Sue Toledo. Tableau systems for first order number theory and certain higher order theories. Lecture notes in mathematics, vol. 447. Springer-Verlag, Berlin, Heidelberg, and New York, 1975, iii + 339 pp.

1980 ◽  
Vol 45 (3) ◽  
pp. 636-638
Author(s):  
J. I. Zucker
Keyword(s):  
New York ◽  
Number Theory ◽  
Higher Order ◽  
Order Number ◽  
First Order ◽  
Tableau Systems ◽  
Lecture Notes

Transfinite recursive progressions of axiomatic theories

1962 ◽  
Vol 27 (3) ◽  
pp. 259-316 ◽  
Author(s):  
Solomon Feferman
Keyword(s):  
Number Theory ◽  
Functional Calculus ◽  
Higher Order ◽  
Order Number ◽  
First Order ◽  
Definable Set ◽  
Recursively Enumerable

The theories considered here are based on the classical functional calculus (possibly of higher order) together with a set A of non-logical axioms; they are also assumed to contain classical first-order number theory. In foundational investigations it is customary to further restrict attention to the case that A is recursive, or at least recursively enumerable (an equivalent restriction, by [1]). For such axiomatic theories we have the well-known incompleteness phenomena discovered by Godei [6]. Quite far removed from such theories are those based on non-constructive sets of axioms, for example the set of all true sentences of first-order number theory. According to Tarski's theorem, there is not even an arithmetically definable set of axioms A which will give the same result (cf. [18] for exposition).

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).

Incompleteness along paths in progressions of theories

1962 ◽  
Vol 27 (4) ◽  
pp. 383-390 ◽  
Author(s):  
S. Feferman ◽  
C. Spector
Keyword(s):  
Number Theory ◽  
Recursive Function ◽  
Order Number ◽  
Basic Logic ◽  
Natural Numbers ◽  
Logical Deduction ◽  
First Order ◽  
Primitive Recursive Function ◽  
Primitive Recursive ◽  
Recursive Definitions

We deal in the following with certain theories S, by which we mean sets of sentences closed under logical deduction. The basic logic is understood to be the classical one, but we place no restriction on the orders of the variables to be used. However, we do assume that we can at least express certain notions from classical first-order number theory within these theories. In particular, there should correspond to each primitive recursive function ξ a formula φ(χ), where ‘x’ is a variable ranging over natural numbers, such that for each numeral ñ, φ(ñ) expresses in the language of S that ξ(η) = 0. Such formulas, when obtained say by the Gödel method of eliminating primitive recursive definitions in favor of arithmetical definitions in +. ·. are called PR-formulas (cf. [1] §2 (C)).

More undecidable lattices of Steinitz exchange systems

2002 ◽  
Vol 67 (2) ◽  
pp. 859-878 ◽  
Author(s):  
L. R. Galminas ◽  
John W. Rosenthal
Keyword(s):  
Number Theory ◽  
Order Theory ◽  
Closed Field ◽  
Order Number ◽  
Infinite Dimensional ◽  
First Order ◽  
First Order Theory ◽  
Finite Dimensional ◽  
Order Arithmetic ◽  
Countably Infinite

AbstractWe show that the first order theory of the lattice <ω(S) of finite dimensional closed subsets of any nontrivial infinite dimensional Steinitz Exhange System S has logical complexity at least that of first order number theory and that the first order theory of the lattice (S∞) of computably enumerable closed subsets of any nontrivial infinite dimensional computable Steinitz Exchange System S∞ has logical complexity exactly that of first order number theory. Thus, for example, the lattice of finite dimensional subspaces of a standard copy of ⊕ωQ interprets first order arithmetic and is therefore as complicated as possible. In particular, our results show that the first order theories of the lattice (V∞) of c.e. subspaces of a fully effective ℵ0-dimensional vector space V∞ and the lattice of c.e. algebraically closed subfields of a fully effective algebraically closed field F∞ of countably infinite transcendence degree each have logical complexity that of first order number theory.

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