Addition in nonstandard models of arithmetic

1972 ◽  
Vol 37 (3) ◽  
pp. 483-486 ◽  
Author(s):  
R. Phillips
Keyword(s):  
Positive Element ◽  
Ring Of Integers ◽  
Nonstandard Model ◽  
Nonstandard Models ◽  
Infinite Height ◽  
Image Position ◽  
Goldbach’S Conjecture ◽  
Models Of Arithmetic ◽  
Goldbach's Conjecture

In [3] Kemeny made the following conjecture: Suppose *Z is a nonstandard model of the ring of integers Z. Letand let F be the subgroup of those cosets ā which contain an element of infinite height in *Z. Kemeny then asked if the ring R = {a: ā ∈ F} is also a nonstandard model of Z. If so then Goldbach's conjecture is false because Kemeny also shows in [3] that Goldbach's conjecture fails in R.The papers [1] and [5] by Gandy and Mendelson show that R is not a nonstandard model of Z but we give here a simpler proof based on Mendelson's paper. Suppose R is a nonstandard model of Z. Then each positive number in R is a sum of four squares. Choose a in R so that a is a positive element of R of infinite height in *Z. Then since a is infinite in *Z, a − 1 is positive. Thus , xi ∈ R for i = 1, …, 4. Now each xi must be of the form ai + ni, where ai has infinite height in *Z and ni, ∈ Z.

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.

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}} $.

A note on the undefinability of cuts

1983 ◽  
Vol 48 (3) ◽  
pp. 564-569 ◽  
Author(s):  
J.B. Paris ◽  
C. Dimitracopoulos
Keyword(s):  
First Order ◽  
Nonstandard Model ◽  
Image Position

The results in this paper were motivated by the following result due to R. Solovay.Theorem 1 (Solovay). Let M be a nonstandard model of Peano's first order axioms P and let I ⊂e M (i.e. ϕ ≠ ⊂ M and I is closed under < and successor). Then for each of the functions we can define J ⊆e I in ‹M, I› such that J is closed under that function. (∣x∣ denotes [log2(x)].)Proof. Just notice that the cuts defined byare successively closed under In view of Theorem 1, the following question was raised by R. Solovay: Can we define J ⊆ I in ‹M, I› such that J is closed under exponentiation? In Theorem 2 we show that the answer is “no”. Theorem 3 is based on Theorem 2 and extends the technique to cuts which are models of subsystems of P.To prove both theorems we shall need an estimate due to R. Parikh (see [1], especially the proof of Theorem 2.2a). For the sake of completeness, and also to introduce some notation we shall sketch Parikh's estimate in the next section. At all times we shall give the easiest estimates which still work rather than the sharpest ones.

Ramification Groups of Abelian Local Field Extensions

1971 ◽  
Vol 23 (2) ◽  
pp. 271-281 ◽  
Author(s):  
Murray A. Marshall
Keyword(s):  
Prime Ideal ◽  
Local Field ◽  
Galois Extension ◽  
Residue Class ◽  
Abelian Extensions ◽  
Ring Of Integers ◽  
Field Extensions ◽  
Valued Field ◽  
Residue Class Field ◽  
Image Position

Let k be a local field; that is, a complete discrete-valued field having a perfect residue class field. If L is a finite Galois extension of k then L is also a local field. Let G denote the Galois group GL|k. Then the nth ramification group Gn is defined bywhere OL, denotes the ring of integers of L, and PL is the prime ideal of OL. The ramification groups form a descending chain of invariant subgroups of G:1In this paper, an attempt is made to characterize (in terms of the arithmetic of k) the ramification filters (1) obtained from abelian extensions L\k.

scholarly journals Homomorphism theories for universal algebras

1968 ◽  
Vol 16 (1) ◽  
pp. 19-35 ◽  
Author(s):  
Hans-Jürgen Hoehnke
Keyword(s):  
Normal Subgroup ◽  
Unit Element ◽  
Group Rings ◽  
Universal Algebras ◽  
Ring Of Integers ◽  
Image Position

It is well-known that a homomorphism ø(A→B) between groups A and B induces a homomorphism ø*(ZA→ZB) between the corresponding group rings ZA and ZB over the ring of integers Z. The identical congruence O on B and the unit element eB of B can be characterised by the equations x–y = 0 and x–eB = 0 (x,y ∈ B) respectively. Similarly the congruence Γø corresponding to ø and the corresponding normal subgroup of A areand {x∈A1 = A,(x–eA)ø = 0} respectively.

Some theorems on R-maximal sets and major subsets of recursively enumerable sets

1971 ◽  
Vol 36 (2) ◽  
pp. 193-215 ◽  
Author(s):  
Manuel Lerman
Keyword(s):  
Turing Degrees ◽  
Elementary Equivalence ◽  
Recursive Functions ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Nonstandard Models ◽  
Relational Systems ◽  
The Relationship ◽  
Models Of Arithmetic

In [5], we studied the relational systems /Ā obtained from the recursive functions of one variable by identifying two such functions if they are equal for all but finitely many х ∈ Ā, where Ā is an r-cohesive set. The relational systems /Ā with addition and multiplication defined pointwise on them, were once thought to be potential candidates for nonstandard models of arithmetic. This, however, turned out not to be the case, as was shown by Feferman, Scott, and Tennenbaum [1]. We showed, letting A and B be r-maximal sets, and letting denote the complement of X, that /Ā and are elementarily equivalent (/Ā ≡ ) if there are r-maximal supersets C and D of A and B respectively such that C and D have the same many-one degree (C =mD). In fact, if A and B are maximal sets, /Ā ≡ if, and only if, A =mB. We wish to study the relationship between the elementary equivalence of /Ā and , and the Turing degrees of A and B.

scholarly journals Distribution of the Largest Strong Goldbach Numbers Generated by Primes

2018 ◽  
Vol 10 (5) ◽  
pp. 1
Author(s):  
Pingyuan Zhou ◽  
Rong Ao
Keyword(s):  
Distribution Curve ◽  
Goldbach’S Conjecture ◽  
Goldbach's Conjecture

Using the first 4000000 primes to find Ln, the largest strong Goldbach number generated by the n-th prime Pn, we generalize a proposition in our previous work (Zhou 2017) and propose that Ln ≈ 2Pn and Ln/2Pn &lt; 1 for sufficiently large Pn but the limit of Ln/2Pn as n → ∞ is 1, Ln ≈ Pn + n log n and Ln/(Pn + n log n) &gt; 1 for sufficiently large Pn but the limit of Ln/(Pn + n log n) as n → ∞ is 1. There are five corollaries of the generalized proposition for getting Ln → ∞ as n → ∞, which is equivalent to Goldbach’s conjecture. If every step in distribution curve of Ln is called a Goldbach step, a study on the ratio of width to height for Goldbach steps supports the existence of above two limits but a study on distribution of Goldbach steps supports an estimation that Q(n) ≈ (1 + 1/log log n)n/log n and the limit of Q(n)/((1 + 1/log log n)n/log n) as n → ∞ is 1, where Q(n) is the number of Goldbach steps, from which we may expect there are infinitely many Goldbach steps to imply Goldbach’s conjecture.

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