A reflection principle and its applications to nonstandard models

1995 ◽  
Vol 60 (4) ◽  
pp. 1137-1152
Author(s):  
James H. Schmerl
Keyword(s):  
Number Theory ◽  
Reflection Principle ◽  
Peano Arithmetic ◽  
Second Order ◽  
Order Number ◽  
Nonstandard Models ◽  
Consistent Extension ◽  
Weak Theories

Some methods of constructing nonstandard models work only for particular theories, such as ZFC, or CA + AC (which is second order number theory with the choice scheme). The examples of this which motivated the results of this paper occur in the main theorems of [5], which state that if T is any consistent extension of either ZFC0 (which is ZFC but with only countable replacement) or CA + AC and if κ and λ are suitably chosen cardinals, then T has a model which is κ-saturated and has the λ-Bolzano-Weierstrass property. (Compare with Theorem 3.5.) Another example is a result from [12] which states that if T is any consistent extension of CA + AC and cf (λ) > ℵ0, then T has a natural λ-Archimedean model. (Compare with Theorem 3.1 and the comments following it.) Still another example is a result in [6] in which it is shown that if a model of Peano arithmetic is expandable to a model of ZF or of CA, then so is any cofinal extension of . (Compare with Theorem 3.10.) Related types of constructions can also be found in [10] and [11].A reflection principle will be proved here, allowing these constructions to be extended to models of many other theories, among which are some exceedingly weak theories and also all of their completions.

scholarly journals About the proof-theoretic ordinals of weak fixed point theories

1992 ◽  
Vol 57 (3) ◽  
pp. 1108-1119 ◽  
Author(s):  
Gerhard Jäger ◽  
Barbara Primo
Keyword(s):  
Fixed Point ◽  
Number Theory ◽  
Second Order ◽  
Order Number ◽  
Natural Numbers

AbstractThis paper presents several proof-theoretic results concerning weak fixed point theories over second order number theory with arithmetic comprehension and full or restricted induction on the natural numbers. It is also shown that there are natural second order theories which are proof-theoretically equivalent but have different proof-theoretic ordinals.

scholarly journals ULTIMATE TRUTH VIS-À-VIS STABLE TRUTH

2008 ◽  
Vol 1 (1) ◽  
pp. 126-142 ◽  
Author(s):  
P. D. WELCH
Keyword(s):  
Number Theory ◽  
Null Hypothesis ◽  
Second Order ◽  
Solution Model ◽  
Ground Model ◽  
Order Number ◽  
Consistency Proof ◽  
Ultimate Truth ◽  
Semantic Paradoxes ◽  
Semantic Values

We show that the set of ultimately true sentences in Hartry Field's Revenge-immune solution model to the semantic paradoxes is recursively isomorphic to the set of stably true sentences obtained in Hans Herzberger's revision sequence starting from the null hypothesis. We further remark that this shows that a substantial subsystem of second-order number theory is needed to establish the semantic values of sentences in Field's relative consistency proof of his theory over the ground model of the standard natural numbers: \Delta _3^1-CA0 (second-order number theory with a \Delta _3^1-comprehension axiom scheme) is insufficient. We briefly consider his claim to have produced a ‘revenge-immune’ solution to the semantic paradoxes by introducing this conditional. We remark that the notion of a ‘determinately true’ operator can be introduced in other settings.

scholarly journals RELATIVE PREDICATIVITY AND DEPENDENT RECURSION IN SECOND-ORDER SET THEORY AND HIGHER-ORDER THEORIES

2014 ◽  
Vol 79 (3) ◽  
pp. 712-732 ◽  
Author(s):  
SATO KENTARO
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Higher Order ◽  
Second Order ◽  
Order Number ◽  
Recursion Scheme ◽  
Transfinite Recursion ◽  
The Given ◽  
The Universe ◽  
Order Set

AbstractThis article reports that some robustness of the notions of predicativity and of autonomous progression is broken down if as the given infinite total entity we choose some mathematical entities other than the traditional ω. Namely, the equivalence between normal transfinite recursion scheme and new dependent transfinite recursion scheme, which does hold in the context of subsystems of second order number theory, does not hold in the context of subsystems of second order set theory where the universe V of sets is treated as the given totality (nor in the contexts of those of n+3-th order number or set theories, where the class of all n+2-th order objects is treated as the given totality).

