The Church-Turing thesis (not to be confused with the Church-Turing theorem) is the empirical claim that to prove whether or not an effective process exists for a given procedure, it suffices to prove whether or not a Turing machine—equivalently, a lambda calculus term or primitive recursive function—exists to perform the same procedure.
The thesis is empirical because it cannot be formally proven, only falsified; “every computational system so far seems computationally equivalent to Turing Machines.”
Claim: The Church-Turing thesis
Every “effectively calculable” function (e.g., function for which there exists an algorithm to compute) is computable by a Turing Machine. That is, if an algorithm exists, then there exists a Turing Machine that can perform that algorithm.
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