In mathematical logic, Gödel's β function is a function used to permit quantification over finite sequences of natural numbers in formal theories of arithmetic. The β function is used, in particular, in showing that the class of arithmetically definable functions is closed under primitive recursion, and therefore includes all primitive recursive functions. == Definition == The β function takes three natural numbers as arguments. It is defined as β(x1, x2, x3) = rem(x1, 1 + (x3 + 1) · x2) ( = rem(x1, (x3 · x2 + x2 + 1) ) where rem(x, y) denotes the remainder after integer division of x by ...

Self-verifying theories are consistent first-order systems of arithmetic much weaker than Peano arithmetic that are capable of proving their own consistency. Dan Willard was the first to investigate their properties, and he has described a family of such systems. According to Gödel's incompleteness theorem, these systems cannot contain the theory of Peano arithmetic, and in fact, not even the weak fragment of Robinson arithmetic; nonetheless, they can contain strong theorems. In outline, the key to Willard's construction of his system is to formalise enough of the Gödel machinery to talk about...

In mathematics, more precisely in Ito calculus, the Euler–Maruyama method, also called simply the Euler method, is a method for the approximate numerical solution of a stochastic differential equation (SDE). It is a simple generalization of the Euler method for ordinary differential equations to stochastic differential equations. It is named after Leonhard Euler and Gisiro Maruyama. Unfortunately the same generalization cannot be done for the other methods from deterministic theory, e.g. Runge–Kutta schemes. Consider the stochastic differential equation (see Itō calculus) with initial condition...

In mathematical logic, the De Bruijn index is a notation invented by the Dutch mathematician Nicolaas Govert de Bruijn for representing terms in the λ calculus with the purpose of eliminating the names of the variable from the notation. Terms written using these indices are invariant with respect to α conversion, so the check for α-equivalence is the same as that for syntactic equality. Each De Bruijn index is a natural number that represents an occurrence of a variable in a λ-term, and denotes the number of binders that are in scope between that occurrence and its corresponding binder. The following...

In non-standard analysis, a monad (also called halo) is the set of points infinitesimally close to a given point. Given a hyperreal number x in R*, the monad of x is the set If x is finite (limited), the unique real number in the monad of x is called the standard part of x. == See also == Infinitesimal == Notes == == References == H. Jerome Keisler: Foundations of Infinitesimal Calculus, available for downloading...