scholarly journals An Arithmetical-like Theory of Hereditarily Finite Sets

2021 ◽  
Author(s):  
Márcia R. Cerioli ◽  
Vitor Krauss ◽  
Petrucio Viana
Keyword(s):  
Order Theory ◽  
Second Order ◽  
Natural Numbers ◽  
Finite Sets ◽  
Homomorphism Theorem ◽  
Usual Pattern

This paper presents the (second-order) theory of hereditarily finite sets according to the usual pattern adopted in the presentation of the (second-order) theory of natural numbers. To this purpose, we consider three primitive concepts, together with four axioms, which are analogous to the usual Peano axioms. From them, we prove a homomorphism theorem, its converse, categoricity, and a kind of (semantical) completeness.

Decidability and undecidability of theories with a predicate for the primes

1993 ◽  
Vol 58 (2) ◽  
pp. 672-687 ◽  
Author(s):  
P. T. Bateman ◽  
C. G. Jockusch ◽  
A. R. Woods
Keyword(s):  
General Result ◽  
Order Theory ◽  
Second Order ◽  
The Other ◽  
Linear Case ◽  
Natural Numbers ◽  
Successor Function ◽  
First Order ◽  
First Order Theory

AbstractIt is shown, assuming the linear case of Schinzel's Hypothesis, that the first-order theory of the structure 〈ω; +, P〉, where P is the set of primes, is undecidable and, in fact, that multiplication of natural numbers is first-order definable in this structure. In the other direction, it is shown, from the same hypothesis, that the monadic second-order theory of 〈ω S, P〉 is decidable, where S is the successor function. The latter result is proved using a general result of A. L. Semënov on decidability of monadic theories, and a proof of Semënov's result is presented.

Categoricity and the natural numbers

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Order Theory ◽  
Second Order ◽  
Order Logic ◽  
Natural Numbers ◽  
Categorical Theory ◽  
First Order Theory ◽  
Standard Models ◽  
Skolem Theorem ◽  
Infinite Model ◽  
Second Order Logic

This chapter focuses on modelists who want to pin down the isomorphism type of the natural numbers. This aim immediately runs into two technical barriers: the Compactness Theorem and the Löwenheim-Skolem Theorem (the latter is proven in the appendix to this chapter). These results show that no first-order theory with an infinite model can be categorical; all such theories have non-standard models. Other logics, such as second-order logic with its full semantics, are not so expressively limited. Indeed, Dedekind's Categoricity Theorem tells us that all full models of the Peano axioms are isomorphic. However, it is a subtle philosophical question, whether one is entitled to invoke the full semantics for second-order logic — there are at least four distinct attitudes which one can adopt to these categoricity result — but moderate modelists are unable to invoke the full semantics, or indeed any other logic with a categorical theory of arithmetic.

scholarly journals There are Infinitely Many Fibonacci Primes

2020 ◽  
pp. 1-17
Author(s):  
Fengsui Liu
Keyword(s):  
Residue Class ◽  
Second Order ◽  
Order Topology ◽  
Sieve Method ◽  
Natural Numbers ◽  
Finite Sets ◽  
Index Set ◽  
Infinite Set ◽  
Novel Algorithm

We invent a novel algorithm and solve the Fibonacci prime conjecture by an interaction between proof and algorithm. From the entire set of natural numbers successively deleting the residue class 0 mod a prime, we retain this prime and possibly delete another one prime retained, then we invent a recursive sieve method, a modulo algorithm on finite sets of natural numbers, for indices of Fibonacci primes. The sifting process mechanically yields a sequence of sets of natural numbers, which converges to the index set of all Fibonacci primes. The corresponding cardinal sequence is strictly increasing. The algorithm reveals a structure of particular order topology of the index set of all Fibonacci primes, then we readily prove that the index set of all Fibonacci primes is an infinite set based on the existing theory of the structure. Some mysteries of primes are hidden in second order arithmetics.

A Model-Theoretic Approach to Ordinal Analysis

1997 ◽  
Vol 3 (1) ◽  
pp. 17-52 ◽  
Author(s):  
Jeremy Avigad ◽  
Richard Sommer
Keyword(s):  
Second Order ◽  
Cut Elimination ◽  
Large Set ◽  
Theoretic Approach ◽  
Natural Numbers ◽  
Finite Sets ◽  
Ordinal Analysis ◽  
Second Order Arithmetic ◽  
Combinatorial Properties ◽  
Order Arithmetic

AbstractWe describe a model-theoretic approach to ordinal analysis via the finite combinatorial notion of an α-large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in nonstandard instances, give rise to models of the theory being analyzed. This method is applied to obtain ordinal analyses of a number of interesting subsystems of first- and second-order arithmetic.

scholarly journals Surface wavepackets subject to an abrupt depth change. Part 1. Second-order theory

2021 ◽  
Vol 915 ◽  
Author(s):  
Yan Li ◽  
Yaokun Zheng ◽  
Zhiliang Lin ◽  
Thomas A.A. Adcock ◽  
Ton S. van den Bremer
Keyword(s):  
Order Theory ◽  
Second Order

Abstract

Invariant relative orbits for satellite constellations: A second order theory

2006 ◽  
Vol 181 (1) ◽  
pp. 6-20 ◽  
Author(s):  
F.A. Abd El-Salam ◽  
I.A. El-Tohamy ◽  
M.K. Ahmed ◽  
W.A. Rahoma ◽  
M.A. Rassem
Keyword(s):  
Order Theory ◽  
Second Order ◽  
Satellite Constellations

How Second-Order Is the Regional Level? An Analysis of Tweets in Simultaneous Campaigns

2017 ◽  
Vol 65 (4) ◽  
pp. 1021-1039
Author(s):  
Nicolas Bouteca ◽  
Evelien D’heer ◽  
Steven Lannoo
Keyword(s):  
Order Theory ◽  
Second Order ◽  
Regional Level ◽  
The Political ◽  
Voting Behaviour ◽  
National Parliament ◽  
Political Experience ◽  
Regional Elections

This article puts the second-order theory for regional elections to the test. Not by analysing voting behaviour but with the use of campaign data. The assumption that regional campaigns are overshadowed by national issues was verified by analysing the campaign tweets of Flemish politicians who ran for the regional or national parliament in the simultaneous elections of 2014. No proof was found for a hierarchy of electoral levels but politicians clearly mix up both levels in their tweets when elections coincide. The extent to which candidates mix up governmental levels can be explained by the incumbency past of the candidates, their regionalist ideology, and the political experience of the candidates.

A second-order theory for resonance capture in corotation centers

1999 ◽  
Vol 47 (5) ◽  
pp. 643-652 ◽  
Author(s):  
C. Beauge ◽  
A. Lemaı̂tre ◽  
S. Jancart
Keyword(s):  
Order Theory ◽  
Second Order ◽  
Resonance Capture
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