Some independence results for Peano arithmetic

1978 ◽  
Vol 43 (4) ◽  
pp. 725-731 ◽  
Author(s):  
J. B. Paris
Keyword(s):  
Significant Contribution ◽  
Peano Arithmetic ◽  
Natural Numbers ◽  
Theoretic Method ◽  
First Order ◽  
Detailed Proof ◽  
Finite Games ◽  
Independence Results ◽  
First Time

In this paper we shall outline a purely model theoretic method for obtaining independence results for Peano's first order axioms (P). The method is of interest in that it provides for the first time elementary combinatorial statements about the natural numbers which are not provable in P. We give several examples of such statements.Central to this exposition will be the notion of an indicator. Indicators were introduced by L. Kirby and the author in [3] although they had occurred implicitly in earlier papers, for example Friedman [1]. The main result on indicators which we shall need (Lemma 1) was proved by Laurie Kirby and the author in the summer of 1976 but it was not until early in the following year that the author realised that this lemma could be used to give independence results.The first combinatorial independence results obtained were essentially statements about certain finite games and consequently were not immediately meaningful (see Example 2). This shortcoming was remedied by Leo Harrington who, upon hearing an incorrect version of our results, noticed a beautifully simply independent combinatorial statement. We outline this result in Example 3. An alternative, more detailed, proof may be found in [5].Clearly Laurie Kirby and Leo Harrington have made a very significant contribution to this paper and we wish to express our sincere thanks to them.

Tarski’s Theorem for Arithmetic

1992 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Peano Arithmetic ◽  
Finite Alphabet ◽  
Universal Quantifier ◽  
Natural Numbers ◽  
Successor Function ◽  
Material Implication ◽  
Second Order Arithmetic ◽  
First Order ◽  
Order Arithmetic ◽  
Concrete Language

In the last chapter, we dealt with mathematical languages in considerable generality. We shall now turn to some particular mathematical languages. One of our goals is to reach Gödel’s incompleteness theorem for the particular system known as Peano Arithmetic. We shall give several proofs of this important result; the simplest one is based partly on Tarski’s theorem, to which we first turn. The first concrete language that we will study is the language of first order arithmetic based on addition, multiplication and exponentiation. [We also take as primitive the successor function and the less than or equal to relation, but these are inessential.] We shall formulate the language using only a finite alphabet (mainly for purposes of a convenient Gödel numbering); specifically we use the following 13 symbols. . . . 0’ ( ) f, υ ∽ ⊃ ∀ = ≤ # . . . The expressions 0, 0′, 0″, 0‴, · · · are called numerals and will serve as formal names of the respective natural numbers 0, 1, 2, 3, · · ·. The accent symbol (also called the prime) is serving as a name of the successor function. We also need names for the operations of addition, multiplication and exponentiation; we use the expressions f′, f″, f‴ as respective names of these three functions. We abbreviate f′ by the familiar “+”; we abbreviate f’’ by the familiar dot and f‴ by the symbol “E”. The symbols ~ and ⊃ are the familiar symbols from prepositional logic, standing for negation and material implication, respectively. [For any reader not familiar with the use of the horseshoe symbol, for any propositions p and q, the propositions p ⊃ q is intended to mean nothing more nor less than that either p is false, or p and q are both true.] The symbol ∀ is the universal quantifier and means “for all.” We will be quantifying only over natural numbers not over sets or relations on the natural numbers. [Technically, we are working in first-order arithmetic, not second-order arithmetic.] The symbol “=” is used, as usual, to denote the identity relation, and “≤” is used, as usual, to denote the “less than or equal to” relation.

The Natural Numbers

2018 ◽  
Author(s):  
Øystein Linnebo
Keyword(s):  
Natural Number ◽  
Peano Arithmetic ◽  
Number Sequence ◽  
Natural Numbers ◽  
Ordinal Numbers ◽  
Cardinal Numbers

How are the natural numbers individuated? That is, what is our most basic way of singling out a natural number for reference in language or in thought? According to Frege and many of his followers, the natural numbers are cardinal numbers, individuated by the cardinalities of the collections that they number. Another answer regards the natural numbers as ordinal numbers, individuated by their positions in the natural number sequence. Some reasons to favor the second answer are presented. This answer is therefore developed in more detail, involving a form of abstraction on numerals. Based on this answer, a justification for the axioms of Dedekind–Peano arithmetic is developed.