MINIMAL TRUTH AND INTERPRETABILITY

2009 ◽  
Vol 2 (4) ◽  
pp. 799-815 ◽  
Author(s):  
MARTIN FISCHER
Keyword(s):  
Peano Arithmetic ◽  
Second Order ◽  
Conservative Extension ◽  
Axiomatic Theories Of Truth ◽  
Second Order Arithmetic ◽  
Theories Of Truth ◽  
Fine Grained ◽  
Order Arithmetic ◽  
Weak Theories

In this paper we will investigate different axiomatic theories of truth that are minimal in some sense. One criterion for minimality will be conservativity over Peano Arithmetic. We will then give a more fine-grained characterization by investigating some interpretability relations. We will show that disquotational theories of truth, as well as compositional theories of truth with restricted induction are relatively interpretable in Peano Arithmetic. Furthermore, we will give an example of a theory of truth that is a conservative extension of Peano Arithmetic but not interpretable in it. We will then use stricter versions of interpretations to compare weak theories of truth to subsystems of second-order arithmetic.

Regularity in models of arithmetic

1984 ◽  
Vol 49 (1) ◽  
pp. 272-280 ◽  
Author(s):  
George Mills ◽  
Jeff Paris
Keyword(s):  
Peano Arithmetic ◽  
Second Order ◽  
Axiom Schema ◽  
First Order ◽  
Nonstandard Models ◽  
Order Arithmetic ◽  
Bounded Sets ◽  
Models Of Arithmetic

AbstractThis paper investigates the quantifier “there exist unboundedly many” in the context of first-order arithmetic. An alternative axiomatization is found for Peano arithmetic based on an axiom schema of regularity: The union of boundedly many bounded sets is bounded. We also obtain combinatorial equivalents of certain second-order theories associated with cuts in nonstandard models of arithmetic.

scholarly journals NOTES ON BOUNDED INDUCTION FOR THE COMPOSITIONAL TRUTH PREDICATE

2017 ◽  
Vol 10 (3) ◽  
pp. 455-480 ◽  
Author(s):  
BARTOSZ WCISŁO ◽  
MATEUSZ ŁEŁYK
Keyword(s):  
Reflection Principle ◽  
Peano Arithmetic ◽  
Truth Predicate ◽  
Base Theory ◽  
Modified Theory ◽  
Induction Scheme

AbstractWe prove that the theory of the extensional compositional truth predicate for the language of arithmetic with Δ0-induction scheme for the truth predicate and the full arithmetical induction scheme is not conservative over Peano Arithmetic. In addition, we show that a slightly modified theory of truth actually proves the global reflection principle over the base theory.

On external Scott algebras in nonstandard models of Peano arithmetic

1996 ◽  
Vol 61 (2) ◽  
pp. 586-607
Author(s):  
Vladimir Kanovei
Keyword(s):  
Peano Arithmetic ◽  
Sufficient Condition ◽  
Elementary Extension ◽  
Necessary And Sufficient Condition ◽  
Nonstandard Models ◽  
Necessary And Sufficient

AbstractWe prove that a necessary and sufficient condition for a countable set of sets of integers to be equal to the algebra of all sets of integers definable in a nonstandard elementary extension of ω by a formula of the PA language which may include the standardness predicate but does not contain nonstandard parameters, is as follows: is closed under arithmetical definability and contains 0(ω) the set of all (Gödel numbers of) true arithmetical sentences.Some results related to definability of sets of integers in elementary extensions of ω are included.

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