arithmetical set (or arithmetic set) is a set of natural numbers that can be defined by a formula of first-order Peano arithmetic. The arithmetical sets...

certain sets based on the complexity of formulas that define them. Any set that receives a classification is called arithmetical. The arithmetical hierarchy...

types of arithmetic. Modular arithmetic operates on a finite set of numbers. If an operation would result in a number outside this finite set then the...

scheme for arithmetical formulas (which is sometimes called the "arithmetical comprehension axiom"). That is, ACA0 allows us to form the set of natural...

An arithmetic progression or arithmetic sequence (AP) is a sequence of numbers such that the difference from any succeeding term to its preceding term...

of memory, but most RISC instruction sets include SIMD or vector instructions that perform the same arithmetic operation on multiple pieces of data at...

collection. The collection is often a set of results from an experiment, an observational study, or a survey. The term "arithmetic mean" is preferred in some mathematics...

variables (that is, no quantifiers over set variables) is called arithmetical. An arithmetical formula may have free set variables and bound individual variables...

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus...

definable in the language of arithmetic is called analytical. Every computable real number is arithmetical, and the arithmetical numbers form a subfield of...

classified into a hierarchy extending the arithmetical hierarchy; the hyperarithmetical sets are exactly the sets that are assigned a rank in this hierarchy...

or lower in the arithmetical hierarchy. Post's theorem shows that, for each n, Thn( N {\displaystyle {\mathcal {N}}} ) is arithmetically definable, but...

Schröder. The Peano axioms define the arithmetical properties of natural numbers, usually represented as a set N or N . {\displaystyle \mathbb {N} .}...

models is known. However, the arithmetical operations are much more complicated. It is easy to see that the arithmetical structure differs from ω + (ω*...

operation has completed, the ALU inputs may be set up for the next ALU operation. A number of basic arithmetic and bitwise logic functions are commonly supported...

other sets. A set may have a finite number of elements or be an infinite set. There is a unique set with no elements, called the empty set; a set with...

of the set {1, 2, 3}, but are not subsets of it; and in turn, the subsets, such as {1}, are not members of the set {1, 2, 3}. Just as arithmetic features...

finite generalized arithmetic progression, or sometimes just generalized arithmetic progression (GAP), of dimension d is defined to be a set of the form {...

independence proof by forcing automatically proves independence from arithmetical statements, other concrete statements, and large cardinal axioms. Some...

ring of integers. The classical objects of interest in arithmetic geometry are rational points: sets of solutions of a system of polynomial equations over...

In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation...

a partial computable function. The set S is Σ 1 0 {\displaystyle \Sigma _{1}^{0}} (referring to the arithmetical hierarchy). There is a partial computable...

In mathematical logic and descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas...

In mathematics, Robinson arithmetic is a finitely axiomatized fragment of first-order Peano arithmetic (PA), first set out by Raphael M. Robinson in 1950...

natural number c where a + c = b. This order is compatible with the arithmetical operations in the following sense: if a, b and c are natural numbers...

In set theory, the intersection of two sets A {\displaystyle A} and B , {\displaystyle B,} denoted by A ∩ B , {\displaystyle A\cap B,} is the set containing...

possesses arithmetical properties similar to those of the multiplicative semigroup of positive integers. Fundamental Theorem of Arithmetic is, in fact...

between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships...

mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations...

its proof, is an arithmetical relation between two numbers. Therefore, there is a statement form Bew(y) that uses this arithmetical relation to state...