Formal Statement
In first-order theories such as HA, which cannot quantify over functions directly, CT is stated as an axiom schema which says that any definable function is computable, using Kleene's T predicate to define computability. For each formula φ(x,y) of two variables, the schema includes the axiom
This axiom asserts that, if for every x there is a y satisfying φ then there is in fact an e which is the Gödel number of a general recursive function that will, for every x, produce such a y satisfying the formula.
In higher-order systems that can quantify over functions directly, CT can be stated as a single axiom which says that every function from the natural numbers to the natural numbers is computable.
Read more about this topic: Church's Thesis (constructive Mathematics)
Famous quotes containing the words formal and/or statement:
“I will not let him stir
Till I have used the approvèd means I have,
With wholesome syrups, drugs, and holy prayers,
To make of him a formal man again.”
—William Shakespeare (1564–1616)
“It is commonplace that a problem stated is well on its way to solution, for statement of the nature of a problem signifies that the underlying quality is being transformed into determinate distinctions of terms and relations or has become an object of articulate thought.”
—John Dewey (1859–1952)