Sentence examples for arithmetical formulas from inspiring English sources

It's as if you had swallowed some science-fiction pill that laid bare the arithmetical formulas underlying everyday perception.

Linda Francis turns sub-molecular theory into elegant, Mondrianesque paintings; Mark Lombardi's delicate drawings chart networks of powerful people and organizations; Francois Morellet's neatly penciled grids follow arithmetical formulas; and John O'Connor's large, exuberant drawings look like the blackboard diagrams of a nutty professor (Johnson).

Moreover, we show how our main results can be used to obtain arithmetical formulas for the number of representations of an integer n as the sums of s squares.

In this paper, we study the differences between Fast and TReX acceleration techniques and show that although Fast remains in Presburger logics while accelerating, TReX can produce 1st order arithmetical formulas even when accelerating functions.

Kleene's function-realizability interpretation, developed to prove the consistency of his formalization FIM of the intuitionistic theory of sequences ("intuitionistic analysis"), changes the interpretation of arithmetical formulas; for example, ¬ ¬∀x (A x) ∨ ¬A x)) is function-realizable for every arithmetical formula A x).

Gödel numbering provides a quantifier-free formula G x) which (numeralwise) expresses the predicate "x is the code of a proof in HA of (0 = 1)." By intuitionistic logic with the decidability of quantifier-free arithmetical formulas, HA proves ∀x ∃yG y) ∨ ¬G x)).

In first-order formalizations of arithmetic, this is formulated as a scheme: for each first-order arithmetical formula of the language of arithmetic with one free variable, one instance of the induction principle is included in the formalization of arithmetic.

Rules of inference also get expressed in the arithmetized form: For example, there is an arithmetical formula M x, y, z) which is true exactly when one has an application of a standard rule of inference "Modus Ponens" at hand; i.e., for some formulas A and B, x = ⌈A⌉, y = ⌈A → B⌉ and z = ⌈B⌉.

A simple arithmetical formula shows that Australia, two-time winners of the tournament, and New Zealand, desperately seeking a second title to go with the one they claimed on home soil 24 years ago, are the best-performing countries at World Cups.

There is an arithmetical formula, call it Fmla(x), which is true of n iff n is a Gödel number of a well-formed formula of the system.

There is also an arithmetical formula M x, y, z) which is true exactly if one has a valid application of the rule of inference modus ponens for some formulas A and B with x = ⌈A⌉, y = ⌈A → B⌉ and z = ⌈B⌉; etc.