### 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-

... (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-

... In mathematical logic, 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-

... 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-

... For example, it can be shown that every ω-model of full 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-

... The first-

**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 ...### Famous quotes containing the words arithmetic and/or order:

“‘Tis no extravagant *arithmetic* to say, that for every ten jokes,—thou hast got an hundred enemies; and till thou hast gone on, and raised a swarm of wasps about thine ears, and art half stung to death by them, thou wilt never be convinced it is so.”

—Laurence Sterne (1713–1768)

“There are instances when we are like horses, we psychologists, and grow restless: we see our own shadow wavering up and down before us. A psychologist must look away from himself in *order* to see anything at all.”

—Friedrich Nietzsche (1844–1900)

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