scholarly journals Non-Linear First-Order Differential Boundary Problems with Multipoint and Integral Conditions

2021 ◽  
Vol 5 (1) ◽  
pp. 15
Author(s):  
Misir J. Mardanov ◽  
Yagub A. Sharifov ◽  
Yusif S. Gasimov ◽  
Carlo Cattani
Keyword(s):  
Integral Equation ◽  
Integral Boundary Conditions ◽  
Contraction Mapping Principle ◽  
Boundary Problems ◽  
Integral Boundary ◽  
Multipoint Problem ◽  
First Order ◽  
Banach Contraction ◽  
First Time ◽  
Banach Contraction Mapping Principle

This paper considers boundary value problem (BVP) for nonlinear first-order differential problems with multipoint and integral boundary conditions. A suitable Green function was constructed for the first time in order to reduce this problem into a corresponding integral equation. So that by using the Banach contraction mapping principle (BCMP) and Schaefer’s fixed point theorem (SFPT) on the integral equation, we can show that the solution of the multipoint problem exists and it is unique.

PAL – Propositional Algorithmic Logic

1981 ◽  
Vol 4 (3) ◽  
pp. 675-760
Author(s):  
Grażyna Mirkowska
Keyword(s):  
Data Structures ◽  
Completeness Theorem ◽  
Natural Numbers ◽  
First Order ◽  
Algorithmic Logic ◽  
Axiomatic Systems ◽  
Nondeterministic Program ◽  
Basic Sets

The aim of propositional algorithmic logic is to investigate the properties of program connectives. Complete axiomatic systems for deterministic as well as for nondeterministic interpretations of program variables are presented. They constitute basic sets of tools useful in the practice of proving the properties of program schemes. Propositional theories of data structures, e.g. the arithmetic of natural numbers and stacks, are constructed. This shows that in many aspects PAL is close to first-order algorithmic logic. Tautologies of PAL become tautologies of algorithmic logic after replacing program variables by programs and propositional variables by formulas. Another corollary to the completeness theorem asserts that it is possible to eliminate nondeterministic program variables and replace them by schemes with deterministic atoms.

The kinetics of the decarboxylation of β-resorcylic acid in catechol

1976 ◽  
Vol 29 (2) ◽  
pp. 443 ◽  
Author(s):  
MA Haleem ◽  
MA Hakeem
Keyword(s):  
Rate Equation ◽  
Kinetic Data ◽  
Reaction Rate ◽  
Complex Mechanism ◽  
First Order ◽  
Absolute Reaction Rate ◽  
First Time ◽  
Kinetics Of ◽  
Absolute Reaction ◽  
Reaction Rate Equation

Kinetic data are reported for the decarboxylation of β-resorcylic acid in resorcinol and catechol for the first time. The reaction is first order. The observation supports the view that the decomposition proceeds through an intermediate complex mechanism. The parameters of the absolute reaction rate equation are calculated.

scholarly journals Quantum Gravity at the Corner

Universe ◽  
2018 ◽  
Vol 4 (10) ◽  
pp. 107 ◽  
Author(s):  
Laurent Freidel ◽  
Alejandro Perez
Keyword(s):  
Quantum Gravity ◽  
Symplectic Form ◽  
Direct Application ◽  
Quantum Geometry ◽  
Quantum Level ◽  
Gravitational System ◽  
Gauge Transformations ◽  
First Order ◽  
Conformal Field ◽  
First Time

We investigate the quantum geometry of a 2d surface S bounding the Cauchy slices of a 4d gravitational system. We investigate in detail for the first time the boundary symplectic current that naturally arises in the first-order formulation of general relativity in terms of the Ashtekar–Barbero connection. This current is proportional to the simplest quadratic form constructed out of the pull back to S of the triad field. We show that the would-be-gauge degrees of freedo arising from S U ( 2 ) gauge transformations plus diffeomorphisms tangent to the boundary are entirely described by the boundary 2-dimensional symplectic form, and give rise to a representation at each point of S of S L ( 2 , R ) × S U ( 2 ) . Independently of the connection with gravity, this system is very simple and rich at the quantum level, with possible connections with conformal field theory in 2d. A direct application of the quantum theory is modelling of the black horizons in quantum gravity.

scholarly journals Measurements of hydrogen cyanide (HCN) and acetylene (C<sub>2</sub>H<sub>2</sub>) from the Infrared Atmospheric Sounding Interferometer (IASI)

