Formal theories of arithmetic

Formal theories of arithmetic are mathematical frameworks that aim to rigorously express and explore the concepts and propositions related to arithmetic using a formal language. These theories typically involve the axiomatization of basic arithmetic operations like addition and multiplication, as well as the properties of numbers, especially the natural numbers. One of the most notable formal theories of arithmetic is Peano Arithmetic (PA), developed by Giuseppe Peano in the late 19th century.

Bounded arithmetic is a branch of mathematical logic that studies systems of arithmetic that restrict the types of quantifiers that can be used in formulas. Unlike classical arithmetic, which may allow for arbitrary quantification over natural numbers, bounded arithmetic restricts quantification to a certain range. Specifically, in bounded arithmetic, quantifiers are typically restricted to bounded formulas, which are those that can quantify only over natural numbers within a specified limit.

The decidability of first-order theories of the real numbers is a significant topic in mathematical logic, particularly concerning model theory and the foundations of mathematics. In general terms, a first-order theory consists of a set of axioms and rules for reasoning about a particular mathematical domain. When we talk about the first-order theory of the real numbers, we typically refer to the standard axioms that describe the real numbers, including properties of addition, multiplication, order, and the completeness property of the reals.

Elementary function arithmetic refers to the basic operations that can be performed on elementary functions, which are a class of functions that include well-known mathematical functions such as polynomials, exponential functions, logarithmic functions, trigonometric functions, and their inverses.

The Existential Theory of the Reals, often discussed in the context of mathematical logic and model theory, relates to the study of the properties of real numbers as represented in certain logical frameworks. It focuses on the notion of whether certain mathematical statements can be expressed as true or false when considering the real numbers. In particular, the existential theory of the reals often examines the sets of real numbers defined by existentially quantified formulas. These formulas are statements that assert the existence of certain elements satisfying given conditions.

Heyting arithmetic is a formal system of arithmetic that is based on intuitionistic logic, reflecting the mathematical philosophy initiated by mathematician L.E.J. Brouwer. It serves as the foundational framework for arithmetic in intuitionistic mathematics, which differs from classical mathematics primarily in its treatment of truth and existence.

Induction, bounding, and the least number principles are fundamental concepts in mathematics, particularly in the realm of number theory and set theory. Here’s a brief overview of each: ### Mathematical Induction Mathematical induction is a method of proof used to establish that a statement is true for all natural numbers. The process consists of two main steps: 1. **Base Case**: Prove that the statement holds for the first natural number (usually 1).

Primitive recursive arithmetic is a formal system used in mathematical logic and the foundations of mathematics. It is a subset of first-order Peano arithmetic, and it focuses on functions that can be defined using a limited type of recursive processes. The key features of primitive recursive arithmetic include: 1. **Primitive Recursive Functions**: The system defines certain functions (called primitive recursive functions) that are built using basic functions and operations in a specific way.

Robinson arithmetic, denoted as \( R \), is a weak system of arithmetic that is part of the field of mathematical logic and foundational studies. It was introduced by the mathematician and logician John Robinson in the 1950s. The key features of Robinson arithmetic include: 1. **Language**: The language of Robinson arithmetic includes a number of basic symbols for logical operations (like conjunction and disjunction), equality, and a unary function symbol typically interpreted as a successor function.

Second-order arithmetic is a foundational system in mathematical logic and set theory that extends first-order arithmetic by allowing quantification over sets of natural numbers, in addition to quantifying over individual natural numbers. In first-order arithmetic, the language contains symbols for natural numbers, addition, multiplication, and logical connectives, as well as quantification over individual natural numbers. A typical axiom system for first-order arithmetic is Peano Arithmetic (PA).

Skolem arithmetic is a branch of mathematical logic that deals with the arithmetic of the natural numbers and is based on the systems introduced by the Norwegian mathematician Thoralf Skolem. It is particularly focused on the study of sequences, functions, and relations that can be defined using certain logical frameworks, including the use of quantifiers. In more formal terms, Skolem arithmetic can be seen as an extension of first-order arithmetic where the focus is on the properties of functions and relations defined on the natural numbers.

Alfred Tarski, a prominent logician and mathematician, developed an axiomatization of the real numbers based on first-order logic. Tarski's approach was notable for its focus on the completeness and consistency of the real number system, as well as its relationship to ordered fields.