Concept
Peano axioms
Related concepts (40)
Non-logical symbol
In logic, the formal languages used to create expressions consist of symbols, which can be broadly divided into constants and variables. The constants of a language can further be divided into logical symbols and non-logical symbols (sometimes also called logical and non-logical constants). The non-logical symbols of a language of first-order logic consist of predicates and individual constants. These include symbols that, in an interpretation, may stand for individual constants, variables, functions, or predicates.
Wilhelm Ackermann
Wilhelm Friedrich Ackermann (ˈækərmən; ˈakɐˌman; 29 March 1896 – 24 December 1962) was a German mathematician and logician best known for his work in mathematical logic and the Ackermann function, an important example in the theory of computation. Ackermann was born in Herscheid, Germany, and was awarded a Ph.D. by the University of Göttingen in 1925 for his thesis Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit, which was a consistency proof of arithmetic apparently without Peano induction (although it did use e.
Diophantine set
In mathematics, a Diophantine equation is an equation of the form P(x1, ..., xj, y1, ..., yk) = 0 (usually abbreviated P(, ) = 0) where P(, ) is a polynomial with integer coefficients, where x1, ..., xj indicate parameters and y1, ..., yk indicate unknowns. A Diophantine set is a subset S of , the set of all j-tuples of natural numbers, so that for some Diophantine equation P(, ) = 0, That is, a parameter value is in the Diophantine set S if and only if the associated Diophantine equation is satisfiable under that parameter value.
Categorical theory
In mathematical logic, a theory is categorical if it has exactly one model (up to isomorphism). Such a theory can be viewed as defining its model, uniquely characterizing the model's structure. In first-order logic, only theories with a finite model can be categorical. Higher-order logic contains categorical theories with an infinite model. For example, the second-order Peano axioms are categorical, having a unique model whose domain is the set of natural numbers In model theory, the notion of a categorical theory is refined with respect to cardinality.
Hilbert's second problem
In mathematics, Hilbert's second problem was posed by David Hilbert in 1900 as one of his 23 problems. It asks for a proof that the arithmetic is consistent – free of any internal contradictions. Hilbert stated that the axioms he considered for arithmetic were the ones given in , which include a second order completeness axiom. In the 1930s, Kurt Gödel and Gerhard Gentzen proved results that cast new light on the problem. Some feel that Gödel's theorems give a negative solution to the problem, while others consider Gentzen's proof as a partial positive solution.
Ordinal analysis
In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory. The field of ordinal analysis was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that the proof-theoretic ordinal of Peano arithmetic is ε0.
Order type
In mathematics, especially in set theory, two ordered sets X and Y are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) such that both f and its inverse are monotonic (preserving orders of elements). In the special case when X is totally ordered, monotonicity of f already implies monotonicity of its inverse. One and the same set may be equipped with different orders.
Tennenbaum's theorem
Tennenbaum's theorem, named for Stanley Tennenbaum who presented the theorem in 1959, is a result in mathematical logic that states that no countable nonstandard model of first-order Peano arithmetic (PA) can be recursive (Kaye 1991:153ff). A structure in the language of PA is recursive if there are recursive functions and from to , a recursive two-place relation
Arithmetices principia, nova methodo exposita
The 1889 treatise Arithmetices principia, nova methodo exposita (The principles of arithmetic, presented by a new method) by Giuseppe Peano is widely considered to be a seminal document in mathematical logic and set theory, introducing what is now the standard axiomatization of the natural numbers, and known as the Peano axioms, as well as some pervasive notations, such as the symbols for the basic set operations ∈, ⊂, ∩, ∪, and A−B.
Ordered ring
In abstract algebra, an ordered ring is a (usually commutative) ring R with a total order ≤ such that for all a, b, and c in R: if a ≤ b then a + c ≤ b + c. if 0 ≤ a and 0 ≤ b then 0 ≤ ab. Ordered rings are familiar from arithmetic. Examples include the integers, the rationals and the real numbers. (The rationals and reals in fact form ordered fields.) The complex numbers, in contrast, do not form an ordered ring or field, because there is no inherent order relationship between the elements 1 and i.