2013 ◽  
Vol 6 (4) ◽  
pp. 917-925 ◽  
Author(s):  
V. Duflot ◽  
D. Hurtmans ◽  
L. Clarisse ◽  
Y. R'honi ◽  
C. Vigouroux ◽  
...  
Keyword(s):  
Fourier Transform ◽  
Hydrogen Cyanide ◽  
Atmospheric Transport ◽  
Trace Gases ◽  
First Order ◽  
Line Mixing ◽  
Atmospheric Sounding ◽  
First Time ◽  
Order Comparison ◽  
Emission Ratios

Abstract. Hydrogen cyanide (HCN) and acetylene (C2H2) are ubiquitous atmospheric trace gases with medium lifetime, which are frequently used as indicators of combustion sources and as tracers for atmospheric transport and chemistry. Because of their weak infrared absorption, overlapped by the CO2 Q branch near 720 cm−1, nadir sounders have up to now failed to measure these gases routinely. Taking into account CO2 line mixing, we provide for the first time extensive measurements of HCN and C2H2 total columns at Reunion Island (21° S, 55° E) and Jungfraujoch (46° N, 8° E) in 2009–2010 using observations from the Infrared Atmospheric Sounding Interferometer (IASI). A first order comparison with local ground-based Fourier transform infraRed (FTIR) measurements has been carried out allowing tests of seasonal consistency which is reasonably captured, except for HCN at Jungfraujoch. The IASI data shows a greater tendency to high C2H2 values. We also examine a nonspecific biomass burning plume over austral Africa and show that the emission ratios with respect to CO agree with previously reported values.

Strong extension axioms and Shelah's zero-one law for choiceless polynomial time

2003 ◽  
Vol 68 (1) ◽  
pp. 65-131 ◽  
Author(s):  
Andreas Blass ◽  
Yuri Gurevich
Keyword(s):  
Fixed Point ◽  
Polynomial Time ◽  
Order Logic ◽  
Complexity Class ◽  
First Order Logic ◽  
Infinitary Logic ◽  
Extensive Discussion ◽  
First Order ◽  
Detailed Proof ◽  
Strong Extension

AbstractThis paper developed from Shelah's proof of a zero-one law for the complexity class “choiceless polynomial time,” defined by Shelah and the authors. We present a detailed proof of Shelah's result for graphs, and describe the extent of its generalizability to other sorts of structures. The extension axioms, which form the basis for earlier zero-one laws (for first-order logic, fixed-point logic, and finite-variable infinitary logic) are inadequate in the case of choiceless polynomial time; they must be replaced by what we call the strong extension axioms. We present an extensive discussion of these axioms and their role both in the zero-one law and in general.

scholarly journals THE MINUSINSK-KHAKAS ARCHAEOLOGICAL EXPEDITION OF 1930 FOR “NOVOEXPORT”

2018 ◽  
pp. 30-35
Author(s):  
A. S. Vdovin ◽  
L. Yu. Kitova ◽  
E. I. Kochkina ◽  
V. A. Konokhov
Keyword(s):  
Cultural Heritage ◽  
State Policy ◽  
Significant Contribution ◽  
Current Paper ◽  
Complete Picture ◽  
Main Method ◽  
Soviet State ◽  
System Method ◽  
First Time

The article presents the results of a study of materials from the Minusinsk-Khakass expedition of 1930 under the leadership of archaeologist V. G. Kartsov. This expedition is connected with the activities of Siberian museums, the Society for Siberian Studies, the office of “Novoexport” and a number of organizations involved in the search for funds to sell historical and cultural heritage abroad. The new Soviet state looked for means for industrialization. In this regard the “Novoeksport” office was created and the program of archaeological researches was developed. The current paper features mostly the work of N. K. Auerbach, V. G. Kartsov and V. P. Levasheva, who became the organizers and participants of the expedition. They made the significant contribution to science and museum work in Siberia. The current research has involved a considerable amount of documents from the central and regional archives, most of which are introduced for the first time into scientific use. The main method used was the historical and system method that gives a chance to show the interaction of researchers, to recreate a complete picture of activity of the expedition, as well as its role for implementation of state policy. The research considerably expands the data on the archaeological researches that Siberian museums conducted for “Novoeksport” in 1920s–1930s.

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