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:
“The manifestation of poetry in external life is formal perfection. True sentiment grows within, and art must represent internal phenomena externally.”
—Franz Grillparzer (1791–1872)
“The new statement will comprise the skepticisms, as well as the faiths of society, and out of unbeliefs a creed shall be formed. For, skepticisms are not gratuitous or lawless, but are limitations of the affirmative statement, and the new philosophy must take them in, and make affirmations outside of them, just as much as must include the oldest beliefs.”
—Ralph Waldo Emerson (1803–1882)