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)
“And weren’t there special cemetery flowers,
That, once grief sets to growing, grief may rest:
The flowers will go on with grief awhile,
And no one seem neglecting or neglected?
A prudent grief will not despise such aids.”
—Robert Frost (1874–1963)