Order Arithmetic

Some articles on order, order arithmetic, arithmetic:

Recursion Theory - Relationships Between Definability, Proof and Computability
... (in terms of the arithmetical hierarchy) of defining that set using a first-order formula ... show that the set of logical consequences of an effective first-order theory is a recursively enumerable set, and that if the theory is strong enough this set will be uncomputable ... Recursion theory is also linked to second order arithmetic, a formal theory of natural numbers and sets of natural numbers ...
Second-order Arithmetic
... In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets ... The standard axiomatization of second-order arithmetic is denoted Z2 ... Second-order arithmetic includes, but is significantly stronger than, its first-order counterpart Peano arithmetic ...
Coding Mathematics in Second-order Arithmetic
... Second-order arithmetic allows us to speak directly (without coding) of natural numbers and sets of natural numbers ... can be coded in the language of second-order arithmetic, although doing so is a bit tricky ...
Subsystems of Second-order Arithmetic - Arithmetical Comprehension
... For example, it can be shown that every ω-model of full second-order arithmetic is closed under Turing jump, but not every ω-model closed under Turing jump is a model of full second-order arithmetic ... We may ask whether there is a subsystem of second-order arithmetic satisfied by every ω-model that is closed under Turing jump and satisfies some other ... formula φ, and the ordinary second-order induction axiom again, we could also choose to include the arithmetical induction axiom scheme, in other words the induction axiom for every arithmetical formula φ, without ...
Definable Functions of Second-order Arithmetic
... The first-order functions that are provably total in second-order arithmetic are precisely the same as those representable in system F (Girard et al ... system F is the theory of functionals corresponding to second-order arithmetic in a manner parallel to how Gödel's system T corresponds to first-order arithmetic in the Dialectica interpretation ...

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