfield [6] constructed a hierarchy for any type two object F in which 2E is recursive by generalizing the hyperarithmetical hierarchy, using a jump operation jF defined by(A similar hierarchy was constructed independently by Hinman [3].) In order to construct a hierarchy for an arbitrary type two F we must first associate with F an operator which, on one hand will always be recursive in F, and on the other hand will generate all the functions recursive in F when iterated over a simultaneously generated set of ordinal notations OF. Clearly the use of 2E in the above definition of jF(f) can be avoided if, instead of diagonalizing F over all functions recursive in f, we only diagonalize F over, say, the functions primitive recursive in f. If furthermore we code in a function which enumerates all functions primitive recursive in f then the resulting operation will certainly generate a primitive recursively expanding hierarchy of functions recursive in F. The problem that remains is whether this hierarchy will exhaust the 1-section of F. But this reduces to an effectiyized version of the following problem: If g is recursive in some level of the hierarchy, is g primitive recursive in some higher level ? An affirmative answer is suggested by the completeness results of Feferman [2], and our main theorem below will be proved by combining his ideas with those of Shoenfield [6]. The result is a hierarchy which applies to all type two objects, and which replaces the notion of recursion in F by the simpler notion of primitive recursion in certain functions generated by F. Unfortunately, in contrast with the Shoenfield hierarchy, the hierarchy developed here cannot always be expected to have the uniqueness property (even w.r.t. ≤T), and for this reason the proof of our main theorem is rather more complicated than the corresponding proof in [6].

A study of cometary eruptions through OH radio-lines

1999 ◽  
Vol 173 ◽  
pp. 249-254
Author(s):  
A.M. Silva ◽  
R.D. Miró
Keyword(s):  
Short Duration ◽  
Growth Curves ◽  
The Other ◽  
Initial Mass ◽  
Radio Lines ◽  
Other Hand ◽  
Sudden Rise

AbstractWe have developed a model for theH2OandOHevolution in a comet outburst, assuming that together with the gas, a distribution of icy grains is ejected. With an initial mass of icy grains of 108kg released, theH2OandOHproductions are increased up to a factor two, and the growth curves change drastically in the first two days. The model is applied to eruptions detected in theOHradio monitorings and fits well with the slow variations in the flux. On the other hand, several events of short duration appear, consisting of a sudden rise ofOHflux, followed by a sudden decay on the second day. These apparent short bursts are frequently found as precursors of a more durable eruption. We suggest that both of them are part of a unique eruption, and that the sudden decay is due to collisions that de-excite theOHmaser, when it reaches the Cometopause region located at 1.35 × 105kmfrom the nucleus.

Life Beyond 1MeV

1980 ◽  
Vol 38 ◽  
pp. 30-33
Author(s):  
K.H. Westmacott
Keyword(s):  
Electron Microscopy ◽  
Past Experience ◽  
The Other ◽  
Good Case ◽  
Other Hand ◽  
The U.S

Life beyond 1MeV – like life after 40 – is not too different unless one takes advantage of past experience and is receptive to new opportunities. At first glance, the returns on performing electron microscopy at voltages greater than 1MeV diminish rather rapidly as the curves which describe the well-known advantages of HVEM often tend towards saturation. However, in a country with a significant HVEM capability, a good case can be made for investing in instruments with a range of maximum accelerating voltages. In this regard, the 1.5MeV KRATOS HVEM being installed in Berkeley will complement the other 650KeV, 1MeV, and 1.2MeV instruments currently operating in the U.S. One other consideration suggests that 1.5MeV is an optimum voltage machine – Its additional advantages may be purchased for not much more than a 1MeV instrument. On the other hand, the 3MeV HVEM's which seem to be operated at 2MeV maximum, are much more expensive.

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