Implicitly Arithmetical Sets
Each arithmetical set has an arithmetical formula which tells whether particular numbers are in the set. An alternative notion of definability allows for a formula that does not tell whether particular numbers are in the set but tells whether the set itself satisfies some arithmetical property.
A set Y of natural numbers is implicitly arithmetical or implicitly arithmetically definable if it is definable with an arithmetical formula that is able to use Y as a parameter. That is, if there is a formula in the language of Peano arithmetic with no free number variables and a new set parameter Z and set membership relation such that Y is the unique set such that holds.
Every arithmetical set is implicitly arithmetical; if X is arithmetically defined by φ(n) then it is implicitly defined by the formula
- .
Not every implicitly arithmetical set is arithmetical, however. In particular, the truth set of first order arithmetic is implicitly arithmetical but not arithmetical.
Read more about this topic: Arithmetical Set
Famous quotes containing the words implicitly and/or sets:
“In short, let it be your maxim through life, to know all you can know, yourself; and never to trust implicitly to the informations of others.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“Almsgiving tends to perpetuate poverty; aid does away with it once and for all. Almsgiving leaves a man just where he was before. Aid restores him to society as an individual worthy of all respect and not as a man with a grievance. Almsgiving is the generosity of the rich; social aid levels up social inequalities. Charity separates the rich from the poor; aid raises the needy and sets him on the same level with the rich.”
—Eva Perón (1919–1952)