In first-order theories such as HA, which cannot quantify over functions directly, CT is stated as an axiom schema saying that any definable function is computable, using Kleene's T predicate to define computability. For each formula φ 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 that is the Gödel number of a general recursive function that will, for every x, produce such a y satisfying the formula, with some u being a Gödel number encoding a verifiable computation bearing witness to the fact that y is in fact the value of that function at x. In higher-order systems that can quantify over functions directly, CT can be stated as a single axiom saying that every function from the natural numbers to the natural numbers is computable.
Relationship to classical logic
The schema form of CT shown above, when added to constructive systems such as HA, implies the negation of the law of the excluded middle. As an example, it is a classical tautology that every Turing machine either halts or does not halt on a given input. Assuming this tautology, in sufficiently strong systems such as HA it is possible to form a function h that takes a code for a Turing machine and returns 1 if the machine halts and 0 if it does not halt. Then, from Church's Thesis one would conclude that this function is itself computable, but this is known to be false, because the Halting problem is not computably solvable. Thus HA and CT disproves some consequence of the law of the excluded middle. The "single axiom" form of CT mentioned above, quantifies over functions and says that every function f is computable. This axiom is consistent with some weak classical systems that do not have the strength to form functions such as the function f of the previous paragraph. For example, the weak classical system is consistent with this single axiom, because has a model in which every function is computable. However, the single-axiom form becomes inconsistent with the law of the excluded middle in any system that has sufficient axioms to construct functions such as the function h in the previous paragraph.
Extended Church's thesis
Extended Church's thesis extends the claim to functions which are totally defined over a certain type of domain. It is used by the school of constructive mathematics founded by Andrey Markov Jr. It can be formally stated by the schema: In the above, is restricted to be almost-negative. For first-order arithmetic, this means cannot contain any disjunction, and existential quantifiers can only appear in front of formulas. This thesis can be characterised as saying that a sentence is true if and only if it is computably realisable. In fact this is captured by the following meta-theoretic equivalences: Here, stands for "". So, it is provable in with that a sentence is true iff it is realisable. But also, is true in with iff is realisable in without. The second equivalence can be extended with Markov's principle as follows: So, is true in with and iff there is a number n which realises in. The existential quantifier has to be outside in this case, because PA is non-constructive and lacks the existence property.
